A fourth-order method for solving singularly perturbed boundary value problems using nonpolynomial splines

Document Type : Research Article

Authors

• S. Khan ¹
• A. Khan ²

¹ Department of Mathematics, Sri Venkateswara College, University of Delhi, New Delhi, India.

² Department of Mathematics, Jamia Millia Islamia, New Delhi, India.

10.22067/ijnao.2022.74627.1091

Abstract

In this paper, a class of second-order singularly perturbed interior layer problems is examined. A nonpolynomial mixed spline is used to develop the tridiagonal scheme. The developed method is second as well as fourth-order accurate based on the parameters. Error analysis is also carried out. The method is shown to converge point-wise to the true solution with higher accuracy. Linear and nonlinear second-order singularly perturbed boundary value problems have been solved by the presented method. Five numerical illustrations are given to demonstrate the applicability of the proposed method. Absolute errors are given in tables, which show that our method is more efficient than previously existing methods.

Keywords

• Singularly perturbed
• Second-order problems
• Nonpolynomial splines
• Convergence

20.1001.1.24236977.2022.12.2.12.8

Main Subjects

• Numerical Analysis

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References

1.Ahmad, N., Siddiqui, Z.U. and Mishra, M.K., Boundary layer flow and heat transfer past a stretching plate with variable thermal conductivity, Int. J. Non-Linear Mech. 45 (2010) 306–309.
2. Al-Said, E.A., Smooth spline solution for a system of second order bound-ary value problems, J. Nat. Geometry 16 (1999) 19–28.
3. Ascher, U.M., Matheij, R.M.M. and Russell, R.D., Numerical solution of boundary value problems for ordinary differential Equations, (Englewood Cliffs, NJ: Prentice-Hall) 1988.
4. Ayalew, M., Kiltu, G.G. and Duressa, G.F., Fitted numerical scheme for second-order singularly perturbed differential-difference equations with mixed shifts, Abstr. Appl. Anal. Volume 2021, Article ID 4573847, 11 pages.
5. Aziz, T. and Khan, A., A spline method for second-order singularly per-turbed boundary-value problems, J. Comput. Appl. Math. 147 (2002) 445–452.

6. Chekole, A.T., Duressa, G.F. and Kiltu, G.G., Non-polynomial septic spline method for singularly perturbed two point boundary value problems of order three, J. Taibah Univ. Sci. 13 (2019) 651–660.
7. El-Zahar, E.R., Approximate analytical solution of singularly perturbed boundary value problems in MAPLE, AIMS Mathematics, 5 (2020) 2272–2284.
8. Gupta, R., Bhagyamma, D. and SharathBabu, K., Numerical solution of singularly perturbed two-point boundary value problem using transforma-tion technique using quadrature method,Turk. J. Comput. Math. Educ. 12 (2021) 2084–2091.
9. Kadalbajoo, M.K. and Bawa, R.K., Variable mesh difference scheme for singularly-perturbed boundary value problems using splines, J. Optim. Theory Appl. 90 (1996) 405–416.
10. Kadalbajoo, M.K. and Reddy, Y.N., Asymptotic and numerical analysis of singular perturbation problems: A survey, Appl. Math. Comput. 30 (1989) 223–259.
11. Keller, H.B., Numerical methods for two point boundary value problems, Blaisdell Publications Co., New York, 1968.
12. Khan, A., Khan I., Aziz, T. and Stojanovic, M., A Variable-Mesh Approx-imation Method for Singularly Perturbed Boundary-Value Problems using Cubic Spline in Tension, Int. J. Comput. Math. 81 (2004) 1513–1518.
13. Kiltu, G.G., Duressa, G.F. and Bullo, T.A., Computational method for singularly perturbed delay differential equations of the reaction-diffusion type with negative shift, ’J. Ocean Eng. Sci. 6 (2021) 285–291.
14. Kumar, M., Mishra, H.K. and Singh, P., A boundary value approach for a class of linear singularly perturbed boundary value problems, Adv. Eng.Softw. 10(2009) 298–304.
15. Nikan, O., Avazzadeh, Z. and Rasoulizadeh, M.N., Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory, Nonlinear Dyn. 106 (2021) 783–813.
16. Rasoulizadeh, M.N., Ebadi, M.J., Avazzadeh, Z. and Nikan, O., An effi-cient local meshless method for the equal width equation in fluid mechan-ics, Eng. Anal. Bound. Elem. 131 (2021) 258–268.
17. Rasoulizadeh, M.N., Nikan, O. and Avazzadeh, Z., The impact of LRBF-FD on the solutions of the nonlinear regularized long wave equation, Math. Sci. 15 (2021) 365–376.

18. Reddy, Y.N. and Chakravarthy, P.P., An initial-value approach for solv-ing singularly perturbed two-point boundary value problems, Appl. Math. Comput. 155 (2004) 95–110.
19. Varga, R.S., Matrix iterative analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
20. Zahra, W.K. and Mhlawy, A.M.El. Numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline, J. King Saud Univ. Sci. 25 (2013) 201–208.