Two‐dimensional collocation method for generalized partial integro‐differential equations of fractional order with applications

Abstract

In this paper, we present a new generalized model of partial integro‐differential equation of fractional order (PIDEFO). The new model of the PIDEFO is defined in terms of the operator presented recently. The operator takes form of the Riemann‐Liouville and Caputo derivatives in special case. The generalized model of the PIDEFO reduces to (1) linear space‐time fractional reaction‐diffusion equation (FRDE), (2) space‐time fractional diffusion equations (STFDE), and (3) time fractional telegraph equations (TFTE) in particular cases. The considered model is solved with collocation method using Jacobi polynomials. For theoretical accuracy, we prove that the Jacobi approximation converges to the exact solution uniformly. Two test examples are considered to verify the accuracy of the proposed method. Further, we present two applications of the proposed model and solve it using the presented method. The obtained numerical results are compared with the existing results. Finally, the numerical stability of the presented method is analyzed under the effect of the random noise in the force term.