A Simple Method of Numerical Integration for a Class of Singularly Perturbed Two Point Boundary Value Problems

Rakesh Ranjan*, H. S. Prasad**, Md. Javed Alam***
*,*** Research Scholar, Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand, India.
** Assistant Professor, Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand, India.
Periodicity:January - March'2018
DOI : https://doi.org/10.26634/jmat.7.1.14031

Abstract

This paper deals with a simple but efficient numerical integration method to solve a class of singularly perturbed twopoint boundary value problems. Using the methods of exact rule of integration with a finite difference approximation of first derivatives, a three-term recurrence relationship is obtained. The authors have employed Thomas algorithm to obtain the solution of the obtained system. Also, the stability and convergence of the proposed scheme are established. Several model example problems have been solved and the results are presented in terms of maximum absolute errors, which show the accuracy and efficiency of the method. The method produces highly accurate results for a fixed value of step size h when the perturbation parameter e tends to zero.

Keywords

Singular Perturbation Problems, Boundary Value Problems, Stability and Convergence, Simpson's Rule.

How to Cite this Article?

Rakesh Ranjan., H. S. Prasad., Md. Javed Alam. (2018). A Simple Method of Numerical Integration for A Class of Singularly Perturbed Two Point Boundary Value Problems. i-manager’s Journal on Mathematics, 7(1), 43-52. https://doi.org/10.26634/jmat.7.1.14031

References

[1]. Bender, C. M., & Orszag, S. A. (1978). Boundary Layer Theory. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York.
[2]. Chakravarthy, P. P., Kumar, S. D., & Rao, R. N. (2015). An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays. Ain Shams Engineering Journal, 8(4), 663-671.
[3]. Gold, R. R., (1962). Magneto-hydrodynamic pipe flow. Part I. Journal of Fluid Mechanics, 13(4), 505-512.
[4]. Kadalbajoo, M. K., & Reddy, Y. N. (1988). An Approximate Method for Solving a Class of Singular Perturbation Problems. Journal of Mathematical Analysis and Applications, 133(3), 306-323.
[5]. Kadalbajoo, M. K., & Aggarwal, V. K., (2005). Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems. Applied Mathematics and Computation, 161(3), 973-987.
[6]. Kadalbajoo, M. K., & Sharma, K. K., (2006). Parameter-uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior. Electronic Transactions on Numerical Analysis, 23,180-201.
[7]. Kadalbajoo, M. K., & Arora, P. (2009). B-spline collocation method for the singular-perturbation problem using artificial viscosity. Computers & Mathematics with Applications, 57(4), 650-663.
[8]. Kadalbajoo, M. K., & Gupta, V. (2010). A brief survey on numerical methods for solving singularly perturbed problems. Applied Mathematics and Computation, 217(8), 3641-3716.
[9]. Kadalbajoo, M. K., & Reddy, Y. (1989). Asymptotic and numerical analysis of singular perturbation problems: A survey. Applied Mathematics and Computation, 30(3), 223-259.
[10]. Kevorkian, J., & Cole, J. D. (1981). Perturbation Methods in Applied Mathematics. Journal of Fluid Mechanics, 148, 500-501.
[10]. Kevorkian, J., & Cole, J. D. (1981). Perturbation Methods in Applied Mathematics. Journal of Fluid Mechanics, 148, 500-501.
[12]. Miller, J. J. (Ed.). (2009). Single Perturbation Problems in Chemical Physics: Analytic and Computational Methods (Vol. 256). John Wiley & Sons
[13]. Miller, J. J., O'Riordan, E., & Shishkin, G. I. (2012). Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific.
[14]. Nayfeh, A. H. (1979). Perturbation Methods. Wiley, New York.
[15]. O'Malley, R. E. (1974). Introduction to Singular Perturbations. Academic Press, New York.
[16]. O'Malley, R. E. (1991). Singular Perturbation Methods for Ordinary Differential Equations. Springer, New York.
[17]. Reddy, Y. N., & Reddy, K. A. (2002). Numerical integration method for general singularly perturbed two point boundary value problems. Applied Mathematics and Computation, 133(2-3), 351-373.
[18]. Roos, H. G., Stynes, M., & Tobiska, L. (1996). Numerical Methods for Singularly Perturbed Differential Methods: Convection-Diffusion and Flow Problems. Springer Verlag Bertin Heidelberg.
[19]. Shanthi, V., Ramanujam, N., & Natesan, S. (2006). Fitted mesh method for singularly perturbed Reaction-convectiondiffusion problems with boundary and interior layers. Journal of Applied Mathematics and Computing, 22(1-2), 49-65.
[20]. Subburayan, V., & Ramanujam, N. (2012). Asymptotic Initial Value Technique for singularly perturbed convection–diffusion delay problems with boundary and weak interior layers. Applied Mathematics Letters, 25(12), 2272- 2278.
If you have access to this article please login to view the article or kindly login to purchase the article

Purchase Instant Access

Single Article

North Americas,UK,
Middle East,Europe
India Rest of world
USD EUR INR USD-ROW
Pdf 35 35 200 20
Online 35 35 200 15
Pdf & Online 35 35 400 25

Options for accessing this content:
  • If you would like institutional access to this content, please recommend the title to your librarian.
    Library Recommendation Form
  • If you already have i-manager's user account: Login above and proceed to purchase the article.
  • New Users: Please register, then proceed to purchase the article.