The Solution of the Nonlinear Mixed Partial Integro-differential Equation via Two-Dimensional Hybrid Functions

In the present paper, a new method is introduced for the approximate solution of two-dimensional nonlinear mixed Volterra–Fredholm partial integro-differential equations with initial conditions using two-dimensional hybrid Taylor polynomials and Block-Pulse functions. For this purpose, operational metrics of product and integration of the cross-product and derivative are introduced that essentially of hybrid functions. The use of these operational matrices simplifies considerably the structure of the computation used for a set of algebraic equations for the solution of partial integro-differential equations. Convergence analysis and some numerical results are presented to illustrate the effectiveness and accuracy of the method. Some figures are plotted to demonstrate the error analysis of the proposed scheme. Use operational matrices of two-dimensional hybrid Taylor polynomials and Block-Pulse functions and obtain high accuracy of the method.