# A new mixed MADM-Collocation approach for solving a class of Lane–Emden singular boundary value problems

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## Abstract

In this paper, a new approach is proposed for solving a class of singular boundary value problems of Lane–Emden type. It is well known that the Adomian decomposition method (ADM) fails to provide a convergent series solution to strongly nonlinear boundary value problem in the wider region and the B-spline collocation method yields unsatisfactory approximation in the presence of singularity. To avoid these shortcomings of both methods, we propose a novel numerical method based on a combination of modified Adomian decomposition method and quintic B-spline collocation method to obtain more accurate solution of the problem under consideration. The principal idea of this approach is to decompose the domain of the problem \(D = [0, 1]\) into two subdomains as \(D = D_{1} U D_{2}= [0, \delta ] U [\delta , 1]\) (\(\delta \) is vicinity of the singularity). In the first domain \(D_1\), the underlying singular boundary value problem is efficiently tackled by a modified Adomian decomposition method. The intent is to apply the ADM in the smaller domain for finding a satisfactory solution. Finally, in the second domain \(D_2\), a collocation approach based on quintic B-spline basis function is designed for solving the resulting regular boundary value problem. The error estimation of the quintic B-spline interpolation is supplemented. In addition, six illustrative examples are presented to demonstrate the applicability and accuracy of the new method. It is shown that the resulting solutions appear to be higher accurate when compared to some existing numerical methods.

## Keywords

Lane–Emden boundary value problem Quintic B-spline collocation method ADM Domain decomposition Error analysis Reaction–diffusion problem Oxygen-diffusion problem## Mathematics Subject Classification

65L10 65L60 34B16## Notes

### Acknowledgements

The author is very grateful to anonymous referees for their valuable suggestions and comments which improved the paper and thankfully acknowledge the financial support provided by the SERB, Department of science and Technology, New Delhi, India in the form of Project No. SB/S4/MS/877/2014.

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