Asymptotic expansions and numerical approximation of nonlinear degenerate boundary-value problems
This paper is concerned with the numerical solution of boundary-value problems for the generalized Emden-Fowler equation \(y''(x) + c x^p y(x)^q = 0, x \in[0,1]\) with \(y(0)=1, y(1)=0\). Here \(c,p,q\) are real constants, \( p \in (-2,-1), q>1,\) and \(c<0\). The approach proposed by the authors combines a Picard iteration in which the step \(y_{\nu -1}(x) \to y_{\nu}(x)\) satisfies the linear equation \( y_{\nu}'' + c q x^q y_{\nu} = - c x^p ( y_{\nu-1}^q - q y_{\nu-1})\) together with finite differences and extrapolation. In order to justify the extrapolation, asymptotic expansions for the Picard iterates around the two end points have been obtained. It is shown that these expansions depend crucially on the fact that \( (p+2)^{-1}\) is an integer or not. Taking into account these results, extrapolation was used to approximate the solution in the cases \((p,q,c)=(-1.5, 2.5, -1), (-1.7, 2.7, -1), (-1.4, 1, -0.6)\). Furthermore a change of the independent variable has been considered to improve the performance of the algorithm. Finally some numerical results are presented to show the improvement of the new method over the standard finite-difference technique for several grids.
scientific article; zbMATH DE number 1311312