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.

Based on the least square method, we proposed a new algorithm to obtain the solution of the second kind regular and weakly singular Volterra-Fredholm integral equations in reproducing kernel spaces. The stability and uniform convergence of the algorithm are investigated in details. Numerical experiments verify the theoretical findings. Meanwhile this method is also applicable to the nonlinear Volterra integral equations. Test problems which have non-smooth solutions are also considered and our proposed method is efficient as some recent Krylov subspace methods such as LSQR and LSMR.