Zelal TEMEL, Musa ÇAKIR

A New Numerical Scheme for Singularly Perturbed Reaction-Diffusion Problems

This study is related to a novel numerical technique for solving the singularly perturbed reaction-diffusion boundary value problems. First, explicit boundaries for the solution of the problem are established. Then, a finite difference scheme is established on a uniform mesh supported by the method of integral identities using the remainder term in integral form and the exponential rules with weight. The uniform convergence and stability of these schemes are investigated concerning the perturbation parameter in the discrete maximum norm. At last, the numerical results that provide theoretical results are presented.

Keywords:

Singular perturbation, Reaction-Diffusion, Uniform mesh Boundary layer, Numerical solution,

PDF

___

[1] Doolan, E. P., Miller, J. J. H., Schilders, W. H. A., Uniform numerical methods for problems with initial and boundary layers, Boole Press, Dublin, (1980).
[2] Farrell, P. A., Miller, J. J. H., O’Riordan, E., Shishkin, G. I., “A Uniformly Convergent Finite Difference Scheme for a Singularly Perturbed Semilinear Equation”, SIAM Journal on Numerical Analysis, 33: 1135-1149, (1996).
[3] Gupta, C. P., 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).
[4] Miller, J. J. H., O’Riordan, E., Shishkin, G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific Publishing, Singapore, (1996).
[5] Roos, H. G., Stynes, M., “Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equation, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 604, (2008).
[6] Cıbık, A. B., Yılmaz, F. N., “Variational multiscale method for the optimal control problems of convection-diffusion-reaction equations”, Turkish Journal of Mathematics, 42: 164-180, (2018).
[7] O'Malley, R. E., Singular Perturbation Methods for Ordinary Differential Equations, Springer Verlag, New York, (1991).
[8] Roos, H. G., Stynes, M., Tobiska, L., Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, second edition, 24, (2008).
[9] Kevorkian, J., and Cole, J. D., Perturbation Methods in Applied Mathematics, Applied Mathematical Sciences, Springer-Verlag, Second Edition, 34, New York, (1981).
[10] Nayfeh, A. H., Perturbation Methods, Pure and Applied Mathematics, John Wiley and Sons, New York, (1973).
[11] Nayfeh, A. H., Introduction to perturbation techniques, John Wiley and Sons, New York, USA (1981).
[12] Smith, D. R., Singular-perturbation theory, Cambridge University Press, New York, USA, (1985).
[13] Farrell, P. A., Hegarty, A. F., Miller, J. J. H., O’Riordan, E., Shishkin, G. I., Robust computational techniques for boundary layers, Chapman Hall/CRC, New York, USA, (2000).
[14] Cakır, M., Amiraliyev, G. M., “Numerical solution of a singularly perturbed three-point boundary value problem”, International Journal of Applied Mathematics, 84: 1465- 1481, (2007).
[15] Cakır, M., Arslan, D., “A numerical Method for Nonlinear Singularly Perturbed Multi-Point Boundary Value Problem”, Journal of Applied Mathematics and Physics, 4: 1143-1156, (2016).
[16] Cakır, M., Arslan, D., “Finite difference method for nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1: 25-39, (2016).
[17] Cakır, M., Arslan, D., “Numerical solution of the nonlocal singularly perturbed problem”, International Journal of Modern Research in Engineering and Technology, 1: 13-24, (2016).
[18] 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).
[19] Cimen, E., Amiraliyev, G. M., “A uniform convergent method for singularly perturbed nonlinear differential-difference equation”, Journal of Informatics and Mathematical Sciences, 9: 191–199, (2017).
[20] Cimen, E., Cakir, M., “Numerical treatment of nonlocal boundary value problem with layer behavior”, Bulletin of the Belgian Mathematical Society-Simon Stevin, 24: 339-352, (2017).
[21] Cakir, M., Amiraliyev, G. M., “A finite difference method for the singularly perturbed problem with nonlocal boundary condition”, Applied Mathematics and Computation, 160: 539-549, (2005).
[22] Jankowski, T., “Existence of solutions of differential equations with nonlinear multipoint boundary conditions”, Computers & Mathematics with Applications, 47: 1095-1103, (2004).
[23] Jankowski, T., “Application of the Numerical-Analytic Method to Systems of Differential Equations with Parameter”, Ukrainian Mathematical Journal, 54: 671-683, (2002).
[24] Amiraliyev, G. M., Mamedov, Y. D., “Difference schemes on the uniform mesh for singularly perturbed pseudo-parabolic equations”, Turkish Journal of Mathematics, 19: 207-222, (1995).