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Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods. (English) Zbl 1187.34012

Summary: The main idea of this paper is to present an algorithm for computing the solutions of singular initial value problems including Lane-Emden-type equations of the form

$\left\{\begin{array}{cc}{u}^{\text{'}\text{'}}\left(x\right)+\frac{{k}_{1}}{a\left(x\right)}\phantom{\rule{0.166667em}{0ex}}{u}^{\text{'}}\left(t\right)+\frac{{k}_{2}}{b\left(x\right)}\phantom{\rule{0.166667em}{0ex}}u\left(t\right)+f\left(x,u\right)=g\left(x\right),\phantom{\rule{1.em}{0ex}}\hfill & 0

where $\alpha ,\beta ,{k}_{1},{k}_{2}$ are real constants, $a\left(x\right)$, $b\left(x\right)$ ar continuous and maybe $a\left(0\right)=0$, $b\left(0\right)=0$, $f\left(x,y\right)$ is a continuous real valued function, and $g\left(x\right)\in C\left(0,1\right]$, i.e., the case when the function $g\left(x\right)$ may be undefined at the origin.

##### MSC:
 34A45 Theoretical approximation of solutions of ODE 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions
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