Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations

<abstract><p>This paper addresses a new spectral collocation method for solving nonlinear fractional quadratic integral equations. The main idea of this method is to construct the approximate solution based on fractional order Chelyshkov polynomials (FCHPs). To this end, first, we introduce these polynomials and express some of their properties. The operational matrices of fractional integral and product are derived. The spectral collocation method is utilized together with operational matrices to reduce the problem to a system of algebraic equations. Finally, by solving this system, the unknown coefficients are computed. Further, the convergence analysis and numerical stability of the method are investigated. The proposed method is computationally simple and easy to implement in computer programming. The accuracy and applicability of the method is presented by some numerical examples.</p></abstract>

Purpose The objectives of this study are threefold: (1) to introduce the tempered Chebyshev wavelet (TCW), (2) to propose the TCW method for solving linear tempered Caputo fractional diffusion-type equations and (3) to develop a fast tempered Chebyshev wavelet (fTCW) method for solving one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations. Design/methodology/approach The fTCW method integrates the TCW method with L2−1σ and sum-of-exponentials approximations of the tempered Caputo fractional time derivative. For this purpose, we introduce the TCW and derive its new operational matrices for tempered fractional integration by using the hypergeometric function. For nonlinear problems, we employ the interpolation technique in conjunction with operational matrices and fast approximations. The efficiency of the fTCW method is demonstrated through comparisons with the exact solution and the solution obtained using the TCW method. Findings We have derived the TCW operational matrix of tempered fractional integration and the TCW operational matrix of tempered fractional integration for boundary value problems. These matrices, in conjunction with the fast approximations of the tempered Caputo fractional time derivative and the interpolation technique, form the basis for the construction of the fTCW method. We present a detailed methodology for solving linear tempered Caputo fractional equations using the TCW method. Additionally, we provide a comprehensive methodology for solving both one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations using the fTCW method. The convergence and error analysis of both methods are thoroughly discussed. Numerical simulations are presented to validate and illustrate the theoretical results. These simulations demonstrate the effectiveness and accuracy of the proposed methods by solving three test problems, including one linear and two nonlinear tempered Caputo fractional diffusion-type equations. The results are compared with analytical solutions or with each other to highlight the efficiency and precision of the TCW and fTCW methods. Originality/value Many engineers and scientists can utilize the presented methods for solving their linear and nonlinear tempered Caputo fractional diffusion-type models.

The main aim of this paper is to use the operational matrices of fractional integration of Haar wavelets to find the numerical solution for a nonlinear system of two-dimensional fractional partial Volterra integral equations. To do this, first we present the operational matrices of fractional integration of Haar wavelets. Then we apply these matrices to solve systems of two-dimensional fractional partial Volterra integral equations (2DFPVIE). Also, we present the error analysis and convergence as well. At the end, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

This essay focuses on studying the nonlinear fractional integral equation. Various methods, including Akbari-Ganji's Method (AGM), Homotopy Perturbation Method (HPM), and Vibrational Iteration Method (VIM), are utilized to obtain its solution. We introduce an innovative approach to obtain rough approximations for fractional differential equations. These equations play a significant role in the field of fluid dynamics and find widespread application. In this article, we have used analytical methods to check the correctness of the answers. Maple mathematical software is used to solve all fractional equations. Ordinary equations and fractional differential equations have connections to entropy, wavelets, and other related concepts. To demonstrate the method, a few examples are employed, chosen for their accuracy and simplicity of implementation. The solutions are explained using convergent series. According to the calculations performed with analytical methods on the fractional integral equations of the nonlinear oscillator, Due to the swing movements of the oscillator, as the swing movement's increase, the velocity gradient becomes an upward trend.

The operational matrices of fractional‐order integration for the Legendre and Chebyshev wavelets are derived. Block pulse functions and collocation method are employed to derive a general procedure for forming these matrices for both the Legendre and the Chebyshev wavelets. Then numerical methods based on wavelet expansion and these operational matrices are proposed. In this proposed method, by a change of variables, the multiorder fractional differential equations (MOFDEs) with nonhomogeneous initial conditions are transformed to the MOFDEs with homogeneous initial conditions to obtain suitable numerical solution of these problems. Numerical examples are provided to demonstrate the applicability and simplicity of the numerical scheme based on the Legendre and Chebyshev wavelets.

Modified Chebyshev wavelet methods for fractional delay-type equations

In this paper, we examined a wide class of the variable order fractional problems such as linear and nonlin-ear fractional variable order differential equations, variable order fractional functional boundary value problems, variable order fractional pantograph differential equations. The proposed method is a collocation method based on the Bessel polynomials and the operational matrix of derivatives, which transformed equations into a system of non-linear algebraic equations to achieve the approximate solution. By using Caputo fractional derivative, the operational matrix of the variable-order fractional derivatives is constructed. The error analysis shows that the method is convergent. Several numerical results confirm the accuracy and efficiency of the proposed method. Keywords: Bessel collocation method, Variable-order fractional operational matrix, Variable order fractional differential equations, Variable order fractional functional boundary value problems, Variable order fractional panto-graph equations.

In this article, the Bielecki metric on the space is used to analyze the different types of stability results of nonlinear fractional integral equation with delay in its corresponding fractional boundary value problems. Sufficient conditions are obtained to prove stability results for fractional nonlinear Volterra and Fredholm integral equations with delay, given by Ulam, Hyer, and Rassias. Further, those stability results are extended to the fractional integral equations where the domain of integration is an unbounded interval. Two numerical examples are provided to assert the obtained stability results.

PurposeThis paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations.Design/methodology/approachBernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed.FindingsTwo illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method.Originality/valueTo the best of the authors’ knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method.

The article proposes a generalized polynomial projection method for solving linear integral equations with a fractional Riemann - Liouville integral in the main part. Fractional-integral equations have numerous applications in various applied problems, in particular, in problems of plasma diagnostics, optical and X-ray diffraction, in ultrasonic measurements in moving media, in metallurgy, biology, microscopy, seismology, in astrophysics and other fields of science.The equation we are considering is an equation of the first kind and therefore, generally speaking, refers to incorrectly set by Hadamard equations. The latter circumstance is connected with the fact that in known function spaces the fractional integral Riemann-Liouville operator is completely continuous. All this imposes its own characteristics on the construction and study of approximate methods for solving integral equations with a fractional integral operator in the main part. It should be noted that the operator of the projection method is not necessarily projective, which allows the construction of computational schemes using the methods of summation of Fourier series and interpolation polynomials. A theoretical and functional substantiation of the proposed projection method was carried out, based on the correct formulation of the equation in a pair of specially selected different Holder spaces. In particular, it proved the unique solvability of the system of linear algebraic equations of the method and the convergence of the constructed approximations to the exact solution in the norm of the space of Holder functions and, as a consequence, in the uniform metric. This also implies the substantiation of specific well-known projection methods such as the Galerkin methods, colocations, and subdomains.

Purpose The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically. Design/methodology/approach For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time. Findings By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated. Originality/value Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

In this paper, we investigate the semi‐Markovian random walk processes with jumps and delaying screen in zero. The Laplace transform on time, Laplace‐ Stieltjes transform on phase of the conditional distribution of semi‐Markovian random walk processes with jumps is found. We get a mathematical modeling of the semi‐Markov random walk processes with a delaying screen in zero, given in general form by means of integral equation. In this paper, the residence time of the system is given by the gamma distribution with the parameters α and β resulting in a fractional order integral equation. The purpose of this paper is to reduce the fractional order integral equation to a fractional order differential equation. Finally, we find the exact solution of fractional order differential equation.

In this article, we have study a 2D Legendre and Chebyshev wavelets collocation scheme for solving a class of linear and nonlinear weakly singular Volterra partial integro-differential equations (PIDEs). The scheme is based on wavelets collocation for PIDEs with uniquely designed matrices over the Hilbert space defined on the domain $$\left( [0, 1]\times [0, 1]\right) $$ . Using piecewise approximation associated with 2D Legendre wavelet, 2D Chebyshev wavelet and its operational matrices, the considered PIDEs will be reduced into the corresponding system of linear and nonlinear algebraic equations. The corresponding linear and nonlinear system of equations solved by collocation scheme and well-known Newton–Raphson scheme at collocation points respectively. In addition, the convergence and error analysis of the numerical scheme is provided under several mild conditions. The numerical results are correlated with the exact solutions and the execution of the proposed scheme is determined by estimating the maximum absolute errors, $$l_{2}\hbox {-}norm$$ errors and $$l_{\infty }\hbox {-}norm$$ errors. The numerical result shows that the scheme is simply applicable, efficient, powerful and very precisely at small number of basis function. The main important applications of the proposed wavelets collocation scheme is that it can be applied on linear as well as nonlinear problems and can be applied on higher order partial differential equations too.

Fractional-order Legendre–Laguerre functions and their applications in fractional partial differential equations

Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration