A meshless technique for nonlinear Volterra-Fredholm integral equations via hybrid of radial basis functions

In this paper, an effective technique is proposed to determine thenumerical solution of nonlinear Volterra-Fredholm integralequations (VFIEs) which is based on interpolation by the hybrid ofradial basis functions (RBFs) including both inverse multiquadrics(IMQs), hyperbolic secant (Sechs) and strictly positive definitefunctions. Zeros of the shifted Legendre polynomial are used asthe collocation points to set up the nonlinear systems. Theintegrals involved in the formulation of the problems areapproximated based on Legendre-Gauss-Lobatto integration rule.This technique is so convenience to implement and yields veryaccurate results compared with the other basis. In addition aconvergence theorem is proved to show the stability of thistechnique. Illustrated examples are included to confirm thevalidity and applicability of the proposed method. The comparisonof the errors is implemented by the other methods in referencesusing both inverse multiquadrics (IMQs), hyperbolic secant (Sechs)and strictly positive definite functions.

Keywords:

Nonlinear Volterra-Fredholm integral equation , Strictly positive , definite functions , Inverse multiquadrics , Hyperbolic secant