In this paper, a new method combining the simplified reproducing kernel method (SRKM) and the homotopy perturbation method (HPM) to solve the nonlinear Volterra-Fredholm integro-differential equations (V-FIDE) is proposed. Firstly the HPM can convert nonlinear problems into linear problems. After that we use the SRKM to solve the linear problems. Secondly, we prove the uniform convergence of the approximate solution. Finally, some numerical calculations are proposed to verify the effectiveness of the approach.

Based on the good properties of reproducing kernel space, a new method com bining the simplified reproducing kernel method (SRKM) and homotopy per turbation method (HPM) for solving the nonlinear Volterra-Fredholm integro differential equations (V-FIDE) is proposed. The HPM can convert nonlinear problems into linear problems. And then using the SRKM to solve linear prob lems. The uniform convergence of the approximate solution is proved. Some numerical examples are prepared to illustrate the efficiency and rapidity of this method.