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Volume 27
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Math. Comput. Appl., Volume 27, Issue 1 (February 2022) – 17 articles

Cover Story (view full-size image): Industry demands simulation-based design processes, but physics simulations act as a bottleneck due to their large computational costs and run time. In most steps of the design process, only an approximate result of what a fully-fledged physics simulation would provide is necessary. In this paper, we present a proof-of-concept study of the application of neural networks for thermal simulations of electronic systems. The goal of such a tool is to provide accurate approximations of a full solution, in order to quickly select promising designs for more detailed investigations. View this paper.
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20 pages, 646 KiB  
Article
On a Modified Weighted Exponential Distribution with Applications
by Christophe Chesneau, Vijay Kumar, Mukti Khetan and Mohd Arshad
Math. Comput. Appl. 2022, 27(1), 17; https://doi.org/10.3390/mca27010017 - 21 Feb 2022
Cited by 6 | Viewed by 4031
Abstract
Practitioners in all applied domains value simple and adaptable lifetime distributions. They make it possible to create statistical models that are relatively easy to manage. A novel simple lifetime distribution with two parameters is proposed in this article. It is based on a [...] Read more.
Practitioners in all applied domains value simple and adaptable lifetime distributions. They make it possible to create statistical models that are relatively easy to manage. A novel simple lifetime distribution with two parameters is proposed in this article. It is based on a parametric mixture of the exponential and weighted exponential distributions, with a mixture weight depending on a parameter of the involved distribution; no extra parameter is added in this mixture operation. It can also be viewed as a special generalized mixture of two exponential distributions. This decision is based on sound mathematical and physical reasoning; the weight modification allows us to combine some joint properties of the exponential and weighted exponential distribution, which are known as complementary in several modeling aspects. As a result, the proposed distribution may have a decreasing or unimodal probability density function and possess the demanded increasing hazard rate property. Other properties are studied, such as the moments, Bonferroni and Lorenz curves, Rényi entropy, stress-strength reliability, and mean residual life function. Subsequently, a part is devoted to the associated model, which demonstrates how it can be used in a real-world statistical scenario involving data. In this regard, we demonstrate how the estimated model performs well using five different estimation methods and simulated data. The analysis of two data sets demonstrates these excellent results. The new model is compared to the weighted exponential, Weibull, gamma, and generalized exponential models for performance. The obtained comparison results are overwhelmingly in favor of the proposed model according to some standard criteria. Full article
(This article belongs to the Special Issue Statistical and Stochastic Approaches for Predictive Maintenance in the Context of Industry 4.0)
►▼ Show Figures

Figure 1

Figure 1
<p>Graphics of the pdf of the MWE distribution.</p> Full article ">Figure 2
<p>Graphics of the hrf of the MWE distribution.</p> Full article ">Figure 3
<p>Graphics of the (<b>a</b>) skewness and (<b>b</b>) kurtosis of the MWE distribution.</p> Full article ">Figure 4
<p>Graphic of the mean residual life function of the MWE distribution.</p> Full article ">Figure 5
<p>The histogram with the fitted pdfs for the data set 1.</p> Full article ">Figure 6
<p>The empirical cdf with the fitted cdfs for the data set 1.</p> Full article ">Figure 7
<p>The histogram with the fitted pdfs for the data set 2.</p> Full article ">Figure 8
<p>The empirical cdf with the fitted cdf for the data set 2.</p> Full article ">
19 pages, 4512 KiB  
Article
The Minimum Lindley Lomax Distribution: Properties and Applications
by Sadaf Khan, Gholamhossein G. Hamedani, Hesham Mohamed Reyad, Farrukh Jamal, Shakaiba Shafiq and Soha Othman
Math. Comput. Appl. 2022, 27(1), 16; https://doi.org/10.3390/mca27010016 - 18 Feb 2022
Cited by 3 | Viewed by 2580
Abstract
By fusing the Lindley and Lomax distributions, we present a unique three-parameter continuous model titled the minimum Lindley Lomax distribution. The quantile function, ordinary and incomplete moments, moment generating function, Lorenz and Bonferroni curves, order statistics, Rényi entropy, stress strength model, and stochastic [...] Read more.
By fusing the Lindley and Lomax distributions, we present a unique three-parameter continuous model titled the minimum Lindley Lomax distribution. The quantile function, ordinary and incomplete moments, moment generating function, Lorenz and Bonferroni curves, order statistics, Rényi entropy, stress strength model, and stochastic sequencing are all carefully examined as basic statistical aspects of the new distribution. The characterizations of the new model are investigated. The proposed distribution’s parameters were evaluated using the maximum likelihood procedures. The stability of the parameter estimations is explored using a Monte Carlo simulation. Two applications are used to objectively assess the new model’s extensibility. Full article
(This article belongs to the Special Issue Computational Mathematics and Applied Statistics)
►▼ Show Figures

Figure 1

Figure 1
<p>Possible figures of the minLLx pdf for parameter values chosen at random.</p> Full article ">Figure 2
<p>Possible figures of the minLLx hrf for parameter values chosen at random.</p> Full article ">Figure 3
<p>Estimated pdf and cdf plots of the minLLx distribution for the first data set.</p> Full article ">Figure 4
<p>Estimated pdf and cdf plots of the minLLx distribution for the second data set.</p> Full article ">
20 pages, 24807 KiB  
Article
Multi-Physics Inverse Homogenization for the Design of Innovative Cellular Materials: Application to Thermo-Elastic Problems
by Matteo Gavazzoni, Nicola Ferro, Simona Perotto and Stefano Foletti
Math. Comput. Appl. 2022, 27(1), 15; https://doi.org/10.3390/mca27010015 - 15 Feb 2022
Cited by 3 | Viewed by 3424
Abstract
We present a new algorithm to design lightweight cellular materials with required properties in a multi-physics context. In particular, we focus on a thermo-elastic setting by promoting the design of unit cells characterized both by an isotropic and an anisotropic behavior with respect [...] Read more.
We present a new algorithm to design lightweight cellular materials with required properties in a multi-physics context. In particular, we focus on a thermo-elastic setting by promoting the design of unit cells characterized both by an isotropic and an anisotropic behavior with respect to mechanical and thermal requirements. The proposed procedure generalizes the microSIMPATY algorithm to a thermo-elastic framework by preserving all the good properties of the reference design methodology. The resulting layouts exhibit non-standard topologies and are characterized by very sharp contours, thus limiting the post-processing before manufacturing. The new cellular materials are compared with the state-of-art in engineering practice in terms of thermo-elastic properties, thus highlighting the good performance of the new layouts which, in some cases, outperform the consolidated choices. Full article
(This article belongs to the Special Issue Computational Methods for Coupled Problems in Science and Engineering)
►▼ Show Figures

Figure 1

Figure 1
<p>Initial guess <math display="inline"><semantics> <msubsup> <mo>ρ</mo> <mi>h</mi> <mn>0</mn> </msubsup> </semantics></math> (<b>left</b>) and corresponding mesh <math display="inline"><semantics> <msubsup> <mi mathvariant="script">T</mi> <mi>h</mi> <mn>0</mn> </msubsup> </semantics></math> (<b>right</b>).</p> Full article ">Figure 2
<p>Design Case 1: density field (<b>top</b>) and associated anisotropic adapted mesh (<b>bottom</b>) for three different global iterations.</p> Full article ">Figure 3
<p>Design Case 1: evolution of the objective functional <math display="inline"><semantics> <mi mathvariant="script">M</mi> </semantics></math> and of the constraints <math display="inline"><semantics> <msub> <mi>c</mi> <mi>i</mi> </msub> </semantics></math> (<b>top</b>); trend of the mesh cardinality <math display="inline"><semantics> <mrow> <mo>#</mo> <msub> <mi mathvariant="script">T</mi> <mi>h</mi> </msub> </mrow> </semantics></math> (<b>bottom</b>) throughout the global iterations <tt>k</tt>.</p> Full article ">Figure 4
<p>Design Cases 1, 2, and 3 (<b>left</b>–<b>right</b>): <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </semantics></math>-cell meta-material.</p> Full article ">Figure 5
<p>Design Case 2: density field (<b>top</b>) and associated anisotropic adapted mesh (<b>bottom</b>) for three different global iterations.</p> Full article ">Figure 6
<p>Design Case 3: density field (<b>top</b>) and associated anisotropic adapted mesh (<b>bottom</b>) for three different global iterations.</p> Full article ">Figure 7
<p>Comparison between the optimized cells delivered by MultiP.microSIMP (<b>top</b>) and by MultiP-microSIMPATY (<b>bottom</b>) for the Design Cases 1, 2, and 3 (from left to right).</p> Full article ">Figure 8
<p>Effect of filtering for the MultiP-microSIMPATY algorithm: density field (<b>left</b>) and associated anisotropic adapted mesh (<b>right</b>) when filtering is applied during the whole optimization process (<b>top</b>) and in the first 25 iterations only (<b>bottom</b>).</p> Full article ">
44 pages, 8758 KiB  
Article
Arbitrarily Accurate Analytical Approximations for the Error Function
by Roy M. Howard
Math. Comput. Appl. 2022, 27(1), 14; https://doi.org/10.3390/mca27010014 - 9 Feb 2022
Cited by 5 | Viewed by 4209
Abstract
A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be improved by utilizing the approximation [...] Read more.
A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be improved by utilizing the approximation erf(x)≈1 for |x|>xo and with xo optimally chosen. Two generalizations are possible; the first is based on demarcating the integration interval into m equally spaced subintervals. The second, is based on utilizing a larger fixed subinterval, with a known integral, and a smaller subinterval whose integral is to be approximated. Both generalizations lead to significantly improved accuracy. Furthermore, the initial approximations, and those arising from the first generalization, can be utilized as inputs to a custom dynamic system to establish approximations with better convergence properties. Indicative results include those of a fourth-order approximation, based on four subintervals, which leads to a relative error bound of 1.43 × 10−7 over the interval [0, ∞]. The corresponding sixteenth-order approximation achieves a relative error bound of 2.01 × 10−19. Various approximations that achieve the set relative error bounds of 10−4, 10−6, 10−10, and 10−16, over [0, ∞], are specified. Applications include, first, the definition of functions that are upper and lower bounds, of arbitrary accuracy, for the error function. Second, new series for the error function. Third, new sequences of approximations for exp(−x2) that have significantly higher convergence properties than a Taylor series approximation. Fourth, the definition of a complementary demarcation function eC(x) that satisfies the constraint eC2(x)+erf2(x)=1. Fifth, arbitrarily accurate approximations for the power and harmonic distortion for a sinusoidal signal subject to an error function nonlinearity. Sixth, approximate expressions for the linear filtering of a step signal that is modeled by the error function. Full article
(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications)
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Figure 1

Figure 1
<p>Graph of the magnitude of the relative error in the approximations, detailed in <xref ref-type="table" rid="mca-27-00014-t001">Table 1</xref>, for erf(<italic>x</italic>).</p> Full article ">Figure 2
<p>Graph of the magnitude of the relative errors in approximations to erf(<italic>x</italic>): zero to tenth order integral spline based series and first, third, ..., fifteenth order Taylor series (dotted).</p> Full article ">Figure 3
<p>Graph of the magnitude of the relative errors associated with the approximation <inline-formula><mml:math id="mm92"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm93"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>exp</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:msqrt><mml:mo>π</mml:mo></mml:msqrt><mml:mi>x</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> along with the relative error in spline approximations of orders 16, 20, 24, 28 and 32.</p> Full article ">Figure 4
<p>Graph of the magnitude of the relative errors in the approximations to erf(<italic>x</italic>), of even orders, as specified by Equation (46). The dotted results are for the fourth order approximation specified by Theorem 1 (Equation (36)).</p> Full article ">Figure 5
<p>Illustration of the crossover point where the magnitude of the relative error in the approximation <inline-formula><mml:math id="mm126"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> equals the magnitude of the relative error in a set order spline approximation.</p> Full article ">Figure 6
<p>Graph of the relationship between the optimum transition point <inline-formula><mml:math id="mm134"><mml:semantics><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>o</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, as defined by Equation (57) for the case of <inline-formula><mml:math id="mm135"><mml:semantics><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> and the order of the spline approximation.</p> Full article ">Figure 7
<p>Graph of the relative errors in the approximations, <inline-formula><mml:math id="mm143"><mml:semantics><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, to erf(<italic>x</italic>), of orders 2, 4, 6, …, 20, based on utilizing the approximation <inline-formula><mml:math id="mm144"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in an optimum manner.</p> Full article ">Figure 8
<p>Graph of the magnitude of the relative error in Taylor series approximations to <inline-formula><mml:math id="mm155"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> that utilize an optimized change to the approximation <inline-formula><mml:math id="mm156"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 9
<p>Graph of the relative errors in spline approximations to <inline-formula><mml:math id="mm178"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, of orders one to six and based on four variable sub-intervals of equal width.</p> Full article ">Figure 10
<p>Graph of the relative errors in approximations to <inline-formula><mml:math id="mm185"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>: first to seventh order spline based series based on four sub-intervals of equal width and with utilization of the approximation <inline-formula><mml:math id="mm186"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> at the optimum transition point.</p> Full article ">Figure 11
<p>Illustration of areas comprising erf(<italic>x</italic>).</p> Full article ">Figure 12
<p>Graph of the relative error bound, versus the order of approximation, for various set resolutions.</p> Full article ">Figure 13
<p>Graph of the relative errors, based on a resolution of <inline-formula><mml:math id="mm226"><mml:semantics><mml:mrow><mml:mo>Δ</mml:mo><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, in second to fourth order approximations to erf(<italic>x</italic>).</p> Full article ">Figure 14
<p>Feedback system with dynamically varying (modulated) feedback.</p> Full article ">Figure 15
<p>Graph of the relative errors in approximations, of orders one to eight, to erf(<italic>x</italic>) as defined in Theorem 7.</p> Full article ">Figure 16
<p>Graph of the magnitude of the relative errors in approximations to exp(−<italic>x</italic><sup>2</sup>), as defined by Equation (103), of orders 0, 2, 4, 6, 8, 10 and 12. The dotted curves are the relative errors associated with Taylor series of orders 1, 3, 5, 7, 9, 11, 13 and 15.</p> Full article ">Figure 17
<p>Relative error in upper and lower bounds to erf(<italic>x</italic>) as, respectively, defined by Equation (110)–(112). The parameters <italic>p</italic> = 1 and <italic>q</italic> = π/4 have been used for the bounds defined by Equation (110).</p> Full article ">Figure 18
<p>Relative error in the approximations <inline-formula><mml:math id="mm321"><mml:semantics><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm322"><mml:semantics><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mo mathvariant="sans-serif">ε</mml:mo><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> to erf(<italic>x</italic>) where the residual function <inline-formula><mml:math id="mm323"><mml:semantics><mml:mrow><mml:msub><mml:mo mathvariant="sans-serif">ε</mml:mo><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> is approximated by the stated order.</p> Full article ">Figure 19
<p>Relative error in the approximations <inline-formula><mml:math id="mm324"><mml:semantics><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm325"><mml:semantics><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mo mathvariant="sans-serif">ε</mml:mo><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> to erf(<italic>x</italic>) where the residual function <inline-formula><mml:math id="mm326"><mml:semantics><mml:mrow><mml:msub><mml:mo mathvariant="sans-serif">ε</mml:mo><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> is approximated by the stated order.</p> Full article ">Figure 20
<p>Graph of the signals <inline-formula><mml:math id="mm336"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>e</mml:mi><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> and erf(<italic>x</italic>)<sup>2</sup>.</p> Full article ">Figure 21
<p>Graph of <inline-formula><mml:math id="mm342"><mml:semantics><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> for the case of <inline-formula><mml:math id="mm343"><mml:semantics><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>o</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and for amplitudes of <italic>a</italic> = 0.5, <italic>a</italic> = 1, <italic>a</italic> = 1.5 and <italic>a</italic> = 2.</p> Full article ">Figure 22
<p>Graph of the input power, output power and ratio of output power to input power as the amplitude of the input signal varies.</p> Full article ">Figure 23
<p>Graph of the variation of harmonic distortion with amplitude.</p> Full article ">Figure 24
<p>Graph of the input signal <inline-formula><mml:math id="mm366"><mml:semantics><mml:mrow><mml:mi>erf</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mo>γ</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm367"><mml:semantics><mml:mrow><mml:mo>γ</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and the corresponding approximation to the output of a second order linear filter with <inline-formula><mml:math id="mm368"><mml:semantics><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm369"><mml:semantics><mml:mrow><mml:mo>τ</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>π</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 25
<p>Graph of the relative errors associated with the output signal, shown in <xref ref-type="fig" rid="mca-27-00014-f024">Figure 24</xref>, for approximations to the error function (Equation (56)) of orders six to twelve which utilize optimum transition points.</p> Full article ">Figure A1
<p>Graphs of <inline-formula><mml:math id="mm433"><mml:semantics><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msubsup><mml:mo mathvariant="sans-serif">ε</mml:mo><mml:mi>n</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> for orders zero, two, four, six and eight.</p> Full article ">Figure A2
<p>Graphs of <inline-formula><mml:math id="mm434"><mml:semantics><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> for orders zero, two, four, six and eight.</p> Full article ">Figure A3
<p>Illustration of the coefficients that potentially are non-zero for a set power of <italic>t</italic>. The illustration is for the case of <italic>n</italic> = 8.</p> Full article ">
3 pages, 222 KiB  
Editorial
Acknowledgment to Reviewers of MCA in 2021
by MCA Editorial Office
Math. Comput. Appl. 2022, 27(1), 13; https://doi.org/10.3390/mca27010013 - 7 Feb 2022
Viewed by 1535
Abstract
Rigorous peer-reviews are the basis of high-quality academic publishing [...] Full article
20 pages, 1505 KiB  
Article
The Unit Teissier Distribution and Its Applications
by Anuresha Krishna, Radhakumari Maya, Christophe Chesneau and Muhammed Rasheed Irshad
Math. Comput. Appl. 2022, 27(1), 12; https://doi.org/10.3390/mca27010012 - 1 Feb 2022
Cited by 17 | Viewed by 3464
Abstract
A bounded form of the Teissier distribution, namely the unit Teissier distribution, is introduced. It is subjected to a thorough examination of its important properties, including shape analysis of the main functions, analytical expression for moments based on upper incomplete gamma function, incomplete [...] Read more.
A bounded form of the Teissier distribution, namely the unit Teissier distribution, is introduced. It is subjected to a thorough examination of its important properties, including shape analysis of the main functions, analytical expression for moments based on upper incomplete gamma function, incomplete moments, probability-weighted moments, and quantile function. The uncertainty measures Shannon entropy and extropy are also performed. The maximum likelihood estimation, least square estimation, weighted least square estimation, and Bayesian estimation methods are used to estimate the parameters of the model, and their respective performances are assessed via a simulation study. Finally, the competency of the proposed model is illustrated by using two data sets from diverse fields. Full article
(This article belongs to the Special Issue Computational Mathematics and Applied Statistics)
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<p>Plots of various shapes of the pdf of the <math display="inline"><semantics> <mrow> <mi>U</mi> <mi>T</mi> <mi>D</mi> </mrow> </semantics></math> for varying values of the parameter <math display="inline"><semantics> <mi>θ</mi> </semantics></math>.</p> Full article ">Figure 2
<p>Plots of various shapes of the hrf of the <math display="inline"><semantics> <mrow> <mi>U</mi> <mi>T</mi> <mi>D</mi> </mrow> </semantics></math> for varying values of the parameter <math display="inline"><semantics> <mi>θ</mi> </semantics></math>.</p> Full article ">Figure 2 Cont.
<p>Plots of various shapes of the hrf of the <math display="inline"><semantics> <mrow> <mi>U</mi> <mi>T</mi> <mi>D</mi> </mrow> </semantics></math> for varying values of the parameter <math display="inline"><semantics> <mi>θ</mi> </semantics></math>.</p> Full article ">Figure 3
<p>Plots of the (<b>a</b>) mean and (<b>b</b>) variance of the <math display="inline"><semantics> <mrow> <mi>U</mi> <mi>T</mi> <mi>D</mi> </mrow> </semantics></math> for varying values of the parameter <math display="inline"><semantics> <mi>θ</mi> </semantics></math>.</p> Full article ">Figure 4
<p>Graphical comparison of the MSEs obtained from ML, LS, and WLS estimation methods for (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.26</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.45</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.60</mn> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> and (<b>f</b>) <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>2.0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>(<b>a</b>) TTT plot and (<b>b</b>) histogram for the flood level data.</p> Full article ">Figure 6
<p>(<b>a</b>) TTT plot and (<b>b</b>) histogram for secondary reactor pumps data.</p> Full article ">
12 pages, 1446 KiB  
Article
Impact of Infective Immigrants on COVID-19 Dynamics
by Stéphane Yanick Tchoumi, Herieth Rwezaura, Mamadou Lamine Diagne, Gilberto González-Parra and Jean Tchuenche
Math. Comput. Appl. 2022, 27(1), 11; https://doi.org/10.3390/mca27010011 - 29 Jan 2022
Cited by 3 | Viewed by 3581
Abstract
The COVID-19 epidemic is an unprecedented and major social and economic challenge worldwide due to the various restrictions. Inflow of infective immigrants have not been given prominence in several mathematical and epidemiological models. To investigate the impact of imported infection on the number [...] Read more.
The COVID-19 epidemic is an unprecedented and major social and economic challenge worldwide due to the various restrictions. Inflow of infective immigrants have not been given prominence in several mathematical and epidemiological models. To investigate the impact of imported infection on the number of deaths, cumulative infected and cumulative asymptomatic, we formulate a mathematical model with infective immigrants and considering vaccination. The basic reproduction number of the special case of the model without immigration of infective people is derived. We varied two key factors that affect the transmission of COVID-19, namely the immigration and vaccination rates. In addition, we considered two different SARS-CoV-2 transmissibilities in order to account for new more contagious variants such as Omicron. Numerical simulations using initial conditions approximating the situation in the US when the vaccination program was starting show that increasing the vaccination rate significantly improves the outcomes regarding the number of deaths, cumulative infected and cumulative asymptomatic. Other factors are the natural recovery rates of infected and asymptomatic individuals, the waning rate of the vaccine and the vaccination rate. When the immigration rate is increased significantly, the number of deaths, cumulative infected and cumulative asymptomatic increase. Consequently, accounting for the level of inflow of infective immigrants may help health policy/decision-makers to formulate policies for public health prevention programs, especially with respect to the implementation of the stringent preventive lock down measure. Full article
(This article belongs to the Collection Mathematical Modelling of COVID-19)
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<p>COVID-19 transmission dynamic model flowchart with inflow of infective immigrants.</p> Full article ">Figure 2
<p>Graphical representation of the sensitivity of the reproduction number <math display="inline"><semantics> <msub> <mi>R</mi> <mn>0</mn> </msub> </semantics></math> using Latin hypercube sampling and the Partial rank correlation coefficients with 10,000 samples.</p> Full article ">Figure 3
<p>Contour plot of the basic reproduction number <math display="inline"><semantics> <msub> <mi>R</mi> <mn>0</mn> </msub> </semantics></math> for different values of the effective contact rate <math display="inline"><semantics> <mi>β</mi> </semantics></math> and the vaccination rate <span class="html-italic">v</span>.</p> Full article ">Figure 4
<p>Contour plot of the basic reproduction number <math display="inline"><semantics> <msub> <mi>R</mi> <mn>0</mn> </msub> </semantics></math> for different values of the vaccination rate <span class="html-italic">v</span> and waning immunity rate <span class="html-italic">w</span>.</p> Full article ">Figure 5
<p>Dynamics of the model sub-populations without inflow of infective immigrants and without vaccination.</p> Full article ">Figure 6
<p>Dynamics of the model sub-populations without inflow of infective immigrants and with vaccination.</p> Full article ">Figure 7
<p>Dynamics of several sub-populations with inflow of infective immigrants and without vaccination.</p> Full article ">Figure 8
<p>Dynamics of several sub-populations with inflow of infective immigrants and with vaccination.</p> Full article ">Figure 9
<p>Number of deaths, cumulative infected, cumulative asymptomatic and cumulative vaccinated when vaccination rate and inflow level of immigrants are varied.</p> Full article ">
24 pages, 1326 KiB  
Article
Numerical and Theoretical Stability Study of a Viscoelastic Plate Equation with Nonlinear Frictional Damping Term and a Logarithmic Source Term
by Mohammad M. Al-Gharabli, Adel M. Almahdi, Maher Noor and Johnson D. Audu
Math. Comput. Appl. 2022, 27(1), 10; https://doi.org/10.3390/mca27010010 - 28 Jan 2022
Cited by 5 | Viewed by 2868
Abstract
This paper is designed to explore the asymptotic behaviour of a two dimensional visco-elastic plate equation with a logarithmic nonlinearity under the influence of nonlinear frictional damping. Assuming that relaxation function g satisfies [...] Read more.
This paper is designed to explore the asymptotic behaviour of a two dimensional visco-elastic plate equation with a logarithmic nonlinearity under the influence of nonlinear frictional damping. Assuming that relaxation function g satisfies g′(t)≤−ξ(t)G(g(t)), we establish an explicit general decay rates without imposing a restrictive growth assumption on the damping term. This general condition allows us to recover the exponential and polynomial rates. Our results improve and extend some existing results in the literature. We preform some numerical experiments to illustrate our theoretical results. Full article
(This article belongs to the Special Issue Mathematical and Computational Modelling in Mechanics of Materials and Structures)
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<p>Test 1: The solution. <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> at fixed values of <span class="html-italic">x</span>.</p> Full article ">Figure 2
<p>Test 2: The solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> at fixed values of <span class="html-italic">x</span>.</p> Full article ">Figure 3
<p>Test 3: The solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> at fixed values of <span class="html-italic">x</span>.</p> Full article ">Figure 4
<p>Test 4: The solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> at fixed values of <span class="html-italic">x</span>.</p> Full article ">Figure 5
<p>Test 1: The energy decay.</p> Full article ">Figure 6
<p>Test 2: The energy decay.</p> Full article ">Figure 7
<p>Test 3: The energy decay.</p> Full article ">Figure 8
<p>Test 4: The energy decay.</p> Full article ">Figure 9
<p>Test 1: The solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Test 2: The solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Test 3: The solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Test 4: The solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">
13 pages, 6095 KiB  
Article
Numerical Study on Mixed Convection Flow and Energy Transfer in an Inclined Channel Cavity: Effect of Baffle Size
by Sivanandam Sivasankaran and Kandasamy Janagi
Math. Comput. Appl. 2022, 27(1), 9; https://doi.org/10.3390/mca27010009 - 23 Jan 2022
Cited by 4 | Viewed by 2259
Abstract
The objective of the current numerical study is to explore the combined natural and forced convection and energy transport in a channel with an open cavity. An adiabatic baffle of finite length is attached to the top wall. The sinusoidal heating is implemented [...] Read more.
The objective of the current numerical study is to explore the combined natural and forced convection and energy transport in a channel with an open cavity. An adiabatic baffle of finite length is attached to the top wall. The sinusoidal heating is implemented on the lower horizontal wall of the open cavity. The other areas of the channel cavity are treated as adiabatic. The governing equations are solved by the control volume technique for various values of relevant factors. The drag force, bulk temperature and average Nusselt number are computed. It is recognised that recirculating eddies beside the baffle become weak or disappear upon increasing the inclination angle of the channel/cavity. The average thermal energy transportation reduces steadily until the Ri = 1 and then it rises for all inclination angles and lengths of the baffle. Full article
(This article belongs to the Special Issue Advances in Computational Fluid Dynamics and Heat & Mass Transfer)
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<p>Physical model.</p> Full article ">Figure 2
<p>Streamlines for various values of <span class="html-italic">Ri</span> and inclination angles with <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>B</mi> </msub> </mrow> </semantics></math> = 0.5.</p> Full article ">Figure 2 Cont.
<p>Streamlines for various values of <span class="html-italic">Ri</span> and inclination angles with <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>B</mi> </msub> </mrow> </semantics></math> = 0.5.</p> Full article ">Figure 3
<p>Isotherms for various values of <span class="html-italic">Ri</span> and inclination angles with <span class="html-italic">L<sub>B</sub></span> = 0.5.</p> Full article ">Figure 3 Cont.
<p>Isotherms for various values of <span class="html-italic">Ri</span> and inclination angles with <span class="html-italic">L<sub>B</sub></span> = 0.5.</p> Full article ">Figure 4
<p>Streamlines for various <span class="html-italic">L<sub>B</sub></span> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> values with <span class="html-italic">Ri</span> = 1.0.</p> Full article ">Figure 4 Cont.
<p>Streamlines for various <span class="html-italic">L<sub>B</sub></span> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> values with <span class="html-italic">Ri</span> = 1.0.</p> Full article ">Figure 5
<p>Isotherms for various <span class="html-italic">L<sub>B</sub></span> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> values with <span class="html-italic">Ri</span> = 1.0.</p> Full article ">Figure 6
<p>Drag force vs. <span class="html-italic">Ri</span> for variations of <span class="html-italic">L<sub>B</sub></span> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> values.</p> Full article ">Figure 7
<p>Average temperature vs. <span class="html-italic">Ri</span> for variations of <span class="html-italic">L<sub>B</sub></span> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> values.</p> Full article ">Figure 7 Cont.
<p>Average temperature vs. <span class="html-italic">Ri</span> for variations of <span class="html-italic">L<sub>B</sub></span> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> values.</p> Full article ">Figure 8
<p>Local Nusselt number for various inclination angles, <span class="html-italic">L<sub>B</sub></span> and <span class="html-italic">Ri</span>.</p> Full article ">Figure 8 Cont.
<p>Local Nusselt number for various inclination angles, <span class="html-italic">L<sub>B</sub></span> and <span class="html-italic">Ri</span>.</p> Full article ">Figure 9
<p>Average <span class="html-italic">Nu</span> vs. <span class="html-italic">Ri</span> for variations of <span class="html-italic">L<sub>B</sub></span> and <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math>.</p> Full article ">Figure 10
<p>Comparison of the average Nu for different lengths of baffle and various inclination angles and Richardson numbers with <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>B</mi> </msub> </mrow> </semantics></math> = 0.</p> Full article ">Figure 10 Cont.
<p>Comparison of the average Nu for different lengths of baffle and various inclination angles and Richardson numbers with <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mi>B</mi> </msub> </mrow> </semantics></math> = 0.</p> Full article ">
25 pages, 583 KiB  
Article
Fractional Modeling of Viscous Fluid over a Moveable Inclined Plate Subject to Exponential Heating with Singular and Non-Singular Kernels
by Aziz Ur Rehman, Muhammad Bilal Riaz, Wajeeha Rehman, Jan Awrejcewicz and Dumitru Baleanu
Math. Comput. Appl. 2022, 27(1), 8; https://doi.org/10.3390/mca27010008 - 19 Jan 2022
Cited by 11 | Viewed by 2628
Abstract
In this paper, a new approach to investigating the unsteady natural convection flow of viscous fluid over a moveable inclined plate with exponential heating is carried out. The mathematical modeling is based on fractional treatment of the governing equation subject to the temperature, [...] Read more.
In this paper, a new approach to investigating the unsteady natural convection flow of viscous fluid over a moveable inclined plate with exponential heating is carried out. The mathematical modeling is based on fractional treatment of the governing equation subject to the temperature, velocity and concentration field. Innovative definitions of time fractional operators with singular and non-singular kernels have been working on the developed constitutive mass, energy and momentum equations. The fractionalized analytical solutions based on special functions are obtained by using Laplace transform method to tackle the non-dimensional partial differential equations for velocity, mass and energy. Our results propose that by increasing the value of the Schimdth number and Prandtl number the concentration and temperature profiles decreased, respectively. The presence of a Prandtl number increases the thermal conductivity and reflects the control of thickness of momentum. The experimental results for flow features are shown in graphs over a limited period of time for various parameters. Furthermore, some special cases for the movement of the plate are also studied and results are demonstrated graphically via Mathcad-15 software. Full article
(This article belongs to the Special Issue Advances in Computational Fluid Dynamics and Heat & Mass Transfer)
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<p>Trace of dimensionless temperature for dissimilar values of <math display="inline"><semantics> <mrow> <mi>P</mi> <msub> <mi>r</mi> <mrow> <mi>e</mi> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math> via CF.</p> Full article ">Figure 2
<p>Trace of dimensionless temperature for dissimilar values of <math display="inline"><semantics> <mrow> <mi>P</mi> <msub> <mi>r</mi> <mrow> <mi>e</mi> <mi>f</mi> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math> via ABC.</p> Full article ">Figure 3
<p>Trace of dimensionless concentration for dissimilar values of <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>c</mi> </mrow> </semantics></math> via CF.</p> Full article ">Figure 4
<p>Trace of dimensionless concentration for dissimilar values of <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>c</mi> </mrow> </semantics></math> via ABC.</p> Full article ">Figure 5
<p>Trace of dimensionless velocity for dissimilar values of <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>c</mi> </mrow> </semantics></math> via CF.</p> Full article ">Figure 6
<p>Trace of dimensionless velocity for dissimilar values of <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>c</mi> </mrow> </semantics></math> via ABC.</p> Full article ">Figure 7
<p>Trace of dimensionless velocity for dissimilar values of <span class="html-italic">N</span> via CF.</p> Full article ">Figure 8
<p>Trace of dimensionless velocity for dissimilar values of <span class="html-italic">N</span> via ABC.</p> Full article ">Figure 9
<p>Trace of dimensionless velocity for dissimilar values of <math display="inline"><semantics> <msub> <mi>R</mi> <mi>c</mi> </msub> </semantics></math> via CF.</p> Full article ">Figure 10
<p>Trace of dimensionless velocity for dissimilar values of <math display="inline"><semantics> <msub> <mi>R</mi> <mi>c</mi> </msub> </semantics></math> via ABC.</p> Full article ">Figure 11
<p>Trace of dimensionless velocity for dissimilar values of <math display="inline"><semantics> <mi>η</mi> </semantics></math> via CF.</p> Full article ">Figure 12
<p>Trace of dimensionless velocity for dissimilar values of <math display="inline"><semantics> <mi>η</mi> </semantics></math> via ABC.</p> Full article ">Figure 13
<p>Comparison of dimensionless velocity profil for dissimilar values of <math display="inline"><semantics> <mi>η</mi> </semantics></math> between CF and ABC.</p> Full article ">
15 pages, 5406 KiB  
Article
Approximating the Steady-State Temperature of 3D Electronic Systems with Convolutional Neural Networks
by Monika Stipsitz and Hèlios Sanchis-Alepuz
Math. Comput. Appl. 2022, 27(1), 7; https://doi.org/10.3390/mca27010007 - 14 Jan 2022
Cited by 6 | Viewed by 3301
Abstract
Thermal simulations are an important part of the design process in many engineering disciplines. In simulation-based design approaches, a considerable amount of time is spent by repeated simulations. An alternative, fast simulation tool would be a welcome addition to any automatized and simulation-based [...] Read more.
Thermal simulations are an important part of the design process in many engineering disciplines. In simulation-based design approaches, a considerable amount of time is spent by repeated simulations. An alternative, fast simulation tool would be a welcome addition to any automatized and simulation-based optimisation workflow. In this work, we present a proof-of-concept study of the application of convolutional neural networks to accelerate thermal simulations. We focus on the thermal aspect of electronic systems. The goal of such a tool is to provide accurate approximations of a full solution, in order to quickly select promising designs for more detailed investigations. Based on a training set of randomly generated circuits with corresponding finite element solutions, the full 3D steady-state temperature field is estimated using a fully convolutional neural network. A custom network architecture is proposed which captures the long-range correlations present in heat conduction problems. We test the network on a separate dataset and find that the mean relative error is around 2% and the typical evaluation time is 35 ms per sample (2 ms for evaluation, 33 ms for data transfer). The benefit of this neural-network-based approach is that, once training is completed, the network can be applied to any system within the design space spanned by the randomized training dataset (which includes different components, material properties, different positioning of components on a PCB, etc.). Full article
(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications)
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<p>Illustration of the automatized workflow: Randomized systems are generated by randomly choosing and placing basic components. After assigning randomized material properties and BC values, the system is voxelized to create a stack of four 3D-images as input for the NN. Solutions for the supervized training procedure are created using FE simulations.</p> Full article ">Figure 2
<p>Architecture used in this work. In each block, <span class="html-italic">k</span> refers to the kernels size, <span class="html-italic">s</span> to the stride, <span class="html-italic">d</span> to the dilations and <span class="html-italic">C</span> to the output channels of each layer (the same for all dimensions). See <a href="#sec2dot2-mca-27-00007" class="html-sec">Section 2.2</a> for further details.</p> Full article ">Figure 3
<p>Schematic representation of the action of a 2D fusion block consisting of two convolutional layers with 3 × 3 kernels and dilations 3 and 1.</p> Full article ">Figure 4
<p>Histogram of the average relative <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> error per test system (<b>top</b>). Below the temperature distributions estimated by the NN (<b>right</b>) and the relative temperature difference (<b>left</b>) for selected systems of the test dataset (the corresponding error bin of the histogram is indicated in brackets, from 0 indicating the lowest mean error to 19 for the worst mean error). The average relative <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> error (<b>top</b>) is the mean of the absolute values of the relative temperature differences (<b>bottom left</b>) per sample.</p> Full article ">Figure 5
<p>FE solution (<b>left</b>), NN prediction (<b>center</b>), and relative <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> error of the temperature distribution (<b>right</b>) on a horizontal cut of a selected system. High predictive errors are mostly found on the surface of the system while the internal temperature distribution is well represented.</p> Full article ">Figure 6
<p>Comparison of the heat equation error (<b>top row</b>), which can be used as error predictor if no FE solution is available, and the <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> error (<b>bottom row</b>) on selected slices from bottom to top (<b>left to right</b>). The heat equation error is able to indicate most of the regions with high error. It illustrates the checkerboard pattern expected from purely convolutional networks. Since the heat equation error is defined via the local imbalance of heat fluxes and sources, the detected errors can be slightly more localized compared to the actual error (e.g., orange points in slice 69, top compared to bottom).</p> Full article ">
25 pages, 4188 KiB  
Article
AutoML for Feature Selection and Model Tuning Applied to Fault Severity Diagnosis in Spur Gearboxes
by Mariela Cerrada, Leonardo Trujillo, Daniel E. Hernández, Horacio A. Correa Zevallos, Jean Carlo Macancela, Diego Cabrera and René Vinicio Sánchez
Math. Comput. Appl. 2022, 27(1), 6; https://doi.org/10.3390/mca27010006 - 13 Jan 2022
Cited by 14 | Viewed by 4757
Abstract
Gearboxes are widely used in industrial processes as mechanical power transmission systems. Then, gearbox failures can affect other parts of the system and produce economic loss. The early detection of the possible failure modes and their severity assessment in such devices is an [...] Read more.
Gearboxes are widely used in industrial processes as mechanical power transmission systems. Then, gearbox failures can affect other parts of the system and produce economic loss. The early detection of the possible failure modes and their severity assessment in such devices is an important field of research. Data-driven approaches usually require an exhaustive development of pipelines including models’ parameter optimization and feature selection. This paper takes advantage of the recent Auto Machine Learning (AutoML) tools to propose proper feature and model selection for three failure modes under different severity levels: broken tooth, pitting and crack. The performance of 64 statistical condition indicators (SCI) extracted from vibration signals under the three failure modes were analyzed by two AutoML systems, namely the H2O Driverless AI platform and TPOT, both of which include feature engineering and feature selection mechanisms. In both cases, the systems converged to different types of decision tree methods, with ensembles of XGBoost models preferred by H2O while TPOT generated different types of stacked models. The models produced by both systems achieved very high, and practically equivalent, performances on all problems. Both AutoML systems converged to pipelines that focus on very similar subsets of features across all problems, indicating that several problems in this domain can be solved by a rather small set of 10 common features, with accuracy up to 90%. This latter result is important in the research of useful feature selection for gearbox fault diagnosis. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2021)
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Figure 1

Figure 1
<p>General ML pipeline and focus of an AutoML system.</p> Full article ">Figure 2
<p>One stage gearbox.</p> Full article ">Figure 3
<p>Details of the simulated faults for the three case studies. (<b>a</b>) Schemes and photography of crack levels. (<b>b</b>) Schemes and photography of pitting levels. (<b>c</b>) Schemes and photography of broken tooth levels.</p> Full article ">Figure 4
<p>Test bed under realistic conditions.</p> Full article ">Figure 5
<p>Schematic experimental setup.</p> Full article ">Figure 6
<p>Samples of a vibration signal for simulated faults for the three case studies.</p> Full article ">Figure 7
<p>Methodological Framework.</p> Full article ">Figure 8
<p>Plots show the impact of total features used by XGBoost classifier for each problem using individual feature lists. The left column (<b>a</b>,<b>c</b>,<b>e</b>) shows the impact on classification accuracy and the right column (<b>b</b>,<b>d</b>,<b>f</b>) shows the impact on hyperparameter optimization with Grid Search.</p> Full article ">Figure 9
<p>Plots show the impact of total features used by XGBoost classifier for each problem using a common feature list based on average feature importance. The left column (<b>a</b>,<b>c</b>,<b>e</b>) shows the impact on classification accuracy and the right column (<b>b</b>,<b>d</b>,<b>f</b>) shows the impact on hyperparameter optimization with Grid Search.</p> Full article ">Figure 10
<p>Accuracy trend by using RF based classifier and feature ranking with RelieF.</p> Full article ">
13 pages, 520 KiB  
Article
Analysis and Detection of Erosion in Wind Turbine Blades
by Josué Enríquez Zárate, María de los Ángeles Gómez López, Javier Alberto Carmona Troyo and Leonardo Trujillo
Math. Comput. Appl. 2022, 27(1), 5; https://doi.org/10.3390/mca27010005 - 13 Jan 2022
Cited by 7 | Viewed by 4134
Abstract
This paper studies erosion at the tip of wind turbine blades by considering aerodynamic analysis, modal analysis and predictive machine learning modeling. Erosion can be caused by several factors and can affect different parts of the blade, reducing its dynamic performance and useful [...] Read more.
This paper studies erosion at the tip of wind turbine blades by considering aerodynamic analysis, modal analysis and predictive machine learning modeling. Erosion can be caused by several factors and can affect different parts of the blade, reducing its dynamic performance and useful life. The ability to detect and quantify erosion on a blade is an important predictive maintenance task for wind turbines that can have broad repercussions in terms of avoiding serious damage, improving power efficiency and reducing downtimes. This study considers both sides of the leading edge of the blade (top and bottom), evaluating the mechanical imbalance caused by the material loss that induces variations of the power coefficient resulting in a loss in efficiency. The QBlade software is used in our analysis and load calculations are preformed by using blade element momentum theory. Numerical results show the performance of a blade based on the relationship between mechanical damage and aerodynamic behavior, which are then validated on a physical model. Moreover, two machine learning (ML) problems are posed to automatically detect the location of erosion (top of the edge, bottom or both) and to determine erosion levels (from 8% to 18%) present in the blade. The first problem is solved using classification models, while the second is solved using ML regression, achieving accurate results. ML pipelines are automatically designed by using an AutoML system with little human intervention, achieving highly accurate results. This work makes several contributions by developing ML models to both detect the presence and location of erosion on a blade, estimating its level and applying AutoML for the first time in this domain. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2021)
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<p>Sections of a wind turbine blade.</p> Full article ">Figure 2
<p>(<b>a</b>) Wind turbine blade design. (<b>b</b>) Area of erosion damage. (<b>c</b>) Blue line represents the clean Airfoil FX 63_137, while the dark line is the eroded blade considering case <math display="inline"><semantics> <mrow> <mi>B</mi> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>(<b>a</b>) Relationship between the drag coefficient (<math display="inline"><semantics> <msub> <mi>C</mi> <mi>D</mi> </msub> </semantics></math>) and the lift coefficient (<math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math>). (<b>b</b>) Lift coefficient <math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math> Vs angle of attack <math display="inline"><semantics> <mi>α</mi> </semantics></math>. (<b>c</b>) Power coefficient <math display="inline"><semantics> <msub> <mi>C</mi> <mi>p</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>B</mi> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and the clean wind turbine blade. (<b>d</b>) Power output <math display="inline"><semantics> <msub> <mi>P</mi> <mi>G</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>B</mi> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and the clean blade.</p> Full article ">
29 pages, 22708 KiB  
Article
Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk
by Dominic P. Clemence-Mkhope and Gregory A. Gibson
Math. Comput. Appl. 2022, 27(1), 4; https://doi.org/10.3390/mca27010004 - 10 Jan 2022
Cited by 2 | Viewed by 2316
Abstract
Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) [...] Read more.
Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index α=1, the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is, in part, an artifact of the CEFD method, as it is absent from the ESDDFD models. Full article
(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)
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Figure 1
<p>Phase portraits (<b>a</b>) CEFD model (20) (<b>b</b>) MCEFD model (16) (<b>c</b>) Model 17 (<b>d</b>) Model 18 (<b>e</b>) model (19).</p> Full article ">Figure 2
<p>CEFD model (21) profiles of (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>w</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.002</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.232</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.002</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.232</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">Figure 4
<p>CEFD model (21); bifurcation of (<b>a</b>) <math display="inline"><semantics> <mi>u</mi> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mi>x</mi> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mi>z</mi> </semantics></math> versus <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.002</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>MCEFD Model (22); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0.232</mn> <mo>,</mo> <mo> </mo> <mn>0.328</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>ESDDFD model (23); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0.232</mn> <mo>,</mo> <mo> </mo> <mn>0.328</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>ESDDFD model (24); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0.232</mn> <mo>,</mo> <mo> </mo> <mn>0.328</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>ESDDFD model (25); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <msub> <mi>α</mi> <mn>5</mn> </msub> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0.232</mn> <mo>,</mo> <mo> </mo> <mn>0.328</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.232</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">Figure 9 Cont.
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.232</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">Figure 10
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.232</mn> </mrow> </semantics></math> for models (22) through (25). <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.232</mn> </mrow> </semantics></math> for (23) through (25).</p> Full article ">Figure 10 Cont.
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.232</mn> </mrow> </semantics></math> for models (22) through (25). <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>1.0</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.232</mn> </mrow> </semantics></math> for (23) through (25).</p> Full article ">Figure 11
<p>CEFD model (21); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 11 Cont.
<p>CEFD model (21); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>MCEFD model (22); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>ESDDFD1 model (23); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 13 Cont.
<p>ESDDFD1 model (23); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>ESDDFD2 model (24); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 15
<p>ESDDFD2 model (25); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 15 Cont.
<p>ESDDFD2 model (25); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>p</mi> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 16
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.002</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1.94</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">Figure 16 Cont.
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.002</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1.94</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">Figure 17
<p>Model (21) phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>u</mi> <mo>,</mo> </mrow> </semantics></math> and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1.94</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 18
<p>CEFD model (21); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math>(<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo> </mo> <mrow> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1.5</mn> <mo>,</mo> <mo> </mo> <mn>2.5</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 19
<p>MCEFD model (22); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math>(<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo> </mo> <mrow> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1.5</mn> <mo>,</mo> <mo> </mo> <mn>2.5</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 20
<p>ESDDFD1 model (23); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math>(<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo> </mo> <mrow> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1.5</mn> <mo>,</mo> <mo> </mo> <mn>2.5</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 21
<p>ESDDFD2 model (24); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math>(<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo> </mo> <mrow> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1.5</mn> <mo>,</mo> <mo> </mo> <mn>2.5</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 22
<p>ESDDFD2 model (25); (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math>(<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo> </mo> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo> </mo> <mrow> <mi>vs</mi> <mo>.</mo> <mo> </mo> </mrow> <mi>k</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>1.5</mn> <mo>,</mo> <mo> </mo> <mn>2.5</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 23
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2.45</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">Figure 23 Cont.
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mo>,</mo> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2.45</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">Figure 24
<p>Model (21) phase portraits; (<b>a</b>)<math display="inline"><semantics> <mrow> <mo> </mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>u</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math>and (<b>c</b>)<math display="inline"><semantics> <mrow> <mo> </mo> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2.45</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 25
<p>Phase portraits <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> <mo> </mo> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>u</mi> <mo>,</mo> <mo> </mo> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> <mo> </mo> <mi>x</mi> <mo>−</mo> <mi mathvariant="normal">y</mi> <mo>−</mo> <mrow> <mi mathvariant="normal">w</mi> <mo> </mo> </mrow> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.002</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.24</mn> </mrow> </semantics></math> for model (21) CEFD.</p> Full article ">Figure 26
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>u</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>w</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math>at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.002</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.24</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">Figure 26 Cont.
<p>Phase portraits (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>u</mi> <mo>,</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>w</mi> <mo>,</mo> <mo> </mo> </mrow> </semantics></math>at <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>0.002</mn> <mo>,</mo> <mo> </mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <msub> <mi>α</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.24</mn> </mrow> </semantics></math> for models (22) through (25).</p> Full article ">
24 pages, 2015 KiB  
Article
Symbolic Computation Applied to Cauchy Type Singular Integrals
by Ana C. Conceição and Jéssica C. Pires
Math. Comput. Appl. 2022, 27(1), 3; https://doi.org/10.3390/mca27010003 - 31 Dec 2021
Cited by 3 | Viewed by 2723
Abstract
The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the [...] Read more.
The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article. Full article
(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications 2021)
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Figure 1

Figure 1
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">Option 1</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 2
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">Input</span> of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mo>−</mo> </msub> </semantics></math>.</p> Full article ">Figure 3
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 4
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">Option 2</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 5
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">input</span> of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mo>−</mo> </msub> </semantics></math>.</p> Full article ">Figure 6
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 7
<p>Part of the structure of the [SInt] algorithm corresponding to <span class="html-italic">Option 3</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 8
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">input</span> of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mo>−</mo> </msub> </semantics></math>.</p> Full article ">Figure 9
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 10
<p>Part of the structure of the [SInt] algorithm corresponding to <span class="html-italic">Option 2</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 11
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 12
<p>Part of the structure of the [SInt] algorithm corresponding to <span class="html-italic">Option 1</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 13
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">input</span> of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mo>−</mo> </msub> </semantics></math>.</p> Full article ">Figure 14
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 15
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">Option 1</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 16
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">Input</span> of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mo>−</mo> </msub> </semantics></math>.</p> Full article ">Figure 17
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 18
<p>Part of the structure of the [SInt] algorithm corresponding to <span class="html-italic">Option 1</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 19
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">input</span> of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mo>−</mo> </msub> </semantics></math>.</p> Full article ">Figure 20
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 21
<p>Part of the structure of the [SInt] algorithm corresponding to <span class="html-italic">Option 1</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 22
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">input</span> of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mo>−</mo> </msub> </semantics></math>.</p> Full article ">Figure 23
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 24
<p>Part of the structure of the [SInt] algorithm corresponding to <span class="html-italic">Option 1</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 25
<p>Part of the structure of the [SInt] algorithm corresponding to the <span class="html-italic">input</span> of <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mo>−</mo> </msub> </semantics></math>.</p> Full article ">Figure 26
<p><span class="html-italic">Output</span> given by the [SInt] algorithm.</p> Full article ">Figure 27
<p>Part of the code structure of the [ARoot] algorithm responsible for the input options for the polynomial <span class="html-italic">p</span>.</p> Full article ">Figure 28
<p>Part of the code structure of the [ARoot] algorithm responsible for the computation of the roots of <span class="html-italic">p</span>.</p> Full article ">Figure 29
<p>Part of the code structure of the [ARoots] algorithm code corresponding to the analysis of the absolute value of the roots of the inputed polynomial.</p> Full article ">Figure 30
<p>Part of the code structure of the [ARoots] algorithm code corresponding to the computation of an approximate value of a desired root.</p> Full article ">Figure 31
<p>Flowchart of the [ARoots] algorithm.</p> Full article ">Figure 32
<p>Part of the structure of the [ARoots] algorithm corresponding to the <span class="html-italic">Option 1</span> to input <span class="html-italic">p</span>.</p> Full article ">Figure 33
<p><span class="html-italic">Output</span> given by the [ARoots] algorithm.</p> Full article ">Figure 34
<p>Part of the structure of the [ARoots] algorithm corresponding to the option to get an approximate root value.</p> Full article ">Figure 35
<p><span class="html-italic">Output</span> given by the [ARoots] algorithm.</p> Full article ">Figure 36
<p>Part of the structure of the [ARoots] algorithm corresponding to <span class="html-italic">Option 1</span> to input <span class="html-italic">p</span>.</p> Full article ">Figure 37
<p><span class="html-italic">Output</span> given by the [ARoots] algorithm.</p> Full article ">Figure 38
<p>Part of the code structure of the [AZeros] algorithm code responsible for the location, relative to the unit circle, of complex numbers in a given list.</p> Full article ">Figure 39
<p>Flowcharts of the [AZeros] and [APoles] algorithms.</p> Full article ">Figure 40
<p>Part of the structure that allows the computation of the zeros and poles of <span class="html-italic">r</span>.</p> Full article ">Figure 41
<p><span class="html-italic">Output</span> given by the [AZeros] and [APoles] algorithms.</p> Full article ">Figure 42
<p>Part of the structure that allows us to compute and locate zeros of <span class="html-italic">r</span>.</p> Full article ">Figure 43
<p><span class="html-italic">Output</span> given by the [AZeros] algorithm.</p> Full article ">Figure 44
<p>Part of the structure that allows us to compute and locate poles of <span class="html-italic">r</span>.</p> Full article ">Figure 45
<p><span class="html-italic">Output</span> given by the [APoles] algorithm.</p> Full article ">Figure 46
<p>Flowchart of the [ASPPlusPMinus] algorithm.</p> Full article ">Figure 47
<p>Box with three options to insert the rational function <span class="html-italic">r</span>.</p> Full article ">Figure 48
<p>Part of the code structure of the [ASPPlusPMinus] algorithm that integrates the [APoles] algorithm and uses <span class="html-italic">Mathematica</span>’s <span class="html-italic">Solve</span> command.</p> Full article ">Figure 49
<p>Part of the structure of the [ASPPlusPMinus] algorithm corresponding to <span class="html-italic">Option 2</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 50
<p><span class="html-italic">Output</span> given by the [ASPPlusPMinus] algorithm.</p> Full article ">Figure 51
<p>Part of the structure of the [ASPPlusPMinus] algorithm corresponding to <span class="html-italic">Option 1</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 52
<p><span class="html-italic">Output</span> given by the [ASPPlusPMinus] algorithm.</p> Full article ">Figure 53
<p>Part of the structure of the [ASPPlusPMinus] algorithm corresponding to <span class="html-italic">Option 1</span> to input <span class="html-italic">r</span>.</p> Full article ">Figure 54
<p><span class="html-italic">Output</span> given by the [ASPPlusPMinus] algorithm.</p> Full article ">Figure 55
<p>Flowchart of the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm.</p> Full article ">Figure 56
<p>Part of the code structure of the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm responsible for validating <math display="inline"><semantics> <msub> <mi>x</mi> <mo>+</mo> </msub> </semantics></math>.</p> Full article ">Figure 57
<p>Part of the code structure of the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm responsible for the input options.</p> Full article ">Figure 58
<p>Part of the structure of the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm responsible for the input options.</p> Full article ">Figure 59
<p><span class="html-italic">Output</span> given by the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm.</p> Full article ">Figure 60
<p>Part of the structure of the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm responsible for the input options.</p> Full article ">Figure 61
<p><span class="html-italic">Output</span> given by the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm.</p> Full article ">Figure 62
<p>Part of the structure of the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm responsible for the input options.</p> Full article ">Figure 63
<p><span class="html-italic">Output</span> given by the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm.</p> Full article ">Figure 64
<p>Part of the structure of the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm responsible for the input options.</p> Full article ">Figure 65
<p><span class="html-italic">Output</span> given by the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm.</p> Full article ">Figure 66
<p>Part of the structure of the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm responsible for the input options.</p> Full article ">Figure 67
<p>Piece of code removed from [SInt] algorithm.</p> Full article ">Figure 68
<p><span class="html-italic">Output</span> given by the [SInt]<math display="inline"><semantics> <msub> <mrow/> <mrow> <mn>2.0</mn> </mrow> </msub> </semantics></math> algorithm.</p> Full article ">
21 pages, 1318 KiB  
Review
Predictive Maintenance in the Automotive Sector: A Literature Review
by Fabio Arena, Mario Collotta, Liliana Luca, Marianna Ruggieri and Francesco Gaetano Termine
Math. Comput. Appl. 2022, 27(1), 2; https://doi.org/10.3390/mca27010002 - 31 Dec 2021
Cited by 34 | Viewed by 25400
Abstract
With the rapid advancement of sensor and network technology, there has been a notable increase in the availability of condition-monitoring data such as vibration, temperature, pressure, voltage, and other electrical and mechanical parameters. With the introduction of big data, it is possible to [...] Read more.
With the rapid advancement of sensor and network technology, there has been a notable increase in the availability of condition-monitoring data such as vibration, temperature, pressure, voltage, and other electrical and mechanical parameters. With the introduction of big data, it is possible to prevent potential failures and estimate the remaining useful life of the equipment by developing advanced mathematical models and artificial intelligence (AI) techniques. These approaches allow taking maintenance actions quickly and appropriately. In this scenario, this paper presents a systematic literature review of statistical inference approaches, stochastic methods, and AI techniques for predictive maintenance in the automotive sector. It provides a summary on these approaches, their main results, challenges, and opportunities, and it supports new research works for vehicle predictive maintenance. Full article
(This article belongs to the Special Issue Statistical and Stochastic Approaches for Predictive Maintenance in the Context of Industry 4.0)
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Figure 1

Figure 1
<p>Maintenance strategies.</p> Full article ">Figure 2
<p>Comparison of reactive maintenance, preventive maintenance, and predictive maintenance on the cost and frequency of maintenance work.</p> Full article ">Figure 3
<p>Schematic representation of predictive maintenance.</p> Full article ">Figure 4
<p>A schematic representation of an artificial neural network.</p> Full article ">Figure 5
<p>Optimal hyperplane for support vector machine with two classes.</p> Full article ">
19 pages, 2561 KiB  
Article
Dynamic and Interactive Tools to Support Teaching and Learning
by Ana C. Conceição
Math. Comput. Appl. 2022, 27(1), 1; https://doi.org/10.3390/mca27010001 - 23 Dec 2021
Cited by 6 | Viewed by 3847
Abstract
The use of technological learning tools has been increasingly recognized as a useful tool to promote students’ motivation to deal with, and understand, mathematics concepts. Current digital technology allows students to work interactively with a large number and variety of graphics, complementing the [...] Read more.
The use of technological learning tools has been increasingly recognized as a useful tool to promote students’ motivation to deal with, and understand, mathematics concepts. Current digital technology allows students to work interactively with a large number and variety of graphics, complementing the theoretical results and often used paper and pencil calculations. The computer algebra system Mathematica is a very powerful software that allows the implementation of many interactive visual applications. The main goal of this work is to show how some new dynamic and interactive tools, created with Mathematica and available in the Computable Document Format (CDF), can be used as active learning tools to promote better student activity and engagement in the learning process. The CDF format allows anyone with a computer to use them, at no cost, even without an active Wolfram Mathematica license. Besides that, the presented tools are very intuitive to use which makes it suitable for less experienced users. Some tools applicable to several mathematics concepts taught in higher education will be presented. This kind of tools can be used either in a remote or classroom learning environment. The corresponding CDF files are made available as supplement of the online edition of this article. Full article
(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications 2021)
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Figure 1

Figure 1
<p>Part of the code of the tool “Riemann Sums”, used in the examples presented in Figures 7 and 14.</p> Full article ">Figure 2
<p>Part of the code of the tool “Christmas Scene Method for identifying and solving exact differential equations”, used in the examples presented in Figures 8, 16 and 17.</p> Full article ">Figure 3
<p>Part of the code of the “F-Logistic” tool [<a href="#B35-mca-27-00001" class="html-bibr">35</a>] responsible for some checkboxes for the plots’ options and for the parameter values (including the choice of styles and sizes).</p> Full article ">Figure 4
<p>Usage example of the F-Exponential tool.</p> Full article ">Figure 5
<p>Image with a dynamic and interactive tool to solve equations.</p> Full article ">Figure 6
<p>Image with four dynamic and interactive tools to compute integrals of rational functions.</p> Full article ">Figure 7
<p>Example of a midpoint Riemann Sum given by the “Riemann Sum” active leaning tool.</p> Full article ">Figure 8
<p>Some of the possible functions for the function <span class="html-italic">M</span> when using the tool Christmas Scene Method for identifying and solving exact differential equations.</p> Full article ">Figure 9
<p>Example of an exercise proposed to encourage discussion, dialogue and reflection, available in the Computable Document Format: What kind of functions will appear if we change dynamically the parameter <span class="html-italic">A</span>?</p> Full article ">Figure 10
<p>Example of a problem that can be projected and asked to be resolved by a student, in a remote or in a classroom learning environment, concerning the invertibility concept.</p> Full article ">Figure 11
<p>A student’s response to a question about invertibility, raised in a classroom learning environment (during the global pandemic caused by the coronavirus SARS-CoV-2).</p> Full article ">Figure 12
<p>Solution of the exercise displayed on <a href="#mca-27-00001-f010" class="html-fig">Figure 10</a> through the technological F-Logistic tool.</p> Full article ">Figure 13
<p>Video on the invertibility concept produced by the author (using the F-Exponential tool [<a href="#B26-mca-27-00001" class="html-bibr">26</a>]) and available in the learning management system moodle of the class.</p> Full article ">Figure 14
<p>Example of three possible short-problems proposed to be solved individually, concerning the Riemann Sum concept.</p> Full article ">Figure 15
<p>Image that illustrate a challenging problem that can be solved as autonomous students’ work.</p> Full article ">Figure 16
<p>Image that illustrate an activity to be done in autonomous students’ work. The student must justify why it is an exact ODE and confirm all the computed integrals.</p> Full article ">Figure 17
<p>Image that illustrate an activity to be done in autonomous students’ work. The student must justify why it is not an exact ODE.</p> Full article ">Figure 18
<p>Image that illustrate part of an evaluation task: graphical representation of the inverse function whose function is drawn and analytical determination of that function.</p> Full article ">Figure 19
<p>Image that illustrate part of another evaluation task: graphical representation of the inverse function whose function is drawn and analytical determination of that function.</p> Full article ">Figure 20
<p>Image that illustrate the solution of the evaluation task that can be associated to <a href="#mca-27-00001-f018" class="html-fig">Figure 18</a>.</p> Full article ">Figure 21
<p>Image that illustrate the solution of the evaluation task that can be associated to <a href="#mca-27-00001-f019" class="html-fig">Figure 19</a>.</p> Full article ">
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