Geodesic
Fitted operator method over Gaussian
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 3, pp. 315-328.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this manuscript, a new exponentially fitted operator strategy for solving a singularly perturbed parabolic partial differential equation with a right boundary layer is considered. We discretize the time variable using the implicit Euler approach and approximate the equation into first order delay differential equation with a small deviating argument using a Taylor series expansion. The two-point Gaussian quadrature formula and linear interpolation are implemented to obtain a tridiagonal system of equations. The tridiagonal system of equations is solved using the Thomas algorithm. Three numerical examples are considered to illustrate the efficiency of the present method and compared with the methods produced by different authors. Convergence of the method is analyzed. The absolute maximum error and rate of convergence are obtained for the model examples. The result shows that the present method is more accurate and $\epsilon$-uniformly convergent for all $\epsilon\leqslant h$. 
@article{SJVM_2022_25_3_a6,
     author = {D. M. Tefera and A. A. Tiruneh and G. A. Derese},
     title = {Fitted operator method over {Gaussian}},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {315--328},
     publisher = {mathdoc},
     volume = {25},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2022_25_3_a6/}
}
TY  - JOUR
AU  - D. M. Tefera
AU  - A. A. Tiruneh
AU  - G. A. Derese
TI  - Fitted operator method over Gaussian
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2022
SP  - 315
EP  - 328
VL  - 25
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2022_25_3_a6/
LA  - ru
ID  - SJVM_2022_25_3_a6
ER  - 
%0 Journal Article
%A D. M. Tefera
%A A. A. Tiruneh
%A G. A. Derese
%T Fitted operator method over Gaussian
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2022
%P 315-328
%V 25
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2022_25_3_a6/
%G ru
%F SJVM_2022_25_3_a6
D. M. Tefera; A. A. Tiruneh; G. A. Derese. Fitted operator method over Gaussian. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 3, pp. 315-328. http://geodesic.mathdoc.fr/item/SJVM_2022_25_3_a6/

[1] E. Bashier, K. Patidar, “A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation”, Appl. Math. Comp., 217:9 (2011), 4728–4739 | DOI | MR

[2] P. Chakravarthy, S. D. Kumar, R. Rao, “An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays”, Ain Shams Engineering J., 8:4 (2017), 663–671 | DOI | MR

[3] P. P. Chakravarthy, K. Kumar, “An adaptive mesh method for time dependent singularly perturbed differential-difference equations”, Nonlinear Engineering, 8 (2019), 328–339 | DOI

[4] C. Clavero, J. L. Gracia, “A higher order uniformly convergent method with Richardson extrapolation in time for singularly perturbed reaction-diffusion parabolic problems”, J. Comp. Appl. Math., 252 (2013), 75–85 | DOI | MR

[5] E. Doolan, J. Miller, W. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980 | MR

[6] M. K. Kadalbajoo, A. Awasthi, “Crank-Nicolson finite difference method based on a midpoint upwind scheme on a non-uniform mesh for time-dependent singularly perturbed convection-diffusion equations”, Intern. J. Comp. Math., 85 (2008), 771–790 | DOI | MR

[7] M. K. Kadalbajoo, A. Awasthi, “The midpoint upwind finite difference scheme for time-dependent singularly perturbed convection-diffusion equations on non-uniform mesh”, Intern. J. Comp. Meth. Engineering Science and Mechanics, 12:3 (2011), 150–159 | DOI | MR

[8] D. Kumar, P. Kumari, “A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay”, J. Appl. Math. Comp., 59 (2019), 179–206 | DOI | MR

[9] K. Kumar, T. Gupta et al, “An Adaptive Mesh Selection Strategy for Solving Singularly Perturbed Parabolic Partial Differential Equations with a Small Delay”, Appl. Math. Scientific Comp., Springer, 2019 | DOI

[10] T. Linß, M. Stynes, “A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems”, Applied Numerical Mathematics, 31 (1999), 255–270 | DOI | MR

[11] J. Miller, E. O'Riordan, G. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, 2012 | MR

[12] R. E. O'Malley, Historical Developments in Singular Perturbations, Springer, 1974 | MR

[13] D. Phaneendra, M. Lalu, “Gaussian quadrature for two-point singularly perturbed boundary value problems with exponential fitting”, Communications in Mathematics and Applications, 10:3 (2019), 447–467 | DOI

[14] Hans-G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, Springer, 2008 | MR

[15] S. Sastry, Introductory Methods of Numerical Analysis, 5th Ed., New Delhi, 2012

[16] J. Singh, S. Kumar, M. Kumar, “A domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problems with time delay”, Numerical Methods for Partial Differential Equations, 34:5 (2018), 1849–1866 | DOI | MR

[17] D. K. Swamy, K. Phaneendra, Y. Reddy, “A fitted non standard finite difference method for singularly perturbed differential difference equations with mixed shifts”, J. Afrikana, 3:4 (2016), 1–20 | MR

[18] A. B. Tesfaye, F. Gemechis, “Fitted operator average finite difference method for solving singularly perturbed parabolic convection-diffusion problems”, Intern. J. Engineering and Appl. Sciences, 11:3 (2019), 414–427 | DOI