Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

Abstract

In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h3). The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.

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:…”

On an approximate solution of a boundary value problem for a nonlinear integro-differential equation

“…Because we cannot determine the exact solutions for most of these equations, "many numerical" or "semi-analytic" techniques are developed to overcome this gap. From these methods, there are the Adomian decomposition method (ADM), homotopy perturbation method (HPM), variational iteration method (VIM), homotopy analysis method (HAM) and many other methods, see Abdou, Soliman, and Abdel-Aty (2020), Hamoud and Ghadle (2018), He (2020aHe ( , 2020b, Mirzaee and Alipour (2019), Rezabeyk, Abbasbandy, and Shivanian (2020), Saeedi, Tari, and Babolian (2020) among others. These techniques can be applied to find approximate solutions for a large class of linear and nonlinear integral equations and many functional equations as well.…”

Time–space Jacobi pseudospectral simulation of multidimensional Schrödinger equation

“…Trofimov and Peskov [40] have also solved the GPE using a conservative finite difference method. Moreover, multidimensional NSE was solved by several other numerical methods, such as riccati expansion method [1], finite difference method [38, 44], finite element method [10] Galerkin finite element method [43], momentum representation method [9], symplectic and multisymplectic methods [2, 39], compact scheme [18], split‐step Fourier scheme [32], compact boundary value method [31], spectral method [3, 28, 29, 30], operational matrix method [23, 27, 37], discrete collocation method based [26], and spline collocation method [22].…”