Search Results (110)

This study presents a mathematical model of a three-dimensional thermoelastic half-space with variable thermal conductivity under the definition of fractional order heat conduction based on the Moor–Gibson–Thompson theorem. The non-dimensional governing equations using Laplace and double Fourier transform methods have been applied to a three-dimensional thermoelastic, isotropic, and homogeneous half-space exposed to a rectangular thermal loading pulse with a traction-free surface. The double Fourier transforms and Laplace transform inversions have been computed numerically. The numerical distributions of temperature increment, invariant stress, and invariant strain have been shown and analysed. The fractional order parameter and the variability of thermal conductivity significantly influence all examined functions and the behaviours of the thermomechanical waves. Classifying thermal conductivity as weak, normal, and strong is crucial and closely corresponds to the actual behaviour of the thermal conductivity of thermoelastic materials. Full article

Long, narrow, deep excavations commonly encountered in practice, such as those for subway stations, require effective groundwater management to prevent disasters in water-rich areas. To achieve this, a cut-off curtain and pumping well are typically employed during long, deep foundation pit dewatering. The unsteady groundwater flow behavior in the confined aquifer must consider the influence of the cut-off curtain during dewatering. This paper establishes a two-dimensional analytical model to describe transient groundwater flow in a confined aquifer with a cut-off curtain. Both the dewatering well pumped at a steady discharge inside the pit and the cut-off curtain are partially penetrating in the anisotropic confined aquifer. With the help of the Laplace and Fourier cosine transformations, the semi-analytical drawdown solution for the model is derived and validated against numerical solution and unsteady pumping test data. It is shown that the inserted cut-off curtain depth and the structural parameters of the pumping well significantly affect the drawdown inside the pit. Sensitivity analysis reveals that, regardless of whether the observation is made inside or outside the curtain, the drawdown is very sensitive to the change in pumping rate, aquifer thickness, storage coefficient, and horizontal hydraulic conductivity. Additionally, drawdown near the cut-off curtain outside the pit is sensitive to the vertical hydraulic conductivity of the aquifer, the width of the pit, and the interception depth of the cut-off curtain, while drawdown far from the curtain outside the pit is not sensitive to the location and length of the well screen. Full article

The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space ${\mathcal{D}}^{\prime }$ while their Fourier transforms belong to ${\mathcal{Z}}^{\prime }.$ Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved. Full article

This study presents an analytical model for two-dimensional pollutant transport within a three-layer composite liner system, which comprises a geomembrane (GM), a geosynthetic clay liner (GCL), and a soil liner (SL), with particular attention to defects in the geomembrane. The model integrates key processes such as convection, diffusion, adsorption, and degradation, offering a more accurate prediction of pollutant behavior. Through Laplace and Fourier transforms, pollutant concentration distributions are derived, providing a comprehensive view of pollutant migration in landfill settings. Verification against COMSOL 6.0 simulations underscores the model’s robustness. Results show that there is an optimal thickness for the SL that balances the effectiveness of pollutant containment and material usage, while higher diffusion coefficients and advection velocity accelerate migration. The degradation of organic pollutants reduces concentrations over time, especially with shorter half-lives. These findings not only improve the design of landfill liners but also support more sustainable waste management practices by reducing the risk of environmental contamination. This research contributes to the development of more effective, long-lasting landfill containment systems, enhancing sustainability in waste management infrastructure. Full article

In this paper, we study the Caputo–Hadamard time-space fractional diffusion equation, where the Caputo derivative is defined in the temporal direction and the Hadamard derivative is defined in the spatial direction separately. We first use the Laplace transform and the modified Fourier transform to study the analytical solution of the Cauchy problem. Then, using the Galerkin finite element method in space, we generate a semi-discrete scheme and study the convergence analysis. Furthermore, using the L1 scheme of the Caputo derivative in time, we construct a fully discrete scheme and then discuss the stability and error estimation in detail. Finally, the numerical experiments are displaced to verify the theoretical results. Full article

This paper presents a detailed investigation into the application of transient Fourier analysis in select electrical engineering contexts. Two novel approaches for addressing transient analysis are introduced. The first approach combines the Fourier series with the Laplace–Carson ($\mathcal{L}\text{-}C$) transform in the complex domain, utilizing complex time vectors to streamline the computation of the original function. The inverse transformation back into the time domain is achieved using the Cauchy-Heaviside ($C\text{-}\mathcal{H}$) method. The second approach applies the Fourier transform ($\mathcal{F}\text{-}{\rm T}$) for the transient analysis of a power converter circuit with both passive and active loads. The method of complex conjugate amplitudes is employed for steady-state analysis. Both contributions represent innovative approaches within this study. The process begins with Fourier series expansions and the computation of Fourier coefficients, followed by solving the system’s steady-state and transient responses. The transient states are then confirmed using the Fourier transform. To validate these findings, the analytical results are verified through simulations conducted in the Matlab/Simulink R2023b environment. Full article

Proteins form the fastest-growing therapeutic class. Due to their intrinsic instability, loss of native structure is common. Structure alteration must be carefully evaluated as structural changes may jeopardize the efficiency and safety of the protein-based drugs. Hydrogen deuterium exchange (HDX) has long been used to evaluate protein structure and dynamics. The rate of exchange constitutes a sensitive marker of the conformational state of the protein and of its stability. It is often monitored by mass spectrometry. Fourier transform infrared (FTIR) spectroscopy is another method with very promising capabilities. Combining protein microarrays with FTIR imaging resulted in high throughput HDX FTIR measurements. BaF₂ slides bearing the protein microarrays were covered by another slide separated by a spacer, allowing us to flush the cell continuously with a flow of N₂ gas saturated with ²H₂O. Exchange occurred simultaneously for all proteins and single images covering ca. 96 spots of proteins that could be recorded on-line at selected time points. Each protein spot contained ca. 5 ng protein, and the entire array covered 2.5 × 2.5 mm². Furthermore, HDX could be monitored in real time, and the experiment was therefore not subject to back-exchange problems. Analysis of HDX curves by inverse Laplace transform and by fitting exponential curves indicated that quantitative comparison of the samples is feasible. The paper also demonstrates how the whole process of analysis can be automatized to yield fast analyses. Full article

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solutions, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively. Full article

Gaussian processes have gained popularity in contemporary solutions for mathematical modeling problems, particularly in cases involving complex and challenging-to-model scenarios or instances with a general lack of data. Therefore, they often serve as generative models for data, for example, in classification problems. However, a common problem in the application of Gaussian processes is their computational complexity. To address this challenge, sparse methods are frequently employed, involving a reduction in the computational domain. In this study, we propose an innovative computational approach for Gaussian processes. Our method revolves around selecting a computation domain based on Chebyshev nodes, with the optimal number of nodes determined by minimizing the degree of the Chebyshev series, while ensuring meaningful coefficients derived from function values at the Chebyshev nodes with fast Fourier transform. This approach not only facilitates a reduction in computation time but also provides a means to reconstruct the original function using the functional series. We conducted experiments using two computational methods for Gaussian processes: Markov chain Monte Carlo and integrated nested Laplace approximation. The results demonstrate a significant reduction in computation time, thereby motivating further development of the proposed algorithm. Full article

This paper explores the thermal behavior of multiple interface cracks situated between a half-plane and a thermal coating layer when subjected to transient thermal loading. The temperature distribution is analyzed using the hyperbolic heat conduction theory. In this model, cracks are represented as arrays of thermal dislocations, with densities calculated via Fourier and Laplace transformations. The methodology involves determining the temperature gradient within the uncracked region, and these calculations contribute to formulating a singular integral equation specific to the crack problem. This equation is subsequently utilized to ascertain the dislocation densities at the crack surface, which facilitates the estimation of temperature gradient intensity factors for the interface cracks experiencing transient thermal loading. This paper further explores how the relaxation time, loading parameters, and crack dimensions impact the temperature gradient intensity factors. The results can be used in fracture analysis of structures operating at high temperatures and can also assist in the selection and design of coating materials for specific applications, to minimize the damage caused by temperature loading. Full article

In the context of strong electromagnetic interference environments, the measurement accuracy of the capacitance parameters of transmission lines under power frequency measurement methods is not high. In this paper, a capacitance parameter anti-interference measurement method for transmission lines based on harmonic components is proposed to overcome the impact of power frequency interference. When applying this method, it is first necessary to open-circuit the end of the line under test. Subsequently, apply voltage to the head end of the tested line through a step-up transformer. Due to the saturation of the transformer during no-load conditions, a large number of harmonics are generated, primarily third harmonic. The third harmonic components of voltage and current on the tested transmission line are extracted using the Fourier transform. The proposed method addresses the influence of line distribution effects by establishing a distributed parameter model for long-distance transmission lines. The relevant transmission matrix for the zero-sequence distributed parameters is obtained by combining Laplace transform and similarity transform to solve the transmission line equations. Using synchronous measurement data from the third harmonic components of voltage and current at both ends of the transmission line, combined with the transmission matrix, this method accurately measures the zero-sequence capacitance parameters. The PSCAD/EMTDC simulation results and field test outcomes have demonstrated the feasibility and accuracy of the proposed method for measuring line capacitance parameters under strong electromagnetic interference. Full article

The infinite series solution to the boundary-value problems of Laplace’s equation with discontinuous Dirichlet boundary conditions was found by using the basic method of separation of variables. The merit of this paper is that the closed-form solution, or the singular similarity solution in the semi-infinite strip domain and the first-quadrant domain, can be generated from the basic infinite series solution in the rectangular domain. Moreover, based on the superposition principle, the infinite series solution in the rectangular domain can be related to the singular similarity solution in the semi-infinite strip domain. It is proven that the analytical source-type singular behavior in the infinite series solution near certain singular points in the rectangular domain can be revealed from the singular similarity solution in the semi-infinite strip domain. By extending the boundary of the rectangular domain, the infinite series solution to Laplace’s equation in the first-quadrant domain can be derived to obtain the analytical singular similarity solution in a direct and much easier way than by using the methods of Fourier transform, images, and conformal mapping. Full article

The central-symmetric time-fractional heat conduction equation with heat absorption is investigated in a solid with a spherical hole under time-harmonic heat flux at the boundary. The problem is solved using the auxiliary function, for which the Robin-type boundary condition with a prescribed value of a linear combination of a function and its normal derivative is fulfilled. The Laplace and Fourier sine–cosine integral transformations are employed. Graphical representations of numerical simulation results are given for typical values of the parameters. Full article

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases. Full article

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform. Full article