In this work, an improved reproducing kernel method to find the numerical solution of Fredholm integro-differential equation type boundary value problems has been developed. Based on the good properties of reproducing kernel function and the conjugate operator, the solution representation is obtained. Meanwhile, we prove that the approximation converges to the exact solution uniformly. After that the convergence estimates are derived.

In this paper, an efficient method based on Quasi-Newton's method and the simpliffied reproducing kernel method is proposed for solving nonlinear singular boundary value problems. For the Quasi-Newton's method the convergence order is studied. The uniform convergence of the numerical solution as well as its derivatives are also proved. Numerical examples are given to demonstrate the efficiency and stability of the proposed method. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.

Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.