The combined reproducing kernel method and Taylor series for solving nonlinear Volterra-Fredholm integro-differential equations

In this letter, the numerical scheme of nonlinear Volterra-Fredholm integro-differential equations is proposed in a reproducing kernel Hilbert space (RKHS). The method is constructed based on the reproducing kernel properties in which the initial condition of the problem is satised. The nonlinear terms are replaced by its Taylor series. In this technique, the nonlinear Volterra-Fredholm integro-differential equations are converted to nonlinear differential equations. The exact solution is represented in the form of series in the reproducing Hilbert kernel space. The approximation solution is expressed by n-term summation of reproducing kernel functions and it is converge to the exact solution. Some numerical examples are given to show the accuracy of the method.