On The Generation for Numerical Solution of Singularly Perturbed Problem with Right Boundary Layer

Year 2020, Volume: 12 Issue: 1, 31 - 38, 29.06.2020

Derya Arslan

Abstract

In this study, we propose an important numerical method for the numerical solution of singularly perturbed convection-diffusion five points boundary value problem using nonuniform mesh. First, we give the some behaviours of the exact solution and its first derivative. We establish finite difference scheme, which is based on interpolating quadrature rules. Then, we prove the convergence of difference scheme and it is uniformly convergent in $ \varepsilon $ perturbation parameter. Furthermore, by a numerical experiment, we demonstrate the efficiency of the proposed method.

Keywords

Singular perturbation, finite difference scheme, nonuniform mesh, uniformly convergence, five point boundary condition, discrete maximum norm

References

• Adzic, N., {\em Spectral approximation and nonlocal boundary value problems}, Novi Sad J. Math., \textbf{30}(2000), 1--10.
• Amiraliyev, G.M., {\em Difference method for a singularly perturbed initial value problem}, Turkish J. Math., \textbf{22}(1998), 283--294.
• Amiraliyev, G.M., Cakir, M., {\em A uniformly convergent difference scheme for singularly perturbed problem with convective term and zeroth order reduced equation}, International Journal of Applied Mathematics, \textbf{2}(2000), 1407--1419.
• Amiraliyev, G.M., Cakir, M., {\em Numerical solution of the singularly perturbed problem with nonlocal condition}, Applied Mathematics and Mechanics (English Edition), \textbf{23}(2002), 755--764.
• Arslan, D., {\em Finite difference method for solving singularly perturbed multi-point boundary value problem}, Journal of the Institute of Natural and Applied Sciences, \textbf{22}(2017), 64--75.
• Arslan, D., {\em Stability and convergence analysis on Shishkin mesh for a nonlinear singularly perturbed problem with three-point boundary condition}, Quaestiones Mathematicae, (2019), 1--14.
• Arslan, D., {\em An approximate solution of linear singularly perturbed problem with nonlocal boundary condition}, Journal of Mathematical Analysis, \textbf{11}(2020), 46--58.
• Arslan, D., {\em A new second-order difference approximation for nonlocal boundary value problem with boundary layers}, Mathematical Modelling and Analysis, \textbf{25}(2020), 257--270.
• Bakhvalov, N.S., {\em On optimization of methods for solving boundary value problems in the presence of a boundary layer}, Zhurnal Vychislitel'noi Matematikii Matematicheskoi Fiziki, \textbf{9}(1969), 841--859.
• Bitsadze, A.V., Samarskii, A.A., {\em On some simpler generalization of linear elliptic boundary value problems}, Doklady Akademii Nauk SSSR, \textbf{185}(1969), 739--740.
• Cakir, M., {\em Uniform second-order difference method for a singularly perturbed three-point boundary value problem}, Advances in Difference Equations, \textbf{2010}(2010), 13 pages, 2010.
• Cakir, M., Amiraliyev, G.M., {\em A numerical method for a singularly perturbed three-point boundary value problem}, Journal of Applied Mathematics, \textbf{2010}(2010), 17 pages, 2010.
• Cakir, M., Arslan, D., {\em A numerical method for nonlinear singularly perturbed multi-point boundary value problem}, Journal of Applied Mathematics and Physics, \textbf{4}(2016), 1143--1156.
• Cakir, M., Arslan, D., {\em Numerical solution of the nonlocal singularly perturbed problem}, Int. Journal of Modern Research in Engineering and Technology, \textbf{1}(2016), 13--24.
• Cakir, M., Arslan, D., {\em Finite difference method for nonlocal singularly perturbed problem}, Int. Journal of Modern Research in Engineering and Technology, \textbf{1}(2016), 25--39.
• Chegis, R., {\em The numerical solution of problems with small parameter at higher derivatives and nonlocal conditions}, Lietuvas Matematica Rinkinys, (in Russian), \textbf{28}(1988), 144--152.
• Cimen, E., Cakir, M., {\em Numerical treatment of nonlocal boundary value problem with layer behaviour}, Bull. Belg. Math. Soc. Simon Stevin, \textbf{24}(2017), 339--352.
• Farell, P.A., Miller, J.J.H., O'Riordan, E., Shishkin, G.I., {\em A uniformly convergent finite difference scheme for a singularly perturbed semi linear equation}, SIAM Journal on Numerical Analysis, \textbf{33}(1996), 1135--1149.
• Gupta, C.P., Trofimchuk, S.I., {\em A sharper condition for the solvability of a three-point second order boundary value problem}, Journal of Mathematical Analysis and Applications, \textbf{205}(1997), 586--597.
• Jankowski, T., {\em Existence of solutions of differential equations with nonlinear multipoint boundary conditions}, Comput. Math. Appl., \textbf{47}(2004), 1095--1103.
• Miller, J.J.H., O'Riordan, E., Shishkin, G.I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996.
• Nayfeh, A.H., Introduction to Perturbation Techniques, Wiley, New York, 1993.