Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods

Abstract

The nonlinear singular initial value problems including generalized Lane–Emden-type equations are investigated by combining homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He’s HPM is based on the use of traditional perturbation method and homotopy technique and can reduce a nonlinear problem to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can overcome the difficulty at the singular point of non-homogeneous, linear singular initial value problems; especially when the singularity appears on the right-hand side of this type of equations, so it can solve powerfully linear singular initial value problems. Therefore, using advantages of these two methods, more general nonlinear singular initial value problems can be solved powerfully. Some numerical examples are presented to illustrate the strength of the method.