A new second-order difference approximation for nonlocal boundary value problem with boundary layers
Abstract
The aim of this paper is to present finite difference method for numerical solution of singularly perturbed linear differential equation with nonlocal boundary condition. Initially, the nature of the solution of the presented problem for the numerical solution is discussed. Subsequently, the difference scheme is established on Bakhvalov-Shishkin mesh. Uniform convergence in the second-order is proven with respect to the ε− perturbation parameter in the discrete maximum norm. Finally, an example is provided to demonstrate the success of the presented numerical method. Thus, it is shown that indicated numerical results support theoretical results.
Keyword : singular perturbation, finite difference method, Bakhvalov-Shishkin mesh, uniformly convergence, nonlocal condition
How to Cite
Arslan, D. (2020). A new second-order difference approximation for nonlocal boundary value problem with boundary layers. Mathematical Modelling and Analysis, 25(2), 257-270. https://doi.org/10.3846/mma.2020.9824
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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G.M. Amiraliyev and M. Cakir. Numerical solution of singularly perturbed problem with nonlocal boundary condition. Appl. Math. Mech., 23(7):755–764, 2002. https://doi.org/10.1007/BF02456971
G.M. Amiraliyev and Y.D. Mamedov. Difference schemes on the uniform mesh for a singularly perturbed pseudo-parabolic equations. Turk. J. Math., 19(1995):207–222, 1995.
D. Arslan. Finite difference method for solving singularly perturbed multi-point boundary value problem. J. Inst. Natural and Appl. Sci., 22(2):64–75, 2017.
N.S. Bakhvalov. Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. Zh. Vychisl. Mat. Mat. Fiz., 9(4):841–859, 1969. https://doi.org/10.1016/0041-5553(69)90038-X
A.V. Bitsadze and A.A. Samarskii. On some simpler generalization of linear elliptic boundary value problems. Doklady Akademii Nauk SSSR, 185:739–740, 1969.
A. Bugajev and R. Čiegis. Comparison of adaptive meshes for a singularly perturbed reaction-diffusion problem. Math. Model. Anal., 17(5):732–748, 2012. https://doi.org/10.3846/13926292.2012.736416
M. Cakir. A numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition. Math. Model. Anal., 21(5):644–658, 2016. https://doi.org/10.3846/13926292.2016.1201702
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M. Cakir and D. Arslan. Finite difference method for nonlocal singularly perturbed problem. Int. J. of Modern Research Eng. Tech., 1(5):25–39, 2016. Available from Internet: http://www.ijmret.org/
M. Cakir and D. Arslan. A numerical method for nonlinear singularly perturbed multi-point boundary value problem. J. Appl. Math. Phys., 4(6):1143–1156, 2016. https://doi.org/10.4236/jamp.2016.46119
M. Cakir and D. Arslan. Numerical solution of the nonlocal singularly perturbed problem. Int. J. of Modern Research Eng. Tech., 1(5):13–24, 2016. Available from Internet: http://www.ijmret.org/
R. Čiegis. Numerical solution of a problem with small parameter for the highest derivative and a nonlocal condition. Liet. Mat. Rink., 28(1):144–152, 1988. https://doi.org/10.1007/BF00972255
R. Čiegis. On the difference schemes for problems with nonlocal boundary conditions. Informatica, 2(2):155–170, 1991.
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. A monotonic finitedifference scheme for a parabolic problem with nonlocal conditions. Differ. Equ., 38(7):1027–1037, 2002. https://doi.org/10.1023/A:1021167932414
E. Cimen and G.M. Amiraliyev. A uniform convergent method for singularly perturbed nonlinear differential-difference equation. Journal of Informatics and Mathematical Sciences, 9(1):191–199, 2017. Available from Internet: www.projecteuclid.org
E. Cimen and M. Cakir. Numerical treatment of nonlocal boundary value problem with layer behaviour. Bull. Belg. Math. Soc. Simon Stevin, 24:339–352, 2017. https://doi.org/10.36045/bbms/1506477685
P.A. Farrel, A.F. Hegarty, J.J.H. Miller, E. ORiordan and G.I. Shishkin. Robust Computational Techniques for Boundary Layers. Chapman Hall/CRC, New York, 2000. https://doi.org/10.1201/9781482285727
C.P. Gupta and S.I. Trofimchuk. A sharper condition for the solvability of a three-point second order boundary value problem. J. Math. Anal. Appl., 205:586–597, 1997. https://doi.org/10.1006/jmaa.1997.5252
D. Herceg and K. Surla. Solving a nonlocal singularly perturbed nonlocal problem by splines in tension. Univ u Novom Sadu Zb Rad Prirod-Mat Fak Ser Math., 21(2):119–132, 1991.
T. Linss. An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem. J. Comput. Appl. Mat., 110(1):93–104, 1999. https://doi.org/10.1016/S0377-0427(99)00198-3
T. Linss. Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem. IMA J. Num. Anal., 20(4):621– 632, 2000. https://doi.org/10.1093/imanum/20.4.621
J.J.H. Miller, E.R. Doolan and W.H.A. Schilders. Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin, 1980.
J.J.H. Miller, E. ORiordan and G.I. Shishkin. Fitted Numerical Methods for Singular Perturbation Problems. World Scientic, Singapore, 1996. https://doi.org/10.1142/2933
A.H. Nayfeh. Introduction to Perturbation Techniques. Wiley, New York, 1993.
R.E. OMalley. Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York, 1991. https://doi.org/10.1007/978-1-46120977-5
M. Stynes, H.G. Roos and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, 2008.
Q. Zheng, X. Li and Y. Gao. Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs. Applied Numerical Mathematics, 91(2015):46–59, 2015. https://doi.org/10.1016/j.apnum.2014.12.010
Q. Zheng, X. Li and Y. Liu. Uniform second-order hybrid schemes on Bakhvalov-Shishkin mesh for quasi-linear convection-diffusion problems. Advanced Materials Research, 871(2014):135–140, 2014. https://doi.org/10.4028/www.scientific.net/AMR.871.135
P. Zhou, Y. Yin and Y. Yang. Finite element superconvergence on BakhvalovShishkin mesh for singularly perturbed problem. Journal on Numerical Methods and Computer Applications, 34(4):257–265, 2013.
G.M. Amiraliyev and M. Cakir. Numerical solution of singularly perturbed problem with nonlocal boundary condition. Appl. Math. Mech., 23(7):755–764, 2002. https://doi.org/10.1007/BF02456971
G.M. Amiraliyev and Y.D. Mamedov. Difference schemes on the uniform mesh for a singularly perturbed pseudo-parabolic equations. Turk. J. Math., 19(1995):207–222, 1995.
D. Arslan. Finite difference method for solving singularly perturbed multi-point boundary value problem. J. Inst. Natural and Appl. Sci., 22(2):64–75, 2017.
N.S. Bakhvalov. Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. Zh. Vychisl. Mat. Mat. Fiz., 9(4):841–859, 1969. https://doi.org/10.1016/0041-5553(69)90038-X
A.V. Bitsadze and A.A. Samarskii. On some simpler generalization of linear elliptic boundary value problems. Doklady Akademii Nauk SSSR, 185:739–740, 1969.
A. Bugajev and R. Čiegis. Comparison of adaptive meshes for a singularly perturbed reaction-diffusion problem. Math. Model. Anal., 17(5):732–748, 2012. https://doi.org/10.3846/13926292.2012.736416
M. Cakir. A numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition. Math. Model. Anal., 21(5):644–658, 2016. https://doi.org/10.3846/13926292.2016.1201702
M. Cakir and G.M. Amiraliyev. Numerical solution of a singularly perturbed three-point boundary value problem. Int. J. Comput. Math., 84(10):1465–1481, 2007. https://doi.org/10.1080/00207160701296462
M. Cakir and D. Arslan. Finite difference method for nonlocal singularly perturbed problem. Int. J. of Modern Research Eng. Tech., 1(5):25–39, 2016. Available from Internet: http://www.ijmret.org/
M. Cakir and D. Arslan. A numerical method for nonlinear singularly perturbed multi-point boundary value problem. J. Appl. Math. Phys., 4(6):1143–1156, 2016. https://doi.org/10.4236/jamp.2016.46119
M. Cakir and D. Arslan. Numerical solution of the nonlocal singularly perturbed problem. Int. J. of Modern Research Eng. Tech., 1(5):13–24, 2016. Available from Internet: http://www.ijmret.org/
R. Čiegis. Numerical solution of a problem with small parameter for the highest derivative and a nonlocal condition. Liet. Mat. Rink., 28(1):144–152, 1988. https://doi.org/10.1007/BF00972255
R. Čiegis. On the difference schemes for problems with nonlocal boundary conditions. Informatica, 2(2):155–170, 1991.
R. Čiegis, A. Štikonas, O. Štikonienė and O. Suboč. A monotonic finitedifference scheme for a parabolic problem with nonlocal conditions. Differ. Equ., 38(7):1027–1037, 2002. https://doi.org/10.1023/A:1021167932414
E. Cimen and G.M. Amiraliyev. A uniform convergent method for singularly perturbed nonlinear differential-difference equation. Journal of Informatics and Mathematical Sciences, 9(1):191–199, 2017. Available from Internet: www.projecteuclid.org
E. Cimen and M. Cakir. Numerical treatment of nonlocal boundary value problem with layer behaviour. Bull. Belg. Math. Soc. Simon Stevin, 24:339–352, 2017. https://doi.org/10.36045/bbms/1506477685
P.A. Farrel, A.F. Hegarty, J.J.H. Miller, E. ORiordan and G.I. Shishkin. Robust Computational Techniques for Boundary Layers. Chapman Hall/CRC, New York, 2000. https://doi.org/10.1201/9781482285727
C.P. Gupta and S.I. Trofimchuk. A sharper condition for the solvability of a three-point second order boundary value problem. J. Math. Anal. Appl., 205:586–597, 1997. https://doi.org/10.1006/jmaa.1997.5252
D. Herceg and K. Surla. Solving a nonlocal singularly perturbed nonlocal problem by splines in tension. Univ u Novom Sadu Zb Rad Prirod-Mat Fak Ser Math., 21(2):119–132, 1991.
T. Linss. An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem. J. Comput. Appl. Mat., 110(1):93–104, 1999. https://doi.org/10.1016/S0377-0427(99)00198-3
T. Linss. Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem. IMA J. Num. Anal., 20(4):621– 632, 2000. https://doi.org/10.1093/imanum/20.4.621
J.J.H. Miller, E.R. Doolan and W.H.A. Schilders. Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin, 1980.
J.J.H. Miller, E. ORiordan and G.I. Shishkin. Fitted Numerical Methods for Singular Perturbation Problems. World Scientic, Singapore, 1996. https://doi.org/10.1142/2933
A.H. Nayfeh. Introduction to Perturbation Techniques. Wiley, New York, 1993.
R.E. OMalley. Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York, 1991. https://doi.org/10.1007/978-1-46120977-5
M. Stynes, H.G. Roos and L. Tobiska. Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, 2008.
Q. Zheng, X. Li and Y. Gao. Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs. Applied Numerical Mathematics, 91(2015):46–59, 2015. https://doi.org/10.1016/j.apnum.2014.12.010
Q. Zheng, X. Li and Y. Liu. Uniform second-order hybrid schemes on Bakhvalov-Shishkin mesh for quasi-linear convection-diffusion problems. Advanced Materials Research, 871(2014):135–140, 2014. https://doi.org/10.4028/www.scientific.net/AMR.871.135
P. Zhou, Y. Yin and Y. Yang. Finite element superconvergence on BakhvalovShishkin mesh for singularly perturbed problem. Journal on Numerical Methods and Computer Applications, 34(4):257–265, 2013.