A High‐Order Local Meshfree Method for Solving the Two‐Dimensional Multiterm Time‐Fractional Partial Integro‐Differential Equation

ABSTRACT The focus of this paper is to present a new fourth‐order local meshfree technique for numerically solving the multiterm time‐fractional partial integro‐differential equation. The time‐fractional derivative term, defined in the Caputo sense, is approximated using the weighted and shifted Grünwald difference (WSGD) formula. Also, a second‐order technique for approximating the integral term in the Riemann–Liouville sense leads to the semidiscrete formulation of the model. Applying the meshless radial basis function (RBF)‐Hermite finite difference (RBF‐HFD) technique for spatial discretization completes the model's discretization process. The novelty of this method lies in the approximation of the Laplace operator using the Hermite RBF interpolation over local stencils. This leads to RBF‐generated Hermite finite difference (RBF‐HFD) formula, which provides a significant improvement in the accuracy and computational efficiency. Its meshfree nature also makes it very suitable for solving partial integro‐differential equations (PIDEs) on irregular domains. In the subsequent discussion, the unconditional stability and convergence of the time‐discrete scheme are proven via the energy method.The accuracy and efficiency of the proposed scheme are investigated through numerical simulations on regular and irregular domains. Furthermore, the numerical outcomes are compared with existing methods in the literature to highlight the accuracy and efficiency of the present technique.