Research Article
BibTex RIS Cite
Year 2020, Volume: 12 Issue: 1, 31 - 38, 29.06.2020

Abstract

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.

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

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.

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.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Derya Arslan 0000-0001-6138-0607

Publication Date June 29, 2020
Published in Issue Year 2020 Volume: 12 Issue: 1

Cite

APA Arslan, D. (2020). On The Generation for Numerical Solution of Singularly Perturbed Problem with Right Boundary Layer. Turkish Journal of Mathematics and Computer Science, 12(1), 31-38.
AMA Arslan D. On The Generation for Numerical Solution of Singularly Perturbed Problem with Right Boundary Layer. TJMCS. June 2020;12(1):31-38.
Chicago Arslan, Derya. “On The Generation for Numerical Solution of Singularly Perturbed Problem With Right Boundary Layer”. Turkish Journal of Mathematics and Computer Science 12, no. 1 (June 2020): 31-38.
EndNote Arslan D (June 1, 2020) On The Generation for Numerical Solution of Singularly Perturbed Problem with Right Boundary Layer. Turkish Journal of Mathematics and Computer Science 12 1 31–38.
IEEE D. Arslan, “On The Generation for Numerical Solution of Singularly Perturbed Problem with Right Boundary Layer”, TJMCS, vol. 12, no. 1, pp. 31–38, 2020.
ISNAD Arslan, Derya. “On The Generation for Numerical Solution of Singularly Perturbed Problem With Right Boundary Layer”. Turkish Journal of Mathematics and Computer Science 12/1 (June 2020), 31-38.
JAMA Arslan D. On The Generation for Numerical Solution of Singularly Perturbed Problem with Right Boundary Layer. TJMCS. 2020;12:31–38.
MLA Arslan, Derya. “On The Generation for Numerical Solution of Singularly Perturbed Problem With Right Boundary Layer”. Turkish Journal of Mathematics and Computer Science, vol. 12, no. 1, 2020, pp. 31-38.
Vancouver Arslan D. On The Generation for Numerical Solution of Singularly Perturbed Problem with Right Boundary Layer. TJMCS. 2020;12(1):31-8.