Robust numerical method for singularly perturbed convection-diffusion type problems with non-local boundary condition
Abstract
This paper presents the study of singularly perturbed differential equations of convection diffusion type with non-local boundary condition. The proposed numerical scheme is a combination of classical finite difference method for the initial boundary condition and nonstandard finite difference method for the differential equations at the interior points. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical examples considered. The method is shown to be first-order convergence independent of the perturbation parameter ε.
Keyword : singular perturbation, boundary value problem, non-standared fitted operator scheme, uniform convergence, non-local boundary condition
How to Cite
Debela, H. G., Woldaregay, M. M., & Duressa, G. F. (2022). Robust numerical method for singularly perturbed convection-diffusion type problems with non-local boundary condition. Mathematical Modelling and Analysis, 27(2), 199–214. https://doi.org/10.3846/mma.2022.14256
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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H.G. Debela. Exponential fitted operator method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition. Abstract and Applied Analysis, 2021:1–9, 2021. https://doi.org/10.1155/2021/5559486
H.G. Debela and G.F. Duressa. Accelerated exponentially fitted operator method for singularly perturbed problems with integral boundary condition. International Journal of Differential Equations, 2020:1–8, 2020. https://doi.org/10.1155/2020/9268181
H.G. Debela and G.F. Duressa. Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition. International Journal for Numerical Methods in Fluids, 92(12):1914– 1926, 2020. https://doi.org/10.1002/fld.4854
H.G. Debela and G.F. Duressa. Fitted operator finite difference method for singularly perturbed differential equations with integral boundary condition. Kragujevac Journal of Mathematics, 47(4):637–651, 2023.
E.P. Doolan, J.J.H. Miller and W.H.A. Schilders. Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin, 1980.
Z. Du and L. Kong. Asymptotic solutions of singularly perturbed second-order differential equations and application to multi-point boundary value problems. Applied Mathematics Letters, 23(9):980–983, 2010. https://doi.org/10.1016/j.aml.2010.04.021
P.A. Farell, 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.
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. Mathematics, 21(2):119–132, 1991.
V.A. Il’in and E.I. Moiseev. Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations, 23(7):803–810, 1987.
V.A. Il’in and E.I. Moiseev. Nonlocal boundary value problem of the second kind for a Sturm-Lliouville operator. Differential Equations, 23:979–987, 1987.
T. Jankowski. Existence of solutions of differential equations with nonlinear multipoint boundary conditions. Computer and Mathematics with Applications, 47(6):1095–1103, 2004. https://doi.org/10.1016/S0898-1221(04)90089-2
M.K. Kadalbajoo and V. Gupta. A brief survey on numerical methods for solving singularly perturbed problems. Applied Mathematics and Computation, 217(8):3641–3716, 2010. https://doi.org/10.1016/j.amc.2010.09.059
J. Kevorkian and J.D. Cole. Multiple Scale and Singular Perturbation Methods. Springer, New York, 1996.
M. Kudu and G.M. Amiraliyev. Finite difference method for a singularly perturbed differential equations with integral boundary condition. International Journal of Mathematics and Computation, 26(3):72–79, 2015.
M. Kumar and C.S. Rao. High order parameter-robust numerical method for singularly perturbed reactiondiffusion problems. Applied Mathematics and Computation, 216(4):1036–1046, 2010. https://doi.org/10.1016/j.amc.2010.01.121
V. Kumar and B. Srinivasan. An adaptive mesh strategy for singularly perturbed convection diffusion problems. Applied Mathematical Modelling, 39(7):2081– 2091, 2015. https://doi.org/10.1016/j.apm.2014.10.019
T. Linß. Layer-adapted meshes for convectiondiffusion problems. Computer Methods in Applied Mechanics and Engineering, 192(9):1061–1105, 2003. https://doi.org/10.1016/S0045-7825(02)00630-8
R.E. Mickens. Advances in the Applications of Nonstandard Finite Difference Schemes. World Scientific, 2005. https://doi.org/10.1142/5884v
A.H. Nayfeh. Perturbation Methods. Wiley, New York, 1985.
O’Malley. Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0977-5
N. Petrovic. On a uniform numerical method for a nonlocal problem. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mathematics, 21(2):133–140, 1991.
H.G. Roos, M. Stynes and L. Tobiska. Robust Numerical Methods Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, 2008. https://doi.org/10.1007/978-3-540-34467-4
D.R. Smith. Singular Perturbation Theory. Cambridge University Press, Cambridge, 1985.
Q. Zheng, X. Li and Y. Gao. Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs. Applied Numerical Mathematics, 91:46–59, 2015. https://doi.org/10.1016/j.apnum.2014.12.010
K. Bansal and K.K. Sharma. Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments. Numerical Algorithms, 75(1):113–145, 2017. https://doi.org/10.1007/s11075-016-0199-3
M. Benchohra and S.K. Ntouyas. Existence of solutions of nonlinear differential equations with nonlocal conditions. Journal of Mathematical Analysis and Applications, 252(1):477–483, 2000. https://doi.org/10.1006/jmaa.2000.7106
A.V. Bitsadze and A.A. Samarskii. On some simpler generalization of linear elliptic boundary value problem. Doklady Akademii Nauk SSSR, 185:739–740, 1969.
M. Çakir. Uniform second-order difference method for a singularly perturbed three-point boundary value problem. Advances in Difference Equations, 102484(2010):1–13, 2010. https://doi.org/10.1155/2010/102484
M. Cakir. A numerical study on the difference solution of singularly perturbed semilinear problem with integral boundary condition. Mathematical Modelling and Analysis, 21(5):644–658, 2016. https://doi.org/10.3846/13926292.2016.1201702
M. Cakir and G.M. Amiraliyev. A finite difference method for the singularly perturbed problem with nonlocal boundary condition. Applied Mathematics and Computation, 160:539–549, 2005. https://doi.org/10.1016/j.amc.2003.11.035
M. Cakir, E. Cimen and G.M. Amiraliyev. The difference schemes for solving singularly perturbed three-point boundary value problem. Lithuanian Mathematical Journal, 60:147–160, 2020. https://doi.org/10.1007/s10986-020-09471-z
Z. Cen. A second-order finite difference scheme for a class of singularly perturbed delay differential equations. International Journal of Computer Mathematics, 87(1):173–185, 2010. https://doi.org/10.1080/00207160801989875
Z. Cen, A. Le and A. Xu. Parameter-uniform hybrid difference for solutions and derivatives in singularly perturbed initial value problems. Journal of Computational and Applied Mathematics, 320:176–192, 2017. https://doi.org/10.1016/j.cam.2017.02.009
E. Cimen and M. Cakir. Numerical treatment of nonlocal boundary value problem with layer behaviour. Bulletin of the Belgian Mathematical Society - Simon Stevin, 24:339–352, 2017. https://doi.org/10.36045/bbms/1506477685
C. Clavero, J.L. Gracia and F. Lisbona. High order methods on Shishkin meshes for singular perturbation problems of convectiondiffusion type. Numerical Algorithms, 22(73):339–352, 1999. https://doi.org/10.1023/A:1019150606200
H.G. Debela. Exponential fitted operator method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition. Abstract and Applied Analysis, 2021:1–9, 2021. https://doi.org/10.1155/2021/5559486
H.G. Debela and G.F. Duressa. Accelerated exponentially fitted operator method for singularly perturbed problems with integral boundary condition. International Journal of Differential Equations, 2020:1–8, 2020. https://doi.org/10.1155/2020/9268181
H.G. Debela and G.F. Duressa. Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition. International Journal for Numerical Methods in Fluids, 92(12):1914– 1926, 2020. https://doi.org/10.1002/fld.4854
H.G. Debela and G.F. Duressa. Fitted operator finite difference method for singularly perturbed differential equations with integral boundary condition. Kragujevac Journal of Mathematics, 47(4):637–651, 2023.
E.P. Doolan, J.J.H. Miller and W.H.A. Schilders. Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin, 1980.
Z. Du and L. Kong. Asymptotic solutions of singularly perturbed second-order differential equations and application to multi-point boundary value problems. Applied Mathematics Letters, 23(9):980–983, 2010. https://doi.org/10.1016/j.aml.2010.04.021
P.A. Farell, 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.
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. Mathematics, 21(2):119–132, 1991.
V.A. Il’in and E.I. Moiseev. Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations, 23(7):803–810, 1987.
V.A. Il’in and E.I. Moiseev. Nonlocal boundary value problem of the second kind for a Sturm-Lliouville operator. Differential Equations, 23:979–987, 1987.
T. Jankowski. Existence of solutions of differential equations with nonlinear multipoint boundary conditions. Computer and Mathematics with Applications, 47(6):1095–1103, 2004. https://doi.org/10.1016/S0898-1221(04)90089-2
M.K. Kadalbajoo and V. Gupta. A brief survey on numerical methods for solving singularly perturbed problems. Applied Mathematics and Computation, 217(8):3641–3716, 2010. https://doi.org/10.1016/j.amc.2010.09.059
J. Kevorkian and J.D. Cole. Multiple Scale and Singular Perturbation Methods. Springer, New York, 1996.
M. Kudu and G.M. Amiraliyev. Finite difference method for a singularly perturbed differential equations with integral boundary condition. International Journal of Mathematics and Computation, 26(3):72–79, 2015.
M. Kumar and C.S. Rao. High order parameter-robust numerical method for singularly perturbed reactiondiffusion problems. Applied Mathematics and Computation, 216(4):1036–1046, 2010. https://doi.org/10.1016/j.amc.2010.01.121
V. Kumar and B. Srinivasan. An adaptive mesh strategy for singularly perturbed convection diffusion problems. Applied Mathematical Modelling, 39(7):2081– 2091, 2015. https://doi.org/10.1016/j.apm.2014.10.019
T. Linß. Layer-adapted meshes for convectiondiffusion problems. Computer Methods in Applied Mechanics and Engineering, 192(9):1061–1105, 2003. https://doi.org/10.1016/S0045-7825(02)00630-8
R.E. Mickens. Advances in the Applications of Nonstandard Finite Difference Schemes. World Scientific, 2005. https://doi.org/10.1142/5884v
A.H. Nayfeh. Perturbation Methods. Wiley, New York, 1985.
O’Malley. Singular Perturbation Methods for Ordinary Differential Equations. Springer Verlag, New York, 1991. https://doi.org/10.1007/978-1-4612-0977-5
N. Petrovic. On a uniform numerical method for a nonlocal problem. Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mathematics, 21(2):133–140, 1991.
H.G. Roos, M. Stynes and L. Tobiska. Robust Numerical Methods Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, 2008. https://doi.org/10.1007/978-3-540-34467-4
D.R. Smith. Singular Perturbation Theory. Cambridge University Press, Cambridge, 1985.
Q. Zheng, X. Li and Y. Gao. Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs. Applied Numerical Mathematics, 91:46–59, 2015. https://doi.org/10.1016/j.apnum.2014.12.010