Analysis and Solution of a Class of Nonlinear Two-Dimensional Volterra–Fredholm Integral Equations via Hybrid of Radial Basis Functions

In this paper, an efficient numerical method is proposed for solving a class of two-dimensional nonlinear Volterra–Fredholm integral equations of the second kind based on two-dimensional radial basis functions (RBFs). This method is based on a hybrid of radial basis functions including the multiquadric and the Gaussian constructed on Legendre–Gauss–Lobatto nodes and weights. The proposed method does not require any background mesh or cell structures, so it is meshless and consequently independent of the geometry of domain. Newton method is employed for solving the nonlinear system obtained with the RBF collocation method. Additionally, a theorem is proved for the convergence analysis. Some numerical examples are presented and the results are compared with the analytical solution to demonstrate the validity and the applicability of the proposed method.