Homotopy Perturbation-reproducing Kernel Method For Solving Second-order Initial Value Problem

The initial value problem of nonlinear equations can be used to solve manyproblems from modern subjects. It is the hot topic concerned by many experts andscholars, since it has an important practical significance. In the development process ofsolving the initial value problems of nonlinear equation, many numerical methodsproposed by experts and scholars, such as Runge-Kutta method, Linear multi-stepmethod, Variational iteration method, Newton method, Euler method, Homotopyperturbation method and so on.Homotopy perturbation method was first proposed by J.H.He in1998, this methodcombines the traditional perturbation method with homotopy technique, overcomingthe shortcoming of perturbation theory, deforming a difficult problem into simplesolving problems. Using this method, the series solution can quickly converge to thetrue solution. A few several terms of the series solution can be used for approximationto the exact solution. Based on the above advantages, this method has been applied tovarious fields.Reproducing kernel method is an analytical technique, using the initial conditionsof the equation to construct a linear operator, then we can solve the simple linearoperator equation instead of the original complex one.However, there are disadvantages of the homotopy perturbation method:(1) Forstrongly nonlinear problems, this method only local converges;(2) Since thecompression operator is difficult to verify, there is no strict convergence proof.Based on the above two points, the traditional homotopy perturbation method ismodified, which means the interval is divided. This new method overcomes theshortcoming of traditional homotopy perturbation method, strict convergence proof isalso given.The purpose of this paper is to apply the modified homotopy perturbation methodto nonlinear second-order Volterra integro-differential equations, combiningReproducing kernel method to solve strongly nonlinear second-order ordinarydifferential equations with initial value problem. The convergence proof of the newmethod is given. Numerical results of every chapter show that the modified homotopyperturbation method is a fast and simple method.