Singular Two-point Boundary Value Problems Research Articles

Solving a class of singular two-point boundary value problems using new effective reproducing kernel technique

Based on reproducing kernel theory, an efficient reproducing kernel technique is proposed for solving a class of singular two-point boundary value problems with Dirichlet boundary conditions. It is implemented as a new reproducing kernel method. In this method, reproducing kernels with Chebyshev polynomials form are used. Convergence analysis and an error estimation for the method in Lw2 space are discussed. The numerical solutions obtained by the method are compared with the numerical results of reproducing kernel method (RKM). The results reveal that the proposed method is quite efficient and accurate.

Numerical behavior of singular two-point boundary value problems in a comparative way

This article concentrates on discovering numerical behavior of the singular two-point boundary value problems through various numerical techniques. This is carried out in a comparative way by mainly using differential quadrature and finite element methods. Also a discussion has been done by means of advantages and disadvantages of the numerical methods of interest.To properly understand the behavior of the physical processes represented by the model equation, the calculated solutions have been discussed in detail.

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

Abstract The classical homotopy perturbation method proposed by J. H. He, Comput. Methods Appl. Mech. Eng. 178, 257 (1999) is useful for obtaining the approximate solutions for a wide class of nonlinear problems in terms of series with easily calculable components. However, in some cases, it has been found that this method results in slowly convergent series. To overcome the shortcoming, we present a new reliable algorithm called the domain decomposition homotopy perturbation method (DDHPM) to solve a class of singular two-point boundary value problems with Neumann and Robin-type boundary conditions arising in various physical models. Five numerical examples are presented to demonstrate the accuracy and applicability of our method, including thermal explosion, oxygen-diffusion in a spherical cell and heat conduction through a solid with heat generation. A comparison is made between the proposed technique and other existing seminumerical or numerical techniques. Numerical results reveal that only two or three iterations lead to high accuracy of the solution and this newly improved technique introduces a powerful improvement for solving nonlinear singular boundary value problems (SBVPs).

A Special Class of Singular Singularly Perturbed Two-Point Boundary Value Problems via Fitted Mesh B-Spline Method

Present paper recapitulates a new method for solving singularly perturbed two-point singular boundary value problem. We use the fitted mesh technique to generate piecewise uniform mesh and use the B-spline method which leads to a tridiagonal linear system. The proposed method is stable everywhere in the solution region including the vicinity of the singularity. We explain the convergence analysis of the method and it is shown that the method has uniform convergence. Numerical results also demonstrated the accuracy of the method.

The existence of positive solutions for the singular two-point boundary value problem

In this paper, we consider the following boundary value problem: $$ \begin{cases} ((-u'(t))^n)'=nt^{n-1}f(u(t)) &\text{for }0 1$. Using the fixed point theory on a cone and approximation technique, we obtain the existence of positive solutions in which $f$ may be singular at $u=0$ or $f$ may be sign-changing.

Numerical solution of singular two-point boundary value problems by the collocation method with the Chebyshev bases

This paper presents a study of the performance of the collocation method using Chebychev basis functions for solving linear and nonlinear singular boundary value problems. The equations are reduced to systems of linear and nonlinear algebraic equations. The numerical experiments show that the proposed method produces high accurate results.

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

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.

Singular Two-Point Boundary Value Problems of Finite Element method

Singular Two-Point Boundary Value Problems of Finite Element method

An improved iterative technique for solving nonlinear doubly singular two-point boundary value problems

This paper presents a new iterative technique for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions. The method is based on the homotopy perturbation method and the integral equation formalism in which a recursive scheme is established for the components of the approximate series solution. This method does not involve solution of a sequence of nonlinear algebraic or transcendental equations for the unknown coefficients as in some other iterative techniques developed for singular boundary value problems. The convergence result for the proposed method is established in the paper. The method is illustrated by four numerical examples, two of which have physical significance: The first problem is an application of the reaction-diffusion process in a porous spherical catalyst and the second problem arises in the study of steady-state oxygen-diffusion in a spherical cell with Michaelis-Menten uptake kinetics.

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 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.

Transformed Hermite functions on a finite interval and their applications to a class of singular boundary value problems

In this paper, a weighted orthogonal system on finite interval based on the transformed Hermite functions is introduced. Some results on approximations using the Hermite functions on finite interval are obtained from corresponding approximations on infinite interval via a conformal map. To illustrate the potential of the new basis, we apply it to the collocation method for solving a class of singular two-point boundary value problems. The numerical results show that our new scheme is very effective and convenient for solving singular boundary value problems.

Optimal L ∞ estimates for Galerkin methods for nonlinear singular two-point boundary value problems

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

A Fifth Order Compact Difference Method for Singularly Perturbed Singular Boundary Value Problems

In this paper, we have developed a fifth order compact difference method for a class of singularly perturbed singular two-point boundary value problems. To avoid the singularity at zero a terminal boundary condition in the implicit form is derived. Using this condition as one of the boundary condition we solve the singularly perturbed singular two-point boundary value problem by the fifth order compact difference scheme. Numerical results are presented to illustrate the proposed method and compared with exact solution.

A new coupled reduced alternating group explicit method for nonlinear singular two-point boundary value problems on a variable mesh

In this paper, we discuss a new coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to solve the nonlinear singular two-point boundary value problems u″ = f(r, u, u′), 0 < r < 1 subject to given natural boundary conditions u(0) = A1, u(1) = A2 where A1 and A2 are finite constants, along with a third-order numerical method on a geometric mesh. The proposed method is applicable to singular and nonsingular problems. We have discussed the convergence of the CRAGE iteration method in detail. The results obtained from the proposed CRAGE iteration method are compared with the results of the corresponding two-parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.

Modified Cubic B-spline Solution of Singular Two-point Boundary Value Problems

In this paper, we use cubic uniform B-spline to solve the numerical solutions of a class of singular twopoint boundary value problems. The analytic solutions, the first derivative/second derivative values of these problems can be well approximated with local truncation error O(h 4 ) at the knots. Numerical experiments are provided to demonstrate that our proposed method is flexible and effective.

Positive Solutions of Sturm-Liouville Problems for Second Order Singular and Impulsive Differential Equations

In this paper,we study the existence of multiple positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive dierential equations.The proof is based on the theory of fixed point index in cones.

Solving Singular Boundary Value Problems by Optimal Homotopy Asymptotic Method

In this paper, optimal homotopy asymptotic method (OHAM) for the semianalytic solutions of nonlinear singular two-point boundary value problems has been applied to several problems. The solutions obtained by OHAM have been compared with the solutions of another method named as modified adomain decomposition (MADM). For testing the success of OHAM, both of the techniques have been analyzed against the exact solutions in all problems. It is proved by this paper that solutions of OHAM converge rapidly to the exact solution and show most effectiveness as compared to MADM.

A Sinc-Galerkin technique for the numerical solution of a class of singular boundary value problems

Singular boundary value problems have received considerable interest in the mathematical applications in different areas of science and engineering. Due to the presence of a singularity, these problems raise difficulties in obtaining their analytic or numerical solutions, and various schemes have been proposed to overcome these difficulties. However, among existing techniques, the Sinc-Galerkin and Sinc-collocation methods are well-suited for handling the singularity and have high performance on boundary value problems with unbounded domains. In this work, the Sinc-Galerkin scheme is implemented to find a numerical solution of singular two-point boundary value problems arising in various physical models. Some properties of the Sinc-Galerkin method required for our subsequent development are given and are utilized to reduce the computation of solution of singular boundary value problem (BVP) to some nonlinear systems of equations. The accuracy and reliability of the proposed method are demonstrated by five test problems arising in physiology and engineering. The results are found to be in good agreement with the numerical/exact/available solutions.

Novel method for solving singular two-point BVPs

In this paper, we present a new algorithm to find exact and approximate analytical solutions of singular two-point boundary-value problems (BVPs) based on Bernstein operational matrix of differentiation. Different from other numerical techniques, Bernstein polynomials and their properties are employed for deriving a general procedure for forming this matrix. The proposed method can be applied to linear and nonlinear problems. The scheme is tested for some examples and the obtained results demonstrate efficiency of the proposed method.