Search Results (127)

The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. To make PGM application easier, explicit formulas for the inner product between these basis functions and their derivatives with second-kind Chebyshev polynomials are derived. This leads to a simplified system of algebraic equations with a recognizable pattern that facilitates effective inversion to produce an approximate spectral solution. Presentations are made regarding the method’s convergence analysis and the computational cost of matrix inversion. The efficiency of the method described in precisely solving the Euler–Bernoulli beam equation under different scenarios has been validated by numerical testing. Additionally, the procedure proposed in this paper is more effective compared to other existing techniques. Full article

This research work introduces a connection of adjoint Bernoulli’s polynomials and a gamma function as a sequence of linear positive operators. Further, the convergence properties of these sequences of operators are investigated in various functional spaces with the aid of the Korovkin theorem, Voronovskaja-type theorem, first order of modulus of continuity, second order of modulus of continuity, Peetre’s K-functional, Lipschitz condition, etc. In the last section, we extend our research to a bivariate case of these sequences of operators, and their uniform rate of approximation and order of approximation are investigated in different functional spaces. Moreover, we construct a numerical example to demonstrate the applicability of our results. Full article

The aim of this paper is to derive formulae for the generating functions of the Bernstein type polynomials. We give a PDE equation for this generating function. By using this equation, we give recurrence relations for the Bernstein polynomials. Using generating functions, we also derive some identities including a symmetry property for the Bernstein type polynomials. We give some relations among the Bernstein type polynomials, Bernoulli numbers, Stirling numbers, Dahee numbers, the Legendre polynomials, and the coefficients of the classical superoscillatory function associated with the weak measurements. We introduce some integral formulae for these polynomials. By using these integral formulae, we derive some new combinatorial sums involving the Bernoulli numbers and the combinatorial numbers. Moreover, we define Bezier type curves in terms of these polynomials. Full article

This paper introduces the operational rule for 2-iterated 2D Appell polynomials and derives its generalized form using fractional operators. It also presents the generating relation and explicit forms that characterize the generalized 2-iterated 2D Appell polynomials. Additionally, it establishes the monomiality principle for these polynomials and obtains their recurrence relations. The paper also establishes corresponding results for the generalized 2-iterated 2D Bernoulli, 2-iterated 2D Euler, and 2-iterated 2D Genocchi polynomials. Full article

This research aims to introduce and examine a new type of polynomial called the ${\Delta }_h$ Legendre–Appell polynomials. We use the monomiality principle and operational rules to define the ${\Delta }_h$ Legendre–Appell polynomials and explore their properties. We derive the generating function and recurrence relations for these polynomials and their explicit formulas, recurrence relations, and summation formulas. We also verify the monomiality principle for these polynomials and express them in determinant form. Additionally, we establish similar results for the ${\Delta }_h$ Legendre–Bernoulli, Euler, and Genocchi polynomials. Full article

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind. Full article

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials. Full article

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order $\alpha$ and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program. Full article

Digital twins are an emerging technology that can be harnessed for the digitalization of the industry. Steel industry systems contain a large number of electro-hydraulic components as proportional valves. An input–output model for a water proportional cartridge valve was derived from physical modeling based on fluid mechanics, dynamics, and electrical principles. The valve is a two-stage valve with two two/two-way water proportional valves as the pilot stage and a marginally stable poppet-type cartridge valve as the main valve. To our knowledge, this is the first time that an input–output model was derived for a two-stage proportional cartridge valve with a marginally stable main valve. The orifice equation, which is based on Bernoulli principles, was approximated by a polynomial, which made the parameter estimation easier and modeling possible without measuring the pressure of the varying control volume, in contrast with previous studies of similar types of valves situated in the pilot stage part of the valve. This work complements previous studies of similar types of valves in two ways: (1) data were collected when the valve was operating in a closed loop and (2) data were collected when the valve was part of a press mill machine in a steel manufacturing plant. Model parameters were identified from data from these operating conditions. The parameters of the input–output model were estimated by convex optimization with physical constraints to overcome the problems caused by poor system excitation. For comparison, a simple linear model was derived and the least squares method was used for the parameter estimation. A thorough estimation of the parameters’ relative errors is presented. The model contains five parameters related to the design parameters of the valve. The modeled position output was in good agreement with experimental data for the training and test data. The model can be used for the real-time monitoring of the valve’s status by the model parameters. One of the model parameters varied linearly with the production cycles. Thus, the aging of the valve can be monitored. Full article

This article is concerned with the Durrmeyer-type generalization of Szász operators, including confluent Appell polynomials and their approximation properties. Also, the rate of convergence of the confluent Durrmeyer operators is found by using the modulus of continuity and Peetre’s $\mathcal{K}$-functional. Then, we show that, under special choices of $A\left(t\right)$, the newly constructed operators reduce confluent Hermite polynomials and confluent Bernoulli polynomials, respectively. Finally, we present a comparison of newly constructed operators with the Durrmeyer-type Szász operators graphically. Full article

We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform. We show some examples of these generalized mathematical entities, which were derived using the computer algebra system Mathematica© (latest v. 14.0). Full article

The objective of this article is to introduce the ${∆}_h$ bivariate Appell polynomials ${}_{{∆}_h}{\mathbb{A}}_s^{\left[r\right]}(\lambda ,\eta ;h)$ and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of ${∆}_h$ bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the ${∆}_h$ bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials ${∆}_h$ are also proved by demonstrating that the ${∆}_h$ bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of ${∆}_h$ bivariate Appell polynomials is provided, and symmetric identities for the ${∆}_h$ bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for ${∆}_h$ bivariate Appell polynomials. Additionally, generating relations for the ${∆}_h$ bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials. Full article

Memristors are devices built on the basis of fourth passive electrical elements in nanosystems. Because of the multitude of technologies used for memristor implementation, it is not always possible to obtain analytical models of memristors. This difficulty can be overcome using behavioral modeling, which is when mathematical models are constructed according to the input–output relationships on the input and output signals. For memristor modeling, piecewise neural and polynomial models with split signals are proposed. At harmonic input signals of memristors, this study suggests that split signals should be formed using a delay line. This method produces the minimum number of split signals and, as a result, simplifies behavioral models. Simplicity helps reduce the dimension of the nonlinear approximation problem solved in behavioral modeling. Based on the proposed method, the piecewise neural and polynomial models with harmonic input signals were constructed to approximate the transfer characteristic of the memristor, in which the current dynamics are described using the Bernoulli differential equation. It is shown that the piecewise neural model based on the feedforward network ensures higher modeling accuracy at almost the same complexity as the piecewise polynomial model. Full article

The purpose of this article is to give relations among the uniform B-splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the uniform B-splines and generating functions for the Bernstein basis functions. We derive some functional equations for these generating functions. Using the higher-order partial derivative equations of these generating functions, we derive both the generalized de Boor recursion relation and the higher-order derivative formula of uniform B-splines in terms of Bernstein basis functions. Using the functional equations of these generating functions, we derive the relations among the Bernstein basis functions, the uniform B-splines, the Apostol-Bernoulli numbers and polynomials, the Aposto–Euler numbers and polynomials, the Eulerian numbers and polynomials, and the Stirling numbers. Applying the p-adic integrals to these polynomials, we derive many novel formulas. Furthermore, by applying the Laplace transformation to these generating functions, we derive infinite series representations for the uniform B-splines and the Bernstein basis functions. Full article

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed. Full article