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Computational treatment of a convection-diffusion type nonlinear system of singularly perturbed differential equations | ||
| Journal of Mathematical Modeling | ||
| مقاله 4، دوره 12، شماره 2، مهر 2024، صفحه 235-246 اصل مقاله (346.78 K) | ||
| نوع مقاله: Research Article | ||
| شناسه دیجیتال (DOI): 10.22124/jmm.2024.25939.2301 | ||
| نویسنده | ||
| Manikandan Mariappan* | ||
| Department of Mathematics, School of Engineering, Presidency University, Bengaluru - 560 064, Karnataka, India | ||
| چکیده | ||
| In this article, a nonlinear system of singularly perturbed differential equations of convection-diffusion type with Dirichlet boundary conditions is considered on the interval $[0,1].$ Both components of the solution of the system exhibit boundary layers near $t = 0.$ A new computational method involving classical finite difference operators, a piecewise-uniform Shishkin mesh and an algorithm based on the continuation method is developed to compute the numerical approximations. The computational method is proved to be first order convergent uniformly with respect to the perturbation parameters. Numerical experiments support the theoretical results. | ||
| کلیدواژهها | ||
| Nonlinear system of singularly perturbed differential equations؛ boundary layers؛ finite difference scheme؛ Shishkin mesh؛ the continuation method؛ parameter-uniform convergence | ||
| مراجع | ||
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[1] E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, Ireland, 1980. [2] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’ Riordan, G.I. Shishkin, Robust Computational Tech- niques for Boundary Layers, Chapman and hall/CRC, Boca Raton, Florida, USA, 2000. [3] P.A. Farrell, E. O’Riordan, G.I. Shishkin, A class of singularly perturbed semilinear differential equations with interior layers, Math. Comput. 74 (2005) 1759–1776. [4] J.L. Gracia, F.J. Lisbona, M. Madaune-Tort, E. O’Riordan, A system of singularly perturbed semi- linear equations, Lect. Notes Comput. Sci. 69 (2009) 163–172. [5] A.F. Hegarty, E. O’Riordan, A numerical method for singularly perturbed convection-diffusion problems posed on smooth domains, J. Sci. Comput. 92 (2022) 84. [6] C. Johnson, R. Rannacher, M. Boman, Numerics and hydrodynamic stability: toward error control in computational fluid dynamics, SIAM J. Numer. Anal. 32 (1995) 1058–1079. [7] S.S. Kalaiselvan, J.J.H. Miller, V. Sigamani, A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations, Math. Commun. 24 (2019) 193–210. [8] N. Kopteva, M. Stynes, A robust adaptive method for a quasi-linear one-dimensional convection- diffusion problem, SIAM J. Numer. Anal. 39 (2002) 1446–1467. [9] N. Kopteva, M. Stynes, Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions, Appl. Numer. Math. 51 (2004) 273–288. [10] M. Mariappan, Computational analysis on a class of singularly perturbed nonlinear differential equations of convection-diffusion type, Submitted to Mediterr. J. Math.. [11] M. Mariappan, A. Tamilselvan, Higher order numerical method for a semilinear system of singu- larly perturbed differential equations, Math. Commun. 26 (2021) 41–52. [12] M. Mariappan, A. Tamilselvan, Higher order computational method for a singularly perturbed nonlinear system of differential equations, J. Appl. Math. Comput. 68 (2022) 1351–1363. [13] J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific Publishing Co., Singapore, New Jersey, London, Hong Kong, 1996. [14] J.M.J. Ortega, W.S. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970 [15] H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equa- tions, Convection-Diffusion and Flow Problems, Springer-Verlag, New York, 1996. [16] L. Shishkina, G.I. Shishkin, Conservative numerical method for a system of semilinear singularly perturbed parabolic reaction-diffusion equations, Math. Modell. Anal. 14 (2009) 211-228 | ||
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