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Pure Mathematics

An accelerated numerical computation for singularly perturbed Robin-type parabolic problems with small parameters

ORCID Icon & ORCID Icon | (Reviewing editor:)
Article: 2354536 | Received 06 Nov 2023, Accepted 08 May 2024, Published online: 22 May 2024

References

  • Bhathawala, P. H., & Verma, A. P. (1975). A two-parameter singular perturbation solution of one dimension flow through unsaturated porous media. Proceedings of the Indian National Science Academy, 43(5), 380–384.
  • Bigge, J., & Bohl, E. (1985). Deformations of the bifurcation diagram due to discretization. Mathematics of Computation, 45(172), 393–403. http://dx.doi.org/10.1090/S0025-5718-1985-0804931-X
  • Bullo, T. A., Degla, G. A., & Duressa, G. F. (2022). Parameter-uniform finite difference method for singularly perturbed parabolic problem with two small parameters. International Journal for Computational Methods in Engineering Sciences and Mechanics, 23(3), 210–218. https://doi.org/10.1080/15502287.2021.1948148
  • Bullo, T. A., Duressa, G. F., & Delga, G. A. (2021a). A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters. European Journal of Computational Mechanics, 30(2–3), 197–222. https://doi.org/10.13052/ejcm2642-2085.30233
  • Bullo, T. A., Duressa, G. F., & Delga, G. A. (2021b). Robust finite difference method for singularly perturbed two-parameter parabolic convection-diffusion problems. International Journal of Computational Methods, 18(2), 2050034. https://doi.org/10.1142/S0219876220500346
  • Chandru, M., Das, P., & Ramos, H. (2018). Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data. Mathematical Methods in the Applied Sciences, 41(14), 5359–5387. https://doi.org/10.1002/mma.5067
  • Chandru, M., Prabha, T., Das, P., & Shanthi, V. (2019). A numerical method for solving boundary and interior layers dominated parabolic problems with discontinuous convection coefficient and source terms. Differential Equations and Dynamical Systems, 27(1–3), 91–112. https://doi.org/10.1007/s12591-017-0385-3
  • Chen, J., & O’Malley, R. E., Jr. (1974). On the asymptotic solution of a two-parameter boundary value problem of chemical reactor theory. SIAM Journal of Applied Mathematics, 26(4), 717–729. https://doi.org/10.1137/0126064
  • Daba, I. T., Duressa, G. F., & Liu, L. (2022). A novel algorithm for singularly perturbed parabolic differential-difference equations. Research in Mathematics, 9(1), 2133211. https://doi.org/10.1080/27684830.2022.2133211
  • Das, P. (2018). A higher order difference method for singularly perturbed parabolic partial differential equations. Journal of Difference Equations and Applications, 24(3), 452–477. https://doi.org/10.1080/10236198.2017.1420792
  • 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
  • Das, P., & Natesan, S. (2013a). Richardson extrapolation method for singularly perturbed convection-diffusion problems on adaptively generated mesh. CMES, 90(6), 463–485. https://doi.org/10.3970/cmes.2013.090.463.html
  • Das, P., & Natesan, S. (2013b). A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value problems. Journal of Applied Mathematics and Computing, 41(1–2), 447–471. https://doi.org/10.1007/s12190-012-0611-7
  • Das, P., Rana, S., & Vigo-Aguiar, J. (2020). Higher order accurate approximations on equidistributed meshes for boundary layer originated mixed type reaction diffusion systems with multiple scale nature. Applied Numerical Mathematic, 148, 79–97. https://doi.org/10.1016/j.apnum.2019.08.028
  • DiPrima, R. C. (1968). Asymptotic methods for an infinitely long slider squeeze-film bearing. Journal of Lubrication Technology, 90(1), 173–183. https://doi.org/10.1115/1.3601534
  • Duressa, G. F., Gelu, F. W., & Kebede, G. D. (2023). A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems. Results in Applied Mathematics, 20, 100405. https://doi.org/10.1016/j.rinam.2023.100405
  • Gelu, F. W., & Duress, G. F. (2022). Parameter-uniform numerical scheme for singularly perturbed parabolic convection-diffusion Robin type problems with a boundary turning point. Results in Applied Mathematics, 15(2), 100324. https://doi.org/10.1016/j.rinam.2022.100324
  • Gelu, F. W., & Duressa, G. F. (2022a). Computational method for singularly perturbed parabolic reaction-diffusion equations with Robin boundary conditions. Journal of Applied Mathematics & Informatics, 40(1–2), 25–45. https://doi.org/10.14317/jami.2022.025
  • Gelu, F. W., & Duressa, G. F. (2023a). Hybrid method for singularly perturbed Robin type parabolic convection–diffusion problems on Shishkin mesh. Partial Differential Equations in Applied Mathematics, 8, 100586. https://doi.org/10.1016/j.padiff.2023.100586
  • Gelu, F. W., & Duressa, G. F. (2023b). A parameter-uniform numerical method for singularly perturbed Robin type parabolic convection-diffusion turning point problems. Applied Numerical Mathematics, 190, 50–64. https://doi.org/10.1016/j.apnum.2023.04.007
  • Gelu, F. W., Duressa, G. F., & Kovtunenko, V. (2021). A uniformly convergent collocation method for singularly perturbed delay parabolic reaction-diffusion problems. Abstract and Applied Analysis, 2021, 1–11. https://doi.org/10.1155/2021/8835595
  • Gelu, F. W., Duressa, G. F., & Shah, F. A. (2022b). A novel numerical approach for singularly perturbed parabolic convection-diffusion problems on layer-adapted meshes. Research in Mathematics, 9(1), 2020400. https://doi.org/10.1080/27658449.2021.2020400
  • Gupta, V., Kadalbajoo, M. K., & Dubey, R. K. (2019). A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters. International Journal of Computer Mathematics, 96(3), 474–499. https://doi.org/10.1080/00207160.2018.1432856
  • Hailu, W. S., Duressa, G. F., & Liu, L. (2022a). Parameter-uniform cubic spline method for singularly perturbed parabolic differential equation with large negative shift and integral boundary condition. Research in Mathematics, 9(1), 2151080. https://doi.org/10.1080/27684830.2022.2151080
  • Hailu, W. S., Duressa, G. F., & Liu, L. (2022b). Uniformly convergent numerical method for singularly perturbed parabolic differential equations with non-smooth data and large negative shift. Research in Mathematics, 9(1), 2119677. https://doi.org/10.1080/27684830.2022.2119677
  • Hassen, Z. I., Duressa, G. F., & Liu, L. (2023). New approach of convergent numerical method for singularly perturbed delay parabolic convection-diffusion problems. Research in Mathematics, 10(1), 2225267. https://doi.org/10.1080/27684830.2023.2225267
  • Hemker, P. W., Shishkin, G. I., & Shishkina, L. P. (2002). High-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with Robin boundary conditions. Computational Methods in Applied Mathematics, 2(1), 3–25. https://doi.org/10.2478/cmam-2002-0001
  • Janani Jayalakshmi, G., & Tamilselvan, A. (2020). An ɛ-uniform method for a class of singularly perturbed parabolic problems with Robin boundary conditions having boundary turning point. Asian-European Journal of Mathematics, 13(1), 2050025. https://doi.org/10.1142/S1793557120500254
  • 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
  • Kadalbajoo, M. K., & Yadaw, A. S. (2012). Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems. International Journal of Computational Methods, 09(4), 1250047–1)-(1250047–16. https://doi.org/10.1142/S0219876212500478
  • Kumar, D., & Deswal, K. (2022). Wavelet-based approximation for two-parameter singularly perturbed problems with Robin boundary conditions. Journal of Applied Mathematics and Computing, 68(1), 125–149. https://doi.org/10.1007/s12190-021-01511-2
  • Kumar, S., Aakansha, S. J., & Ramos, H. (2023). Parameter-uniform convergence analysis of a domain decomposition method for singularly perturbed parabolic problems with Robin boundary conditions. Journal of Applied Mathematics and Computing, 69, 2239–2261. https://doi.org/10.1007/s12190-022-01832-w
  • Mbroh, N. A., Noutchie, S. C. O., & Massoukou, R. Y. M. (2020). A uniformly convergent finite difference scheme for Robin type singularly perturbed parabolic convection diffusion problem. Mathematics and Computers in Simulation, 174, 218–232. https://doi.org/10.1016/j.matcom.2020.03.003
  • Mekonnen, T. B., Duressa, G. F., & Liu, L. (2020). Computational method for singularly perturbed two-parameter parabolic convection-diffusion problems. Cogent Mathematics & Statistics, 7(1), 1829277. https://doi.org/10.1080/25742558.2020.1829277
  • Mekonnen, T. B., Duressa, G. F., & Liu, L. (2021). Uniformly convergent numerical method for two-parametric singularly perturbed parabolic convection-diffusion problems. Journal of Applied and Computational Mechanics, 7(1), 1829277–1829545. https://doi.org/10.1080/25742558.2020.1829277
  • Mickens, R. E. (1994). Nonstandard finite difference models of differential equations. World Scientific.
  • Mickens, R. E. (2005). Advances in the applications of nonstandard finite difference schemes. World Scientific.
  • Munyakazi, J. B. (2015). A robust finite difference method for two-parameter parabolic convection-diffusion problems. Applied Mathematics and Information Sciences, 9(6), 2877–2883. https://doi.org/10.12785/amis/090614
  • O’Malley, R. E., Jr. (1967a). Singular perturbations of boundary value problems for linear ordinary differential equations involving two-parameters. Journal of Mathematical Analysis and Applications, 19(2), 291–308. https://doi.org/10.1016/0022-247X(67)90124-2
  • O’Malley, R. E., Jr. (1967b). Two-parameter singular perturbation problems for second-order equations. Journal of Mathematics and Mechanics, 16(10), 1143–1164. https://www.jstor.org/stable/45277141
  • O’Malley, R. E., Jr. (1969). Boundary value problems for linear systems of ordinary differential equations involving many small parameters. Journal of Mathematics and Mechanics, 18, 835–856. https://www.jstor.org/stable/24901722
  • O’Riordan, E., Pickett, M. L., & Shishkin, G. I. (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
  • Patidar, K. C. (2008). A robust fitted operator finite difference method for a two-parameter singular perturbation problem 1. Journal of Difference Equations and Applications, 14(12), 1197–1214. https://doi.org/10.1080/10236190701817383
  • Patidar, K. C., & Sharma, K. K. (2006). Uniformly convergent nonstandard finite difference methods for singularly perturbed differential-difference equations with delay and advance. International Journal Numerical Methods in Engineering, 66(2), 272–296. https://doi.org/10.1002/nme.1555
  • Saini, S., Das, P., & Kumar, S. (2023). Computational cost reduction for coupled system of multiple scale reaction diffusion problems with mixed type boundary conditions having boundary layers. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A Matemáticas, 117(2), 66. https://doi.org/10.1007/s13398-023-01397-8
  • Saini, S., Das, P., & Kumar, S. (2024). Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time. Applied Numerical Mathematics, 196, 1–21. https://doi.org/10.1016/j.apnum.2023.10.003
  • Selvi, P. A., & Ramanujam, N. (2017). A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition. Applied Mathematics and Computation, 296, 101–115. https://doi.org/10.1016/j.amc.2016.10.027
  • Srivastava, H. M., Nain, A. K., Vats, R. K., & Das, P. (2023). A theoretical study of the fractional-order p-Laplacian nonlinear Hadamard type turbulent flow models having the Ulam–Hyers stability. Revista de la Real Academia de Ciencias Exactas, Físicas Y Naturales, Serie A Matemáticas, 117(4), 160. https://doi.org/10.1007/s13398-023-01488-6
  • Sumit, K., Kuldeep, S., & Kumar, M. (2020). A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem. Computational and Applied Mathematics, 39, 209. https://doi.org/10.1007/s40314-020-01236-1
  • Sunil, K., Sumit, & Ramos, H. (2021). Parameter-uniform approximation on equidistributed meshes for singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions. Applied Mathematics Computation, 392, 125677. https://doi.org/10.1016/j.amc.2020.125677
  • Vasil’eva, A. B. (1963). Asymptotic methods in the theory of ordinary differential equations containing small parameters in front of the highest derivatives. USSR Computational Mathematics and Mathematical Physics, 3(4), 823–863. https://doi.org/10.1016/0041-5553(63)90381-1