A Novel Numerical Approach for Solving Convection-Diffusion Problem with Boundary Layer Behavior

Year 2020, , 152 - 162, 01.03.2020

Derya Arslan Musa Çakır Younis Masiha

https://doi.org/10.35378/gujs.542897

Cited By: 1

Abstract

This study deals with a new approach method for solving singularly perturbed boundary value problem of convection-diffusion type. Firstly, bounds on the solution and its derivative of solution to be used later in the article are obtained. This robust method is constructed with fitted difference scheme on a uniform mesh. It is proved that the presented method is first-order convergent with respect to the perturbation parameter ε in the discrete maximum norm. Two examples are given to illustrate the efficiency of the method. The numerical results are presented in tables and figures.

Keywords

Singular perturbation, Finite difference scheme, Uniform mesh, Uniform convergence

References

• [1] Amiraliyev, G.M. and Mamedov, Y.D., “Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations”, Tr. J. of Math., 19(1995):207-222, (1995). Google scholar.[2] Amiraliyev, G.M., “Difference method for a singularly perturbed initial value problem”, Turkish Journal of Mathematics, 22: 283-294, (1998). Google scholar.[3] Amiraliyev, G.M. and Çakır, M., “A uniformily convergent difference scheme for singularly perturbed problem with convective term and zeroth order reduced equation”, International Journal of Applied Mathematics, 2(12): 1407-1419, (2000). Google scholar.[4] Amiraliyev, G.M. and Çakır, M., “Numerical solution of the singularly perturbed problem with nonlocal boundary condition”, Applied Mathematics and Mechanics, 23: 755 764, (2002). Google scholar. [5] Bakhvalov, N.S., “On optimization of methods for solving boundary-value problems in the presence of a boundary layer”, The use of special transformation the numerical solution of bounary-layer problems, Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki. 9(4): 841-859, (1969). Google scholar.
• [6] Nayfeh, A.H., Introduction to Perturbation Techniques, Wiley, New York, (1993).[7] Bitsadze, A.V. and Samarskii, A.A., 1969. On some simpler generalization of linear elliptic boundary value problems. Doklady Akademii Nauk SSSR. 185: 739-740. Google scholar.[8] Cakır, M., “Uniform second-order difference method for a singularly perturbed three-point boundary value problem”, Hindawi Publishing Corporation Advances in Difference Equations, 13 pages, (2010). Google scholar.[9] Cakır, M. and Amiraliyev, G.M., “Numerical solution of a singularly perturbed three-point boundary value problem”, International Journal of Applied Mathematics, 84: 1465- 1481, (2007). Google scholar.[10] Cakır, M. and Amiraliyev, G.M., “A numerical method for a singularly perturbed three-point boundary value problem”, Hindawi Publishing Corporation Journal of Applied Math, 17 pages, (2010). Google scholar.[11] Cakır, M. and Arslan, D., “A numerical method for nonlinear singularly perturbed multi-point boundary value problem”, Journal of Applied Mathematics and Physics, 4: 1143-1156, (2016). Google scholar.[12] Cakır, M. and Arslan, D., “Finite difference method for nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1(5): 25-39, (2016). Google scholar.[13] Cakır, M. and Arslan, D., “Numerical solution of the nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1(5):13-24, (2016). Google scholar.[14] Chegis, R., “The Numerical solution of problems with small parameter at higher derivatives and nonlocal conditions”, Lietuvos Matematikos Rinkinys. (in Russian), 28: 144-152, (1988). Google scholar. [15] Cimen, E. and Amiraliyev, G.M. “A uniform convergent method for singularly perturbed nonlinear differential-difference equation”, Journal of Informatics and Mathematical Sciences, 9: 191–199, (2017). Google scholar.[16] Cimen, E. and Cakir, M. “Numerical treatment of nonlocal boundary value problem with layer behaviour”, Bull. Belg. Math. Soc. Simon Stevin, 24, (2017). Google scholar.[17] Cimen, E. “A priori estimates for solution of singularly perturbed boundary value problem with delay in convection term”, J. Math. Anal., 8: 202–211, (2017). Google scholar.[18] Doolan, E.P., Miller, J.J.H. And Schilders, W.H.A., Uniform Numerical Methods for Problem with Initial and Boundary Layers, Boole Press, Dublin, (1980). Google scholar.
• [19] Farrel, P.A., Miller, J.J.H., O’Riordan, E. and Shishkin, G.I., “A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation”, SIAM Journal on Numerical Analysis, 33:1135-1149, (1996). Google scholar.[20] Gupta, C.P. and Trofimchuk, S.I., “A sharper condition for the solvability of a three-point second order boundary value problem”, Journal of Mathematical Analysis and Applications, 205: 586–597, (1997). Google scholar.[21] Miller, J.J.H., O’Riordan, E. and Shishkin, G.I., Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, (1996). Google scholar.[22] Roos, H.G., Stynes, M. and Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equation: Convection-Diffusion and Flow Problems, Springer-Verlag, Berlin, 604, (2008). Google scholar.[23] Jankowski, T., “Existence of solutions of differential equations with nonlinear multipoint bondary conditions”, Comput. Math. Appl., 47:1095-1103, (2004). Google scholar.[24] O'Malley, R.E., Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York (1991). Google scholar.[25] Stynes, M, Roos, H.G and Tobiska, L., Robust Numerical Methods for Singularly Perturbed Dif- ferential Equations, Springer-Verlag, Berlin (2008). Google scholar.[26] Geng, Z. and Tang, Q., Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problem, Applied Mathematics Letters, 62, 1-6, (2016). Google scholar.