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A new numerical approach for solving a class of singular two-point boundary value problems

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Abstract

In this paper, we consider the following class of singular two-point boundary value problem posed on the interval x 𝜖 (0, 1]

$$\begin{array}{@{}rcl@{}} (g(x)y^{\prime})^{\prime}=g(x)f(x,y),\\ y^{\prime}(0)=0,\mu y(1)+\sigma y^{\prime}(1)=B. \end{array} $$

A recursive scheme is developed, and its convergence properties are studied. Further, the error estimation of the method is discussed. The proposed scheme is based on the integral equation formalism and optimal homotopy analysis method in which a recursive scheme is established without any undetermined coefficients. The original differential equation is transformed into an equivalent integral equation to remove the singularity. The integral equation is then made free of undetermined coefficients by imposing the boundary conditions on it. Finally, the integral equation without any undetermined coefficients is efficiently treated by using optimal homotopy analysis method for finding the numerical solution. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. The present method is applied to obtain numerical solution of singular boundary value problems arising in various physical models, and numerical results show the advantages of our method over the existing methods.

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Acknowledgments

The authors 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. The authors are highly grateful to the referees for their constructive comments.

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  1. VNIT, Nagpur, Maharashtra, India

    Pradip Roul & Dipak Biswal

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  1. Pradip Roul
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  2. Dipak Biswal
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Correspondence to Pradip Roul.

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Roul, P., Biswal, D. A new numerical approach for solving a class of singular two-point boundary value problems. Numer Algor 75, 531–552 (2017). https://doi.org/10.1007/s11075-016-0210-z

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