A numerical solution study on singularly perturbed convection-diffusion nonlocal boundary problem

Year 2019, Volume: 68 Issue: 2, 1482 - 1491, 01.08.2019

Derya Arslan Musa Cakir

https://doi.org/10.31801/cfsuasmas.540631

Cited By: 1

Abstract

This important numerical method is given for the numerical solution of singularly perturbed convection-diffusion nonlocal boundary value problem. First, the behavior of the exact solution is analyzed, which is needed for analysis of the numerical solution in later sections. Next, uniformly convergent finite difference scheme on a Shishkin mesh is established, which is based on the method of integral identities with the use exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. It is shown that the method is first order accurate expect for a logarithmic factor, in the discrete maximum norm. Finally, the numerical results are presented in table and graphs, and these results reveal the validity of the theoretical results of our method.

Keywords

Singular perturbation, finite difference scheme, Shishkin mesh, uniformly convergence, nonlocal condition

References

• Adzic, N., Spectral approximation and nonlocal boundary value problems, Novi Sad J. Math. 30 (3), (2000), 1-10.
• Amiraliyev, G. M., Difference method for a singularly perturbed initial value problem, Turkish J. Math., 22, (1998), 283-294.
• Amiraliyev, G. M. and Cakir, M., A uniformly convergent difference scheme for singularly perturbed problem with convective term and zeroth order reduced equation, International Journal of Applied Mathematics, 2 (12), (2000), 1407-1419.
• Amiraliyev, G. M. and Cakir, M., Numerical solution of the singularly perturbed problem with nonlocal condition, Applied Mathematics and Mechanics (English Edition), 23 (7), (2002), 755-764.
• Arslan D., Finite difference method for solving singularly perturbed multi-point boundary value problem, Journal of the Institute of Natural and Applied Sciences, 22 (2), (2017), 64-75.
• Bakhvalov, N. S., On optimization of methods for solving boundary value problems in the presence of a boundary layer, Zhurnal Vychislitel'noi Matematikii Matematicheskoi Fiziki, 9 (4), (1969), 841-859.
• Bitsadze, A. V. and Samarskii, A. A., On some simpler generalization of linear elliptic boundary value problems, Doklady Akademii Nauk SSSR, 185, (1969), 739-40.
• Cakir, M. and Amiraliyev, G. M., Numerical solution of the singularly perturbed three-point boundary value problem, International Journal of Computer Mathematics, 84 (10), (2007), 1465-1481.
• Cakir, M., Uniform second-order difference method for a singularly perturbed three-point boundary value problem, Advances in Difference Equations, (2010), 13 pages.
• Cakir, M. and Amiraliyev, G. M., A numerical method for a singularly perturbed three-point boundary value problem, Journal of Applied Mathematics, (2010), 17 pages.
• Cakir, M. and Arslan, D. A numerical method for nonlinear singularly perturbed multi-point boundary value problem, Journal of Applied Mathematics and Phyics, 4, (2016), 1143-1156.
• Cakir, M. and Arslan, D. Numerical solution of the nonlocal singularly perturbed problem, Int. Journal of Modern Research in Engineering and Technology, 1, (5), (2016), 13-24.
• Cakir, M. and Arslan, D., Finite difference method for nonlocal singularly perturbed problem, Int. Journal of Modern Research in Engineering and Technology, 1, (5), (2016), 25-39.
• Chegis, R., The numerical solution of problems with small parameter at higher derivatives and nonlocal conditions, Lietuvas Matematica Rinkinys, (in Russian), 28, (1988), 144-152.
• Farell, 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, (1996), 1135-1149.
• 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, (1997), 586-597.
• Herceg, D. and Surla, K., Solving a nonlocal singularly perturbed nonlocal problem by splines in tension, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Math., 21, (2), (1991), 119-132.
• Miller, J. J. H., O'Riordan, E. and 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.