Layer adapted finite element method for two-parameter singularly perturbed delay parabolic problems

References

• Ayele, M. A., Tiruneh, A. A., & Derese, G. A. (2023). Fitted cubic spline scheme for two-parameter singularly perturbed time-delay parabolic problems. Results in Applied Mathematics, 18, 100361. https://doi.org/10.1016/j.rinam.2023.100361
   Google Scholar
• Bashier, E., & Patidar, K. (2011). A second-order fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. Journal of Difference Equations and Applications, 17(5), 779–794. https://doi.org/10.1080/10236190903305450
   Web of Science ®Google Scholar
• Bhathawala, P., & Verma, A. (1975). A two-parameter singular perturbation solution of one dimension flow through unsaturated porous media. Applications of Mathematics, 43(5), 380–384.
   Google Scholar
• Bullo, T., Duressa, G., & Degla, G. (2019). Higher order fitted operator finite difference method for two-parameter parabolic convection-diffusion problems. International Journal of Engineering and Applied Sciences, 11(4), 455–467. https://doi.org/10.24107/ijeas.644160
   Google Scholar
• Das, A., & Natesan, S. (2015). Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection–diffusion problems on Shishkin mesh. Applied Mathematics and Computation, 271, 168–186. https://doi.org/10.1016/j.amc.2015.08.137
   Web of Science ®Google Scholar
• Das, P., & Mehrmann, V. (2016). Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters. BIT Numerical Mathematics, 56(1), 51–76. https://doi.org/10.1007/s10543-015-0559-8
   Web of Science ®Google Scholar
• Feng, J. Q. (1990). A method of multiple-parameter perturbations with an application to drop oscillations in an electric field. Quarterly of Applied Mathematics, 48(3), 555–567. https://doi.org/10.1090/qam/1074971
   Web of Science ®Google Scholar
• Gao, X. W., Zhu, Y. M., & Pan, T. (2023). Finite line method for solving high-order partial differential equations in science and engineering. Partial Differential Equations in Applied Mathematics, 7, 100477. https://doi.org/10.1016/j.padiff.2022.100477
   Google Scholar
• Govindarao, L., Sahu, S. R., & Mohapatra, J. (2019). Uniformly convergent numerical method for singularly perturbed time delay parabolic problem with two small parameters. Iranian Journal of Science & Technology, Transactions A: Science, 43(5), 2373–2383. https://doi.org/10.1007/s40995-019-00697-2
   Google Scholar
• Grossmann, C., Roos, H. G., & Stynes, M. (2007). Numerical treatment of partial differential equations (Vol. 154). Springer.
   Google Scholar
• Heywood, J. G., & Rannacher, R. (1990). Finite-element approximation of the nonstationary navier–Stokes problem. Part IV: Error analysis for second-order time discretization. SIAM Journal on Numerical Analysis, 27(2), 353–384. https://doi.org/10.1137/0727022
   Web of Science ®Google Scholar
• Jha, A., & Kadalbajoo, M. K. (2015). A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems. International Journal of Computer Mathematics, 92(6), 1204–1221. https://doi.org/10.1080/00207160.2014.928701
   Web of Science ®Google Scholar
• Kadalbajoo, M., & Yadaw, A. S. (2012). Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems. International Journal of Computational Methods, 9(4), 1250047. https://doi.org/10.1142/S0219876212500478
   Web of Science ®Google Scholar
• Kadalbajoo, M. K., & Gupta, V. (2010). A brief survey on numerical methods for solving singularly perturbed problems. Applied Mathematics and Computation, 217(8), 3641–3716. https://doi.org/10.1016/j.amc.2010.09.059
   Web of Science ®Google Scholar
• Kumar, S., Kumar, M., et al. (2020). A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem. Computational and Applied Mathematics, 39(3), 1–25. https://doi.org/10.1007/s40314-020-01236-1
   Google Scholar
• Li, Z., Qiao, Z., & Tang, T. (2017). Numerical solution of differential equations: Introduction to finite difference and finite element methods. Cambridge University Press.
   Google Scholar
• Mbroh, N. A., Noutchie, S. C. O., & Massoukou, R. Y. M. (2020). A robust method of lines solution for singularly perturbed delay parabolic problem. Alexandria Engineering Journal, 59(4), 2543–2554. https://doi.org/10.1016/j.aej.2020.03.042
   Web of Science ®Google Scholar
• Miller, J., O’Riordan, E., Shishkin, G., & Shishkina, L. (1998). Fitted mesh methods for problems with parabolic boundary layers. In Mathematical Proceedings of the Royal Irish Academy (pp. 173–190). https://www.jstor.org/stable/20459730
   Google Scholar
• Mohye, M. A., Munyakazi, J. B., & Dinka, T. G. (2023). A nonstandard fitted operator finite difference method for two-parameter singularly perturbed time-delay parabolic problems. Frontiers in Applied Mathematics and Statistics, 9, 1222162. https://doi.org/10.3389/fams.2023.1222162
   Google Scholar
• Munyakazi, J. B. (2015). A robust finite difference method for two-parameter parabolic convection-diffusion problems. Applied Mathematics and Information Sciences, 9(6), 2877.
   Google Scholar
• Negero, N. T., & Duressa, G. F. (2021). A method of line with improved accuracy for singularly perturbed parabolic convection–diffusion problems with large temporal lag. Results in Applied Mathematics, 11, 100174. https://doi.org/10.1016/j.rinam.2021.100174
   Google Scholar
• Negero, N. T., Duressa, G. F., Rathour, L., & Mishra, V. N. (2023). A novel fitted numerical scheme for singularly perturbed delay parabolic problems with two small parameters. Partial Differential Equations in Applied Mathematics, 8, 100546. https://doi.org/10.1016/j.padiff.2023.100546
   Google Scholar
• O’Malley, R. (1967). Two-parameter singular perturbation problems for second-order equations. Journal of Mathematics and Mechanics, 16(10), 1143–1164. https://www.jstor.org/stable/45277141
   Google Scholar
• O’Riordan, E., Pickett, M., & Shishkin, G. (2006). Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems. Mathematics of Computation, 75(255), 1135–1154. https://doi.org/10.1090/S0025-5718-06-01846-1
   Web of Science ®Google Scholar
• Rajan, M., & Reddy, G. (2022). A generalized regularization scheme for solving singularly perturbed parabolic PDEs. Partial Differential Equations in Applied Mathematics, 5, 100270. https://doi.org/10.1016/j.padiff.2022.100270
   Google Scholar
• Reddy, J. N. (2019). Introduction to the finite element method. McGraw-Hill Education.
   Google Scholar
• Salama, A., & Al-Amery, D. (2017). A higher order uniformly convergent method for singularly perturbed delay parabolic partial differential equations. International Journal of Computer Mathematics, 94(12), 2520–2546. https://doi.org/10.1080/00207160.2017.1284317
   Web of Science ®Google Scholar
• Singh, S., Kumari, P., & Kumar, D. (2022). An effective numerical approach for two parameter time-delayed singularly perturbed problems. Computational and Applied Mathematics, 41(8), 337. https://doi.org/10.1007/s40314-022-02046-3
   Google Scholar
• Surana, K. S., & Reddy, J. N. (2016). The finite element method for boundary value problems: Mathematics and computations. CRC press.
   Google Scholar
• Woldaregay, M. M., Aniley, W. T., Duressa, G. F., & Macias-Diaz, J. E. (2021). Novel numerical scheme for singularly perturbed time delay convection-diffusion equation. Advances in Mathematical Physics, 2021, 1–13. https://doi.org/10.1155/2021/6641236
   Web of Science ®Google Scholar
• Worku, S. W., Duressa, G. F., et al. (2023). A numerical approach for diffusion-dominant two-parameter singularly perturbed delay parabolic differential equations. International Journal of Mathematics and Mathematical Sciences, 2023, 1–17. https://doi.org/10.1155/2023/6620669
   Google Scholar