A novel algorithm for singularly perturbed parabolic differential-difference equations

References

• Andre Longtin, J. G. M. (1988). Complex oscillations in the human pupil light reflex with mixed and delayed feedback. Mathematical Biosciences, 90(1–2), 183–16. https://doi.org/10.1016/0025-5564(88)90064-8
   Web of Science ®Google Scholar
• Bansal, K., & Kapil, K. (2017). Sharma Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments. Numerical Algorithms, 75(1), 113–145. https://doi.org/10.1007/s11075-016-0199-3
   Web of Science ®Google Scholar
• Bansal, K., & Pratima Rai, K. K. S. (2017). Numerical treatment for the class of time dependent singularly perturbed parabolic problems with general shift arguments. Differential Equations and Dynamical Systems, 25(2), 327–436. https://doi.org/10.1007/s12591-015-0265-7
   Web of Science ®Google Scholar
• Culshaw, R. V., & Ruan, S. (2000). A delay-differential equation model of HIV infection of CD4+ T-cells. Mathematical Biosciences, 165(1), 27–39. https://doi.org/10.1016/S0025-5564(00)00006-7
   PubMed Web of Science ®Google Scholar
• Daba, I. (2021a). Takele and Duressa, gemechis file, 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), e3418. https://doi.org/10.1002/cnm.3418
   Web of Science ®Google Scholar
• Daba, I. (2021b). Takele and Duressa, gemechis file,hybrid algorithm for singularly perturbed delay parabolic partial differential equations. Applications and Applied Mathematics: An International Journal (AAM), 16(1), 21. http://pvamu.edu/aam.
   Web of Science ®Google Scholar
• Daba, I. (2021c). Takele and Duressa, Gemechis file, 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
   Google Scholar
• Daba, I. (2021d). Takele and Duressa, Gemechis file, uniformly convergent numerical scheme for a singularly perturbed differential-difference equations arising in computational neuroscience. Journal of Applied Mathematics and Informatics, 39(5–6), 655–676. https://doi.org/10.1002/cmm4.1178
   Web of Science ®Google Scholar
• Daba, I. (2022a). Takele and Duressa, Gemechis file. Collocation Method Using Artificial Viscosity for Time Dependent Singularly Perturbed differential–difference Equations Mathematics and Computers in Simulation, 192, 201–220,. https://doi.org/10.1002/cmm4.1178
   Google Scholar
• Daba, I. (2022b). Takele and Duressa, Gemechis File,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. https://cmde.tabrizu.ac.ir/article_12797.html
   Web of Science ®Google Scholar
• Derstine, M. W., Gibbs, H. M., Hopf, F. A., & Kaplan, D. L. (1982). Bifurcation gap in a hybrid optically bistable system. Physical Review A, 26(6), 3720–3722. https://doi.org/10.1103/PhysRevA.26.3720
   Web of Science ®Google Scholar
• Edward, P. D., John, J. H., & Miller, Willy, H. A. (1980). Schilders, Uniform numerical methods for problems with initial and boundary layers. Boole Press.
   Google Scholar
• Glizer, V. Y. (2000). Asymptotic solution of a boundary-value problem for linear singularly-perturbed functional differential equations arising in optimal control theory. Journal of Optimization Theory and Applications, 106(2), 309–335. https://doi.org/10.1023/A:1004651430364
   Web of Science ®Google Scholar
• Glizer, V. Y. (2003). Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution. Journal of Mathematical Analysis and Applications, 278(2), 409–433. https://doi.org/10.1016/S0022-247X(02)00715-1
   Web of Science ®Google Scholar
• John Mallet-Paret, R. D. (1989). Nussbaum. A differential-delay Equation Arising in Optics and Physiology, SIAM Journal on Mathematical Analysis, 20(2), 249–292. https://doi.org/10.1137/0520019
   Google Scholar
• John Wilbur, W., & Rinzel, J. (1982). An analysis of Stein’s model for stochastic neuronal excitation. Biological Cybernetics, 45(2), 107–114. https://doi.org/10.1007/BF00335237
   Web of Science ®Google Scholar
• Khan, A., Khan, I., & Tariq Aziz, M. (2004). Stojanovic. A variable-mesh Approximation Method for Singularly Perturbed boundary-value Problems Using Cubic Spline in Tension, International Journal of Computer Mathematics, 81(12), 1513–1518. https://doi.org/10.1080/00207160412331284169
   Google Scholar
• Kostikov, Y. A., & Romanenkov, A. M. (2020). Approximation of the multidimensional optimal control problem for the heat equation (applicable to computational fluid dynamics (CFD)). Civil Engineering Journal, 6(4), 743–768. https://doi.org/10.28991/cej-2020-03091506
   Google Scholar
• Kot, M. (2001). Elements of mathematical ecology. Cambridge University Press.
   Google Scholar
• Kuang, Y. (1993). Delay differential equations: With applications in population dynamics. cademic press.
   Google Scholar
• 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
   Web of Science ®Google Scholar
• Kumar, M., & Gupta, Y. (2010). Methods for solving singular boundary value problems using splines: A review. Journal of Applied Mathematics and Computing, 32(1), 265–278. https://doi.org/10.1007/s12190-009-0249-2
   Google Scholar
• Ladyzhenskaia, O. A., Solonnikov, V. A., & Nina, N. (1968). Ural’tseva, Linear and quasi-linear equations of parabolic type, 23. American Mathematical Soc.
   Google Scholar
• Lánskỳ, P. (1984). On approximations of Stein’s neuronal model. Journal of Theoretical Biology, 107(4), 631–647. https://doi.org/10.1016/S0022-5193(84)80136-8
   Web of Science ®Google Scholar
• Mackey, M. C., & Glass, L. (1977). Oscillation and chaos in physiological control systems. Science, 197(4300), 287–289. https://doi.org/10.1126/science.267326
   PubMed Web of Science ®Google Scholar
• Mayer, H., Zaenker, K. S., & An Der Heiden, U. (1995). A basic mathematical model of the immune response, Chaos: An interdisciplinary. Journal of Nonlinear Science, 5(1), 155–161. https://aip.scitation.org/doi/abs/10.1063/1.166098
   Google Scholar
• MihajloviC, G., & Zivkovic, M. (2020). Sieving extremely wet earth mass by means of oscillatory transporting platform. Emerging Science Journal, 4(3), 172–182. https://doi.org/10.28991/esj-2020-01221
   Google Scholar
• Murray, J. D. (2007). Mathematical biology: I. An Introduction, 17. https://link.springer.com/content/pdf/10.1007/b98868.pdf%20http://www.ulb.tu-darmstadt.de/tocs/127987207.pdf
   Google Scholar
• Musila, M., & Lánskỳ, 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
   PubMed Web of Science ®Google Scholar
• Nageshwar Rao, R., & Pramod 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
   Web of Science ®Google Scholar
• Nelson, P. W., & Perelson, A. S. (2002). Mathematical analysis of delay differential equation models of HIV-1 infection. Mathematical Biosciences, 179(1), 73–94. https://doi.org/10.1016/S0025-5564(02)00099-8
   PubMed Web of Science ®Google Scholar
• Nichols, N. K. (1989). On the numerical integration of a class of singular perturbation problems. Journal of Optimization Theory and Applications, 60(3), 439–452. https://doi.org/10.1007/BF00940347
   Web of Science ®Google Scholar
• O’Malley, R. E. (1991). Singular perturbation methods for ordinary differential equations. New York: Springer.
   Google Scholar
• Pramod Chakravarthy, P., Dinesh Kumar, S., & Nageshwar Rao, R. (2017). Numerical solution of second order singularly perturbed delay differential equations via cubic spline in tension. International Journal of Applied and Computational Mathematics, 3(3), 1703–1717. https://doi.org/10.1007/s40819-016-0204-5
   Google Scholar
• Ramesh, V. P., & Priyanga, B. (2019). Higher order uniformly convergent numerical algorithm for time-dependent singularly perturbed differential-difference equations, (pp. 1–25). Differential Equations and Dynamical Systems.
   Google Scholar
• Ravi Kanth, A. S. V., Murali, P., & Mohan, K. (2017). A numerical approach for solving singularly perturbed convection delay problems via exponentially fitted spline method. Calcolo, 54(3), 943–961. https://doi.org/10.1007/s10092-017-0215–6
   Web of Science ®Google Scholar
• Stein, R. B. (1967). Some models of neuronal variability. Biophysical Journal,7(1), 7(1), 37–68. https://doi.org/10.1016/S0006-3495(67)86574-3
   PubMed Web of Science ®Google Scholar
• Tene, T., Guevara, M., Svozil, J., Tene-Fernandez, R., & Gomez, C. V. (2021). Analysis of covid-19 outbreak in Ecuador using the logistic model. Emerging Science Journal, 5, 105–118. https://doi.org/10.28991/esj-2021-SPER-09
   Google Scholar
• Vikas Gupta, M. K., & Kadalbajoo, R. K. (2019). Dubey. 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
   Google Scholar
• Wazewska-Czyzewska, M., & Lasota, A. (9176). Mathematical models of the red cell system. Matematyta Stosowana, 6(1), 25–40.
   Google Scholar
• Woldaregay, M. M., & Duressa, G. F. (2019). Parameter uniform numerical method for singularly perturbed parabolic differential difference equations. Journal of the Nigerian Mathematical Society, 38(2), 223–245. https://ojs.ictp.it/jnms/index.php/jnms/article/view/474
   Google Scholar
• Woldaregay, M. M., & Duressa, G. F. (2020). Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations. ITM Web of Conferences, 34, 02011. https://doi.org/10.1051/itmconf/20203402011
   Google Scholar
• Woldaregay, M. M., & Duressa, G. F. (2022). Uniformly convergent numerical method for singularly perturbed delay parabolic differential equations arising in computational neuroscience. Kragujevac Journal of mathematics, 46(1), 65–84.
   Web of Science ®Google Scholar