A detailed study on 2D Volterra–Fredholm integro-differential equations involving the Caputo fractional derivative

Abstract

In the present research, the two-dimensional Volterra–Fredholm integro-differential (2D-VFID) equations of fractional order are studied through utilizing a new scheme based on the two-dimensional normalized Müntz–Legendre​ polynomials (2D-NMLPs). An important advantage of the suggested basis over other bases is that the approximate solution can be written in terms of fractional or integer powers because of the parameter α. First, operational matrices of Riemann–Liouville fractional (RLF) integral and Caputo fractional (CF) derivative based on the 2D-NMLPs are extracted; then, based on the resulting operational matrices, the approximate solution of 2D-VFID equations of fractional order is constructed by solving the system of linear/nonlinear algebraic equations. The stability, error bound, and convergence of the scheme are discussed in a detailed manner and to illustrate the accuracy of the scheme in handling the 2D-VFID equations of fractional order, some examples are presented, confirming the good performance of the scheme.