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Computational method for a nonlinear system of two-parameter singular perturbation problems modeling reaction-convection-diffusion processes
https://doi.org/10.26907/0021-3446-2025-8-34-55
Abstract
This article aims at the construction and analysis of a computational method for a system of two-parameter singularly perturbed second-order nonlinear differential equations with prescribed boundary conditions modeling reaction-convection-diffusion processes. A fitted mesh method consists of a classical finite difference scheme together with a Shishkin mesh constructed to solve the system. The fitted mesh method is proved to be convergent essentially first-order uniformly with respect to the perturbation parameters. An algorithm using the continuation method is designed to compute the numerical approximations. Numerical experiments support the theoretical results. Since there is no literature on systems of two-parameter singularly perturbed nonlinear differential equations, the present study reveals the characteristics of such systems and contributes to their numerical solution.
References
1. Bhol E. Finite Modele Gewöhnlicher Randwertaufgaben (Teubner, Stuttgart, 1981).
2. Chen J., O'Malley R.E. "On the asymptotic solution of a two parameter boundary value problem of chemical reactor theory", SIAM J. Appl. Math. 26, 717-729 (1974).
3. O'Malley R.E. "Two-parameter singular perturbation problems for second order equation", J. Math. Mech. 16, 1143-1164 (1967).
4. Shivaranjani Nagarajan "A parameter robust fitted mesh finite difference method for a system of two reaction-convection-diffusion equations", Appl. Numerical Math. 179, 87-104 (2022).
5. Aarthika K., Shanthi V., Higinio Ramos "A computational approach for a two-parameter singularly perturbed system of partial differential equations with discontinuous coefficients", Appl. Math. Comput. 434 (1), 1-15 (2022).
6. Rejla Vulanovic "A higher order scheme for quasilinear boundary value problems with two small parameters", Computing 67, 287-303 (2001).
7. Zahra W.K., Ashraf M. El Mhlawy "Numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline", J. King Saud Univ. - Sci. 25 (3), 201-208 (2013).
8. Manikandan Mariappan "A fitted mesh method for a class of two parameter nonlinear singular perturbation problems", J. Math. Model. 12 (1), 33-49 (2024).
9. Ortega J.M.J., Rheinboldt W.S. Iterative solution of nonlinear equations in several variables (Acad. Press, New York, 1970).
10. Manikandan Mariappan, Ayyadurai Tamilselvan "An efficient numerical method for a nonlinear system of singularly perturbed differential equations arising in a two-time scale system", J. Appl. Math. Comput. 68, 1069-1086 (2022).
11. Saravana Sankar Kalaiselvan, Miller J.J.H., Valarmathi Sigamani "A parameter uniform numerical method for a singularly perturbed two-parameter delay differential equation", Appl. Numerical Math. 145, 90-110 (2019).
12. Miller J.J.H, O'Riordan E., Shishkin G.I. Fitted Numerical Methods for Singular Perturbation Problems (World Sci. Publ. Co., Singapore, New Jersey, London, Hong Kong, 1996).
13. Farrell P.A., Hegarty A.F., Miller J.J.H., O'Riordan E., Shishkin G.I. Robust computational techniques for boundary layers (Chapman and hall/CRC, Boca Raton, Florida, USA, 2000).
Review
For citations:
Mariappan M. Computational method for a nonlinear system of two-parameter singular perturbation problems modeling reaction-convection-diffusion processes. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2025;1(8):34-55. (In Russ.) https://doi.org/10.26907/0021-3446-2025-8-34-55
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