Abstract
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.
Highlights
Hybrid Legendre polynomials and block-pulse functions approach were used by Maleknejad [9]
Bildik [6] used the modified decomposition method to obtain the approximate solution of nonlinear Volterra-Fredholm integro-differential equations (V-FIDE)
The simplified reproducing kernel method (SRKM) and the homotopy perturbation method (HPM) were successfully applied to figure out the nonlinear V-FIDE by getting the uniform approximate solution
Summary
This article mainly discusses the nonlinear V-FIDE: Y u(x) + H(u(x)) = y(x), u(a) = α,. In order to obtain accurate numerical solutions more quickly, many methods for solving such problems have been proposed in recent years. Hybrid Legendre polynomials and block-pulse functions approach were used by Maleknejad [9]. Bildik [6] used the modified decomposition method to obtain the approximate solution of nonlinear V-FIDE. Ghasemi [8] formulated homotopy perturbation method for solving nonlinear equations. Because the traditional reproducing kernel method needs orthogonalization, the calculation method is complex and time-consuming. This article discusses the nonlinear V-FIDE by using SRKM and HPM in the reproducing kernel space, so that the equation can achieve higher accuracy.
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