Two-dimensional shifted Legendre polynomials operational matrix method for solving the two-dimensional integral equations of fractional order

Operational matrices based on hybrid functions for solving general nonlinear two-dimensional fractional integro-differential equations

“…Also, the proposed method was evaluated by solving seven provided numerical examples. The numerical results obtained by the proposed method were more accurate than the numerical results obtained by Hesameddini and Shahbazi (2018); Ezzati (2016, 2017). The obtained results introduce the presented method as a powerful mathematical tool for solving general nonlinear two-dimensional fractional integro-differential equations with lower numbers of bases than the other methods.…”

“…In Tables 5 and 6, respectively, we report the exact and approximate solutions and also the absolute errors for N = 2, M = 2, 3 at some selected nodes. These tables state that usingn = N 2 M 2 = 36 numbers of bases, we obtain more accurate results than the 2D-SLPOM and 2D-BPFs methods reported by Hesameddini and Shahbazi (2018); Najafalizadeh and Ezzati (2016), respectively, that usedn = (N + 1) 2 = 129 2 = 16641 2D-SLPOM andn = m 2 = 128 2 = 16384 2D-BPFs to solve this problem. Figures 5 and 6 illustrate the accuracy and efficiency of the presented method.…”

“…Example 3 Consider the following two-dimensional fractional Fredholm integral equation studied by Hesameddini and Shahbazi (2018); Najafalizadeh and Ezzati (2016):…”

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:…”

Optimal Solution of the Fractional Order Breast Cancer Competition Model

“…Sun et al [43] came with an algorithm based on improved Legendre orthonormal basis for solving second-order Boundary value problems. Hesameddini and Shahbazi [44] approximated the unknown functions based on the two-dimensional shifted Legendre polynomials operational matrix method for the numerical solution of twodimensional fractional integral equations. Guorong et al [45] proposed the Legendre orthogonal polynomials method to calculate the acoustic reflection and transmission coefficients at liquid/solid interfaces.…”