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Numerical solution of doubly singular boundary value problems by finite difference method

Publication Type : Journal Article

Publisher : Springer International Publishing

Source : Springer International Publishing

Campus : Coimbatore

School : School of Engineering

Center : Amrita Innovation & Research

Department : Mechanical

Verified : Yes

Year : 2020

Abstract : In this paper, we propose a computational technique based on a combination of optimal homotopy analysis method (OHAM) and an iterative finite difference method (FDM) for a class of derivative-dependent doubly singular boundary value problems: $$\begin{aligned} (p(x)y^{\prime })^{\prime }= & {} q(x)f(x,y(x),y^{\prime }(x)),\quad 0\le x\le 1, \\ y^{\prime }(0)= & {} 0,~~ {\alpha }y(1)+{\beta }y^{\prime }(1)=B \end{aligned}$$or $$\begin{aligned} y(0)=A,~~{\alpha }y(1)+{\beta }y^{\prime }(1)=B. \end{aligned}$$The principal idea of this approach is to decompose the domain of the problem \(D=[0,1]\) into two subdomains as \(D=D_{1}\cup D_{2}=[0,\gamma ]\cup [\gamma ,1]\) (\(\gamma \) is the vicinity of the singularity). In the first domain \(D_{1},\) we use OHAM to overcome the singularity behaviour at \(x=0\). In the second domain \(D_{2}\), a FDM is designed for solving the resulting regular boundary value problem. Convergence analysis of the method is carried out. Three nonlinear examples are considered to demonstrate the performance and accuracy of the proposed method. It is shown that the computational order of convergence of the FDM is two.

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