A meshless local Galerkin method for solving a class of nonlinear time-dependent mixed integral equations on non-rectangular 2D domains

This article describes an effective method for the numerical solution of nonlinear time-dependent Volterra-Fredholm integral equations derived from simulating the spatio-temporal spread of an epidemic. At first, the proposed method applies the piecewise linear interpolation technique to discretize the temporal direction. Then, to obtain a full-discrete scheme, the moving least squares (MLS) approach as shape functions in the discrete Galerkin scheme is used to approximate the solution on the domain space. The MLS involves a weighted local least square given on a set of scattered data to estimate multivariate functions. By inheriting the features of the MLS, the offered method can be flexibly employed on non-rectangular domains without any mesh generation on the solution domain. The method’s algorithm is uncomplicated and direct, making it effortlessly executable on a standard PC with regular specifications. The error analysis and convergence rate of the proposed scheme are also discussed. The efficiency and accuracy of the new method are tested by some mixed integral equations on various non-rectangular domains together with an application of them.