Numerical Solution of Boundary Layer Problem Using Second Order Variable Mesh Method
In this paper, a second-order finite difference method on non-uniform grid is proposed for the solution of singularly perturbed boundary value problems. Replace the derivatives of the problem with highorder finite differences on a non-uniform grid to get a discrete equation. This equation can be effectively solved by tridiagonal method. This method performs convergence analysis and the method produces second-order consistent convergence. The numerical experiments are used to illustrate the method. The absolute error has been proposed to compare with other methods in the literature to prove the rationality of the method.
A. Awoke and Y. N. Reddy, Solving singularly perturbed differential difference equations via fitted method, Applications and Applied Mathematics: An International Journal 8 (2013), 318 – 332, URL: https://www.pvamu.edu/mathematics/wp-content/uploads/sites/49/19_andargie-aam-r528-aa-073012-ready-to-post-06-25-13.pdf.
J. Bigge and E. Bohl, Deformations of the bifurcation diagram due to discretisation, Mathematics of Computation 45 (1985), 393 – 403, DOI: 10.2307/2008132.
E. P. Doolan, J. J. H. Miller and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin (1980).
H. M. Habib and E. R. El-Zahar, An algorithm for solving singular perturbation problems with mechanization, Applied Mathematics and Computation 188 (2007), 286 – 302, DOI: 10.1016/j.amc.2006.09.132.
M. K. Kadalbajoo and R. K. Bawa, Variable mesh difference scheme for singularly-perturbed boundary value problems using splines, Journal of Optimization Theory and Applications 90 (1996), 405 – 416, DOI: 10.1007/BF02190005.
M. K. Kadalbajoo and K. C. Patidar, "-Uniformly convergent fitted mesh finite difference methods for general singular perturbation problems, Applied Mathematics and Computation 179 (2006), 248 – 266, DOI: 10.1016/j.amc.2005.11.096.
J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York (1996), DOI: 10.1007/978-1-4612-3968-0.
M. K. Kadalbajoo and D. Kumar, A brief survey on numerical methods for solving singularly perturbed problems, Applied Mathematics and Computation 217 (2010), 3641 – 3716, DOI: 10.1016/j.amc.2010.09.059.
M. K. Kadalbajoo and K. K. Sharma, A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations, Applied Mathematics and Computation 197 (2008), 692 – 707, DOI: 10.1016/J.amc.2007.08.089.
R. Mohammadi, Numerical solution of general singular perturbation boundary value problems based on adaptive cubic spline, TWMS Journal of Pure and Applied Mathematics 3 (2012), 11 – 21, http://static.bsu.az/w24/pp11-21%20Mohammadi.pdf.
S. Natesan and N. Ramanujam, A “booster method” for singular perturbation problems arising in chemical reactor theory, Applied Mathematics and Computation 100 (1999), 27 – 48, DOI: 10.1016/S0096-3003(98)00014-9.
R. E. O’Malley, Introduction to Singular Perturbations, New York, Academic Press (1974), URL: https://www.elsevier.com/books/introduction-to-singular-perturbations/omalley/978-0-12-525950-7.
R. N. Rao and P. P. Chakravarty, A finite difference method for singularly perturbed differential difference equations with layer and oscillatory behavior, Applied Mathematical Modelling 37 (2013), 5743 – 5755, DOI: 10.1016/j.apm.2012.11.004.
H. G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly-perturbed Differential Equations, Springer, New York (1996), DOI: 10.1007/978-3-662-03206-0.
K. Surla, Z. Uzelac and L. Teofanov, The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem, Mathematics and Computers in Simulation 79 (2009), 2490 – 2505, DOI: 10.1016/j.matcom.2009.01.007.
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