Numerical scheme for solving singular fractional partial integro‐differential equation via orthonormal Bernoulli polynomials

Abstract

In this paper, an efficient matrix method based on 2D orthonormal Bernoulli polynomials are developed to obtain numerical solution of weakly singular fractional partial integro‐differential equations (FPIDEs). Operational matrix of integration, almost operational matrix of integration, and product operational matrix are employed to transform the solution of Volterra singular FPIDEs to the system of linear or nonlinear algebraic equations. The obtained algebraic system can be easily solved by using an appropriate numerical method. Also, some useful theorems concerning the convergence and error analysis associated to the mentioned approach are proven in the current paper. Finally, some numerical examples are included to illustrate the accuracy, efficiency, and applicability of the proposed approach.

Piecewise barycentric interpolating functions for the numerical solution of Volterra integro‐differential equations

“…18 In most cases, it is difficult or impossible to obtain an analytical solution for the integral equation (1). Therefore, in recent decades, there has been much attention to constructing some numerical techniques to achieve the approximate solutions with high accuracy for the VIDEs such as the homotopy perturbation method, 19 variational iteration method (VIM), 20,21 operational matrix method based on orthogonal polynomials 1,[22][23][24][25][26][27] operational tau method, 28,29 collocation approach, [30][31][32] block-pulse functions method, 33,34 radial basis function method, 3,35 wavelet method, 36,37 hat functions method, 38 least squares method, 10,39,40 hybrid functions method, [41][42][43][44] fixed point method, 45 successive approximations method, [46][47][48] Euler polynomials, 49,50 delta functions, 51 Taylor method, 52 and so on.…”

A numerical solution of an inverse diffusion problem based on operational matrices of orthonormal polynomials

“…In previous studies, 60,62 the authors applied the Jacobi spectral method for solving the time‐fractional differential equation and approximated the heat temperature distribution and the surface heat flux histories. A great number of papers concerning the application of the operational matrix methods for solving the system of nonlinear stochastic Itô–Volterra integral equations, stochastic differential equations, stochastic quadratic equations, fractional partial integro‐differential equations, complex differential equations, and the system of nonlinear ordinary differential equation was proposed in the literature 47–53,57,58,65 . In previous studies, 45,46 the application of operational matrices corresponding the hybrid and triangular functions was presented for solving nonlinear and fractional integral equations.…”

A new and efficient numerical method based on shifted fractional‐order Jacobi operational matrices for solving some classes of two‐dimensional nonlinear fractional integral equations

“…Therefore, finding efficient numerical methods to approximate the solutions of these equations has become the main objective of many mathematicians. Some of these methods include Legendre wavelets [17], higher-order finite element method [18], generalized differential transform method [27], shifted Legendre polynomials [16,21,25], hybrid of block-pulse functions and shifted Legendre polynomials operational matrix method [31], Müntz-Legendre wavelets [32], fractional-order orthogonal Bernstein polynomials [38], delta functions operational matrix method [39], hybrid of block-pulse and parabolic functions [37], hat functions [35,40], two-dimensional orthonormal Bernstein polynomials [41][42][43], two-dimensional block-pulse operational matrix method [44], homotopy analysis method [47], Haar wavelet [4,49], orthonormal Bernoulli polynomials [52], shifted Jacobi polynomials [20,54,56], Bernstein polynomials [30,55], the second kind Chebyshev wavelets [51], etc. In this research study, some classes of two-dimensional nonlinear fractional integral equations of the second kind are considered in the following forms:…”