A localized meshless technique for solving 2D nonlinear integro-differential equation with multi-term kernels

Abstract

This paper studies an accurate localized meshless collocation approach for solving two-dimensional nonlinear integro-differential equation (2D-NIDE) with multi-term kernels. The proposed strategy discretizes the unknown solution in two phases. First, the semi-discrete scheme is obtained by using backward Euler finite difference (FD) approach and the first-order convolution quadrature rule for the first order temporal derivative and the Riemann-Liouville (R-L) fractional integral, respectively. Second, the spatial discretization is established by means of the local radial basis function based on partition of unity (LRBF-PU) in the space variable and its partial derivatives. Furthermore, the unconditionally stable result and first-order convergence of the time semi-discrete scheme in L2-norm are proved by the energy method. It is shown that the proposed method is accurate and that the numerical results support the theoretical analysis.