Pradip Roul

A novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems

This paper deals with two new recursive numerical schemes for the solution of a class of doubly singular two-point boundary value problems with different types of boundary conditions. We first transform the original problem into an equivalent integral equation to overcome the singular behaviour at the origin and then establish recursive schemes by employing Homotopy perturbation method for the solution of the resultant equation. We establish the convergence results of the new numerical methods. Four numerical examples are considered to demonstrate the efficiency and accuracy of the proposed methods. The present methods are shown their advantage over some existing methods (Chawla and Katti, 1982; Pandey and Singh, 2007; Pandey and Singh, 2004; Caglar et?al., 2009; Kumar and Aziz, 2004; Khuri and Sayfy, 2010) developed for the solution of singular two-point boundary value problems. Moreover, the new methods are used to obtain the approximate solutions of the singular boundary value problems arising in various physical problems.

Erratum to: New approach for solving a class of singular boundary value problem arising in various physical models

A new efficient recursive numerical scheme is presented for solving a class of singular two-point boundary value problems that arise in various physical models. The approach is based on the homotopy perturbation method in which we establish a recursive scheme without any undetermined coefficients to approximate the singular boundary value problems. The convergence analysis of the present method is discussed. Several numerical examples are provided to show the efficiency of our method for obtaining approximate solutions and to analyze its accuracy. The numerical results reveal that the present method yields a very rapid convergence of the solution without requiring much computational effort. The approximate solution obtained by the present method shows its superiority over existing methods. The Mathematica codes for numerical computation of singular boundary value problems are provided in the paper.

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

AbstractIn this paper, we consider the following class of singular two-point boundary value problem posed on the interval x𝜖 (0, 1] (g(x)y′)′=g(x)f(x,y),y′(0)=0,μy(1)+σy′(1)=B.

A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems

\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 new approximate method and its convergence for a strongly nonlinear problem governing electrohydrodynamic flow of a fluid in a circular cylindrical conduit

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.

An optimal iterative algorithm for solving Bratu-type problems

ABSTRACT Recently, Caglar et al. [B-spline method for solving Bratus problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891] proposed a numerical technique based on cubic B-spline for solving a Bratu-type problem. This method provides a second-order convergent approximation to the solution of the problem. In this paper, we develop a high-order numerical method based on quartic B-spline collocation approach for the Bratu-type and Lane–Emden problems. The error analysis of the quartic B-spline interpolation is carried out. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and a numerical method based on cubic B-spline approach. Comparison reveals that our method produces more accurate results than the method proposed by Caglar et al. [B-spline method for solving Bratus problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891].

A fast-converging iterative scheme for solving a system of Lane–Emden equations arising in catalytic diffusion reactions

ABSTRACT This paper is concerned with design and implementation of a computational technique for the efficient solution of a class of singular boundary value problems. The method is based on a modified homotopy analysis method. The method is illustrated by six examples, two of which arise in chemical engineering: the first problem arises in the study of thermal explosions, while the second problem arises in the study of heat and mass transfer within the porous catalyst particles. Numerical results reveal that our method provides better results as compared to some existing methods. Furthermore, it is a powerful tool for dealing with different types of problems with strong nonlinearity.

A New Homotopy Perturbation Scheme for Solving Singular Boundary Value Problems Arising in Various Physical Models

Abstract This paper is concerned with the construction and convergence analysis of two B-spline collocation methods for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The first method is based on uniform mesh, while the second method is based on non-uniform mesh. For the second method, we use a grading function to construct the non-uniform grid. We prove that the method based on uniform mesh is of second-order accuracy and the method based on non-uniform mesh is of fourth-order accuracy. Three nonlinear examples with derivative dependent source functions are considered to verify the performance and theoretical rate of convergence of present methods. Moreover, we consider some special cases of the problem under consideration in order to compare our methods with other existing methods. It is shown that our second method based on cubic B-spline basis functions has the same order of convergence as quartic B-spline collocation method [1]. Moreover, our methods yield more accurate results and are computationally attractive than the methods developed in [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] . The proposed methods are applied on three real-life problems, the first problem describes the distribution of radial stress on a rotationally shallow membrane cap, the second problem arises in the study of thermal explosion in cylindrical vessel and the third problem arises in astronomy.