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

Abstract

In the present paper, an attempt was made to develop a numerical method for solving a general form of two-dimensional nonlinear fractional integro-differential equations using operational matrices. Our approach is based on the hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials. Error bound and convergence analysis of the proposed method are discussed. We prove that the order of convergence of our method is O(1 2 2M−1 N M M!). The presented method is tested by seven test problems to demonstrate the accuracy and computational efficiency of the proposed method and to compare our results with other well-known methods. The comparison highlighted that the proposed method exhibits superior performance than the existing methods, even using a few numbers of bases.

A new operational vector approach for time‐fractional subdiffusion equations of distributed order based on hybrid functions

“…Proof From Maleknejad et al 59 (see Theorem 6, page 16), we can write $$$$ {\left\Vert \frac{\partial \mathcal{V}}{\partial t}-{\left(\frac{\partial \mathcal{V}}{\partial t}\right)}_{k,M}\right\Vert}_2\le \frac{\lambda_3\sqrt{t_f}}{2^{\left(k-1\right)M}M! }, $$$$ $$$$ {\left\Vert \frac{\partial^3\mathcal{V}}{\partial {x}^2\partial t}-{\left(\frac{\partial^3\mathcal{V}}{\partial {x}^2\partial t}\right)}_{k,M}\right\Vert}_2\le \frac{\lambda_4\sqrt{t_f}}{2^{\left(k-1\right)M}M!$$…”

“…While several numerical techniques have been proposed to solve some different problems (see, for instance, earlier researches [52][53][54][55][56][57][58][59][60][61][62] and references therein), there have been few research studies that developed numerical methods to solve the NTFSDEs of distributed order (see earlier studies, 39,63,64 ).…”

“…Proof Let $$$$ {}_0^C{D}_t^{\mu}\mathcal{V}=\frac{\partial^{\mu}\mathcal{V}}{\partial {t}^{\mu }}. $$$$ From Maleknejad et al 59 (see Theorem 6, page 16), we can write $$$$ {\left\Vert \frac{\partial^{\mu}\mathcal{V}}{\partial {t}^{\mu }}-{\left(\frac{\partial^{\mu}\mathcal{V}}{\partial {t}^{\mu }}\right)}_{k,M}\right\Vert}_2\le \frac{\lambda_1\sqrt{t_f}}{2^{\left(k-1\right)M}M! },\kern0.3em \operatorname{}\kern0.80em \operatorname{}\operatorname{}\mu \in \left(0,1\right), $$$$ $$$$ {\left\Vert \frac{\partial^2\mathcal{V}}{\partial {x}^2}-{\left(\frac{\partial^2\mathcal{V}}{\partial {x}^2}\right)}_{k,M}\right\Vert}_2\le \frac{\lambda_2\sqrt{t_f}}{2^{\left(k-1\right)M}M!}.$$…”

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:…”

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

“…Therefore, the development of effective and easy‐to‐use numerical schemes for solving such equations acquires an increasing interest. While several numerical techniques have been proposed to solve many different problems (see, for instance [22–49], and references therein), there have been few research studies that developed numerical methods to solve DOFDEs (see [50–58]). The development, however, for efficient numerical methods to solve DOFDEs is still an important issue [51].…”