An exponentially fitted method for two parameter singularly perturbed parabolic boundary value problems
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
This article devises an exponentially fitted method for the numerical solution of two parameter singularly perturbed parabolic boundary value problems. The proposed scheme is able to resolve the two lateral boundary layers of the solution. Error estimates show that the constructed scheme is parameter-uniformly convergent with a quadratic numerical rate of convergence. Some numerical test examples are taken from recently published articles to confirm the theoretical results and demonstrate a good performance of the current scheme.
Abstract
This article devises an exponentially fitted method for the numerical solution of two parameter singularly perturbed parabolic boundary value problems. The proposed scheme is able to resolve the two lateral boundary layers of the solution. Error estimates show that the constructed scheme is parameter-uniformly convergent with a quadratic numerical rate of convergence. Some numerical test examples are taken from recently published articles to confirm the theoretical results and demonstrate a good performance of the current scheme.
참고문헌 (32)
- P. H. Bhathawala, "A two-parameter singular perturbation solution of one dimension ow through unsaturated porous media", Appl. Math, 43, pp.380~384 (1975)
- T. A. Bullo, "Higher order fitted operator finite difference method for two-parameter parabolic convection-diffusion problems", Inter. J. Engin. App. Sci, 11, pp.455~467 (2019)
- T. A. Bullo, "Robust finite difference method for sin-gularly perturbed two-parameter parabolic convection-diffusion problems", Int. J. Comput. Methods, 18(2), pp.2050034~2050050 (2021)[DOI]
- P. P. Chakravarthy, "Numerical study of a singularly perturbed two parameter problems on a modified Bakhvalov mesh", Comput. Math. Math. Phys, 60(11), pp.1778~1786 (2020)[DOI]
- I. T. Daba, "Extended cubic B-spline collocation method for singu-larly perturbed parabolic differential-difference equation arising in computational neu-roscience", Int. J. Numer. Methods Biomed. Eng, 37(2), pp.e3418~e3437 (2021)[DOI]
- I. T. Daba, "A robust computational method for singularly per-turbed delay parabolic convection-diffusion equations arising in the modeling of neu-ronal variability", Comput. Methods Differ. Equ, 10(2), pp.475~488 (2022)[DOI]
- P. Das, "A higher order difference method for singularly perturbed parabolic partial differential equations", J. Difference Equ. Appl, 24(3), pp.452~477 (2018)[DOI]
- P. Das, "Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters", BIT, 56(1), pp.51~76 (2016)[DOI]
- E. P. Doolan, "Uniform Numerical Methods for Problems with Initial and Boundary Layers" (1980), Boole Press
- K. Ganesh, "Computational technique for two parameter singu-larly perturbed parabolic convection-diffusion problem", J. Math. Comput. Sci, 10, pp.1251~1261 (2020)
- V. Gupta, "A parameter-uniform higher order nite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters", Int. J. Comput. Math, 96(3), pp.474~499 (2019)[DOI]
- S. Y. Hahn, "An upwind finite element method for elec-tromagnetic field problems in moving media", Inter. J. Numer. Meth. Engin, 24, pp.2071~2086 (1987)
- M. K. Kadalbajoo, "Crank-Nicolson finite difference method based on a midpoint upwind scheme on a non-uniform mesh for time-dependent singularly per-turbed convection-diffusion equations", Int. J. Comput. Math, 85(5), pp.771~790 (2008)[DOI]
- M. K. Kadalbajoo, "Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems", Int. J. Comput. Methods, 9(4, 1250047), pp.16~ (2012)[DOI]
- V. Kumar, "A novel adaptive mesh strategy for singularly perturbed parabolic convection diffusion problems", Differ. Equ. Dyn. Syst, 27(1-3), pp.203~220 (2019)[DOI]
- T. Lin, "Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters", J. Math. Anal. Appl, 289(2), pp.355~366 (2004)[DOI]
- T. B. Mekonnen, "Computational method for singularly perturbed two-parameter parabolic convection-diffusion problems", Cogent Math. & Stat, 7, pp.1829277~ (2020)
- T. B. Mekonnen, "Uniformly convergent numerical method for two-parametric singularly perturbed parabolic convection-diffusion problems", J. Appl. Comput. Mech, 7, pp.535~545 (2021)
- T. B. Mekonnen, "A fitted mesh cubic spline in tension method for singularly perturbed problems with two parameters", Int. J. Math. Math. Sci, 2022, pp.5410754~5410764 (2022)[DOI]
- J. B. Munyakazi, "A robust finite difference method for two-parameter parabolic convection-diffusion problems", Appl. Math. & Info. Sci, 9, pp.2877~ (2015)
- N. K. Nichols, "On the numerical integration of a class of singular perturbation prob-lems", J. Optim. Theory Appl, 60(3), pp.439~452 (1989)[DOI]
- R. E. O'Malley, Jr., "Singular perturbations of boundary value problems for linear ordi-nary differential equations involving two parameters", J. Math. Anal. Appl, 19, pp.291~308 (1967)[DOI]
- R. E. O'Malley, Jr., "Topics in singular perturbations", Adv. in Math, 2, pp.365~470 (1968)[DOI]
- R. E. O'Malley, Jr., "Singular perturbation methods for ordinary differential equations, Applied Mathematical Sciences, 89" (1991), Springer-Verlag
- E. O'Riordan, "Numerical approximations to the scaled first derivatives of the solution to a two parameter singularly perturbed problem", J. Comput. Appl. Math, 347, pp.128~149 (2019)[DOI]
- E. O'Riordan, "Singularly perturbed problems modeling reaction-convection-diffusion processes", Comput. Methods Appl. Math, 3(3), pp.424~442 (2003)[DOI]
- E. O'Riordan, "Parameter-uniform finite differ-ence schemes for singularly perturbed parabolic diffusion-convection-reaction problems", Math. Comp, 75(255), pp.1135~1154 (2006)[DOI]
- R. Ranjan, "A novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts", J. Appl. Math. Comput, 65(1-2), pp.403~427 (2021)[DOI]
- H. G. Roos, "Layer-adapted meshes : Milestones in 50 years of history", arXiv : 1909. 08273, ~ (2019)
- M. Shivhare, "A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters", J. Math. Chem, 59(1), pp.186~215 (2021)[DOI]
- S. Yuzbasl, "Numerical solutions of singularly perturbed one-dimensional parabolic convection-diffusion problems by the Bessel collocation method", Appl. Math. Comput, 220, pp.305~315 (2013)[DOI]
- Jin Zhang, "High-order finite element method on a Bakhvalov-type mesh for a singularly perturbed convection–diffusion problem with two parameters", Applied Mathematics and Computation, 397, pp.125953~ (2021), Elsevier BV[DOI]
