Abstract
In this paper, we use hybrid parabolic and block-pulse functions (2D-PBPFs) to provide an approximate solution of nonlinear partial mixed Volterra–Fredholm integro-differential equations of fractional order. To reach this goal, we present the Volterra integral operational matrix, operational matrix of fractional integral and operational matrix of mixed Volterra–Fredholm integral by 2D-PBPFs. Using the proposed method, nonlinear partial mixed Volterra–Fredholm integro-differential equations of fractional order become into a nonlinear system of algebraic equations. Moreover, we provide some theorems for convergence analysis and we demonstrate that the convergence order of the suggested approximate approach is $$O(h^{3})$$. Finally, we solve two numerical examples to prove the accuracy of the proposed method.
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