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