698
Views
0
CrossRef citations to date
0
Altmetric
Pure Mathematics

Fitted numerical scheme for singularly perturbed parabolic differential- difference with time lag

& ORCID Icon | (Reviewing editor:)
Article: 2286670 | Received 15 Sep 2023, Accepted 10 Nov 2023, Published online: 19 Dec 2023

References

  • 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, 91–112. https://doi.org/10.1007/s12591-017-0385-3
  • Daba, I. T, & Duressa, G. F. (2021). A hybrid numerical scheme for singularly perturbed parabolic differential-difference equations arising in the modeling of neuronal variability. Computational and Mathematical Methods, 3(5), e1178. https://doi.org/10.1002/cmm4.1178
  • Daba, I. T., & Duressa, G. F. (2021a). Extended cubic B-spline collocation method for singularly perturbed parabolic differential-difference equation arising in computational neuroscience. International Journal for Numerical Methods in Biomedical Engineering, 37(2). https://doi.org/10.1002/cnm.3418
  • Daba, I. T., & Duressa, G. F. (2021b). Hybrid algorithm for singularly perturbed delay parabolic partial differential equations. Applications and Applied Mathematics: An International Journal (AAM), 16(1), 21.
  • Daba, I. T., & Duressa, G. F. (2021c). Uniformly convergent numerical scheme for a singularly perturbed differential-difference equations arising in computational neuroscience. Journal of Applied Mathematics & Informatics, 39(5–6), 655–676.
  • Daba, I. T., & Duressa, G. F. (2022a). Collocation method using artificial viscosity for time dependent singularly perturbed differential–difference equations. Mathematics and Computers in Simulation, 192, 201–220. https://doi.org/10.1016/j.matcom.2021.09.005
  • Daba, I. T., & Duressa, G. F. (2022b). A novel algorithm for singularly perturbed parabolic differential-difference equations. Research in Mathematics, 9(1), 2133211. https://doi.org/10.1080/27684830.2022.2133211
  • Daba, I. T., & Duressa, G. F. (2022c). A robust computational method for singularly perturbed delay parabolic convection-diffusion equations arising in the modeling of neuronal variability. Computational Methods for Differential Equations, 10(2), 475–488.
  • Das, P., & Natesan, S. (2012). Higher-order parameter uniform convergent schemes for Robin type reaction-diffusion problems using adaptively generated grid. International Journal of Computational Methods, 9(4), 1250052. https://doi.org/10.1142/S0219876212500521
  • Das, P., & Natesan, S. (2013). A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value problems. Journal of Applied Mathematics and Computing, 41, 447–471. https://doi.org/10.1007/s12190-012-0611-7
  • Das, P., Rana, S., & Ramos, H. (2019). Homotopy perturbation method for solving Caputo-type fractional-order Volterra-fredholm integro-differential equations. Computational and Mathematical Methods, 1(5), e1047. https://doi.org/10.1002/cmm4.1047
  • Das, P., Rana, S., & Ramos, H. (2020). A perturbation-based approach for solving fractional-order Volterra-fredholm integro differential equations and its convergence analysis. International Journal of Computer Mathematics, 97(10), 1994–2014. https://doi.org/10.1080/00207160.2019.1673892
  • Das, P., Rana, S., & Ramos, H. (2022). On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis. Journal of Computational and Applied Mathematics, 404, 113116. https://doi.org/10.1016/j.cam.2020.113116
  • 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 Mathematics, 148, 79–97. https://doi.org/10.1016/j.apnum.2019.08.028
  • Das, P., & Vigo-Aguiar, J. (2019). Parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter. Journal of Computational and Applied Mathematics, 354, 533–544. https://doi.org/10.1016/j.cam.2017.11.026
  • Das, P. (2015). Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. Journal of Computational and Applied Mathematics, 290, 16–25. https://doi.org/10.1016/j.cam.2015.04.034
  • 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. (2019). An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh. Numerical Algorithms, 81(2), 465–487. https://doi.org/10.1007/s11075-018-0557-4
  • Doolan, E. P., Miller, J. J. H., & Schilders, W. H. A. (1980). Uniform numerical methods for problems with initial and boundary layers. Boole Press.
  • Govindarao, L., & Mohapatra, J. (2020). Numerical analysis and simulation of delay parabolic partial differential equation involving a small parameter. Engineering Computations, 37(1), 289–312. https://doi.org/10.1108/EC-03-2019-0115
  • 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
  • Govindarao, S. R., Mohapatra, J., & Sahu, L. (2019). Uniformly convergent numerical method for singularly perturbed two parameter time delay parabolic problem. International Journal of Applied and Computational Mathematics, 5(3), 1–9. https://doi.org/10.1007/s40819-019-0672-5
  • 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
  • Kumar, D. (2018). An implicit scheme for singularly perturbed parabolic problem with retarded terms arising in computational neuroscience. Numerical Methods for Partial Differential Equations, 34(6), 1933–1952. https://doi.org/10.1002/num.22269
  • Musila, M., & Lánsk’y, P. (1991). Generalized stein’s model for anatomically complex neurons. BioSystems, 25(3), 179–191. https://doi.org/10.1016/0303-2647(91)90004-5
  • Nageshwar Rao, R., & Chakravarthy, P. (2019). Fitted numerical methods for singularly perturbed one-dimensional parabolic partial differential equations with small shifts arising in the modelling of neuronal variability. Differential Equations and Dynamical Systems, 27(1–3), 1–18. https://doi.org/10.1007/s12591-017-0363-9
  • Negero, N. T., & Duressa, G. F. (2022). An efficient numerical approach for singularly perturbed parabolic convection-diffusion problems with large time-lag. Journal of Mathematical Modeling, 10(2), 173–190.
  • Nichols, N. (1989). On the numerical integration of a class of singular perturbation problems. Journal of Optimization Theory and Applications, 60(3), 2050034. https://doi.org/10.1007/BF00940347
  • O’Malley, J. R. E. (1974). Introduction to singular perturbations. Technical report. New York Univ Ny Courant Inst of Mathematical Sciences. https://apps.dtic.mil/sti/citations/AD0787833.
  • Ramesh, V. P., & Priyanga, B. (2019). Higher order uniformly convergent numerical algorithm for time-dependent singularly perturbed differential-difference equations. Differential Equations and Dynamical Systems, 29(1), 239–263. https://doi.org/10.1007/s12591-019-00452-4
  • Reddy, N. R., & Mohapatra, J. (2023). A robust numerical scheme for singularly perturbed delay parabolic initial-boundary value problems involving mixed space shifts. Computational Methods for Differential Equations, 11(1), 42–51.
  • Roos, H.-G., Martin, S., & Tobiska, L. (2008 24). Robust numerical methods for singularly perturbed differential equations: Convection-diffusion-reaction and flow problems. Springer Science & Business Media.
  • Sahu, S. R., & Mohapatra, J. (2021). Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shift. Journal of Applied Analysis, 28(1), 121–134. https://doi.org/10.1515/jaa-2021-2064
  • 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
  • Shakti, D., Mohapatra, J., Das, P., & Vigo-Aguiar, J. (2022). A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction?diffusion problems with arbitrary small diffusion terms. Journal of Computational and Applied Mathematics, 404, 113167. https://doi.org/10.1016/j.cam.2020.113167
  • Shiromani, R., Shanthi, V., & Das, P. (2023). A higher order hybrid-numerical approximation for a class of singularly perturbed two-dimensional convection-diffusion elliptic problem with non-smooth convection and source terms. Computers & Mathematics with Applications, 142, 9–30. https://doi.org/10.1016/j.camwa.2023.04.004
  • Shishkin, G. I., & Shishkina, L. P. (2008). Difference methods for singular perturbation problems. CRC Press.
  • 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., & Kumar, M. (2020). A robust numerical method for a two-parameter singularly perturbed time delay parabolic problem. Computational & Applied Mathematics, 39(3). https://doi.org/10.1007/s40314-020-01236-1
  • Tian, H. (2002). The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag. Journal of Mathematical Analysis and Applications, 270(1), 143–149. https://doi.org/10.1016/S0022-247X(02)00056-2
  • Wang, P. K. C. (1963). Asymptotic stability of a time-delayed diffusion system. Journal of Applied Mechanics, 30(4), 500–504. https://doi.org/10.1115/1.3636609
  • Zahra, W. K., & Ashraf, M. E. M. (2013). Numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline. Journal of King Saud University-Science, 25(3), 201–208. https://doi.org/10.1016/j.jksus.2013.01.003