Solving singularly perturbed delay differential equations via fitted mesh and exact difference method

References

• Baker, C. T. H., Bocharov, G. A., & Rihan, F. A., A report on the use of delay differential equations in numerical modeling in the biosciences, Citeseer, 1999.
   Google Scholar
• Bellen, A., & Zennaro, M. (2003). Numerical methods for delay differential equations. Clarendon Press.
   Google Scholar
• Bushnaq, S., Shah, K., Tahir, S., Ansari, K. J., Sarwar, M., & Abdeljawad, T. (2022). Computation of numerical solutions to variable order fractional differential equations by using non-orthogonal basis. AIMS Mathematics, 7(6), 10917–13. https://doi.org/10.3934/math.2022610
   Web of Science ®Google Scholar
• Elsheikh, S., Ouifki, R., & Patidar, K. C. (2014). A non-standard finite difference method to solve a model of HIV–Malaria co-infection. Journal of Difference Equations and Applications, 20(3), 354–378. https://doi.org/10.1080/10236198.2013.821116
   Web of Science ®Google Scholar
• He, X., & Wang, K. (2018). New finite difference methods for singularly perturbed convection-diffusion equations. Taiwanese Journal of Mathematics, 22(4), 949–978. https://doi.org/10.11650/tjm/171002
   Web of Science ®Google Scholar
• Izadi, M., Yüzbaşı, S., & Ansari, K. J. (2021). Application of Vieta–Lucas series to solve a class of multi-pantograph delay differential equations with singularity. Symmetry, 13(12), 2370. https://doi.org/10.3390/sym13122370
   Google Scholar
• Kadalbajoo, M. K., Patidar, K. C., & Sharma, K. K. (2006). ε-uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs. Applied Mathematics and Computation, 182(1), 119–139. https://doi.org/10.1016/j.amc.2006.03.043
   Web of Science ®Google Scholar
• Kadalbajoo, M. K., & Ramesh, V. P. (2007). Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy. Applied Mathematics and Computation, 188(2), 1816–1831. https://doi.org/10.1016/j.amc.2006.11.046
   Web of Science ®Google Scholar
• Kellogg, T., & Tsan, A. (1978). Analysis of some difference approximations for a singular perturbation problem without turning point. Mathematics of Computation, 32(144), 1025–1039. https://doi.org/10.1090/S0025-5718-1978-0483484-9
   Web of Science ®Google Scholar
• Kuang, Y. (1993). Delay differential equations: With applications in population dynamics. Academic Press.
   Google Scholar
• Kumar, D., & Kadalbajoo, M. K. (2012). Numerical treatment of singularly perturbed delay differential equations using B-Spline collocation method on Shishkin mesh. Jnaiam, 7(3–4), 73–90.
   Google Scholar
• Lopez, A. G. (2020). On an electrodynamic origin of quantum fluctuations. Nonlinear Dynamics, 102(1), 621–634. https://doi.org/10.1007/s11071-020-05928-5
   Web of Science ®Google Scholar
• Mahaffy, P. R., A’Hearn, M. F., Atreya, S. K., Bar-Nun, A., Bruston, P., Cabane, M., Carignan, G. R., Coll, P., Crifo, J. F., Ehrenfreund, P., Harpold, D., Gorevan, S., Israel, G., Kasprzak, W., Mumma, M. J., Niemann, H. B., Owen, T., Raulin, F., Riedler, W., … Strazzulla, G. (1999). The Champollion cometary molecular analysis experiment. Advances in Space Research, 23(2), 349–359. https://doi.org/10.1016/S0273-1177(99)00056-3
   Web of Science ®Google Scholar
• Makroglou, A., Li, J., & Kuang, Y. (2006). Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: An overview. Applied Numerical Mathematics, 56(3–4), 559–573. https://doi.org/10.1016/j.apnum.2005.04.023
   Web of Science ®Google Scholar
• Mickens, R. E. (2002). Non-standard finite difference schemes for differential equations. Journal of Difference Equations and Applications, 8(9), 823–847. https://doi.org/10.1080/1023619021000000807
   Web of Science ®Google Scholar
• Mickens, R. E. (2005). Advances in the applications of Non-standard finite difference schemes. World Scientific.
   Google Scholar
• Miller, J. J., O’Riordan, E., & Shishkin, G. I. (2012). Fitted numerical methods for singular perturbation problems: Error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific.
   Google Scholar
• Patidar, K. C. (2005). On the use of non-standard finite difference methods. Journal of Difference Equations and Applications, 11(8), 735–758. https://doi.org/10.1080/10236190500127471
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Patidar, K. C. (2016). Non-standard finite difference methods: Recent trends and further developments. Journal of Difference Equations and Applications, 22(6), 817–849. https://doi.org/10.1080/10236198.2016.1144748
   Web of Science ®Google Scholar
• Roos, H. G., Stynes, M., & Tobiska, L. R. (2008). Numerical methods for singularly perturbed differential equations. Springer Science & Business Media.
   Google Scholar
• Salpeter, E. E., & Salpeter, S. R. (1998). Mathematical model for the epidemiology of tuberculosis, with estimates of the reproductive number and infection-delay function. American Journal of Epidemiology, 147(4), 398–406. https://doi.org/10.1093/oxfordjournals.aje.a009463
   Web of Science ®Google Scholar
• Shah, K., Arfan, M., Ullah, A., Al-Mdallal, Q., Ansari, K. J., & Abdeljawad, T. (2022). Computational study on the dynamics of fractional order differential equations with applications. Chaos, Solitons, and Fractals, 157, 111955. https://doi.org/10.1016/j.chaos.2022.111955
   Web of Science ®Google Scholar
• Sitthiwirattham, T., Gul, R., Shah, K., Mahariq, I., Soontharanon, J., Khursheed, K., & Ansari, K. J. (2021). Study of implicit-impulsive differential equations involving Caputo-Fabrizio fractional derivative. AIMS Mathematics, 7(3), 4017–4037. https://doi.org/10.3934/math.2022222
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Usta, F., Akyiğit, M., Say, F., & Ansari, K. J. (2022). Bernstein operator method for approximate solution of singularly perturbed Volterra integral equations. Journal of Mathematical Analysis and Applications, 507(2), 125828. https://doi.org/10.1016/j.jmaa.2021.125828
   Web of Science ®Google Scholar
• Woldaregay, M. M., & Duressa, G. F. (2021a). Robust mid-point upwind scheme for singularly perturbed delay differential equations. Computational and Applied Mathematics, 40(5), 1–12. https://doi.org/10.1007/s40314-021-01569-5
   Google Scholar
• Woldaregay, M. M., & Duressa, G. F. (2021b). Robust numerical scheme for solving singularly perturbed differential equations involving small delays. Applied Mathematics E-Notes, 21, 622–633. http://www.math.nthu.edu.tw/amen/
   Web of Science ®Google Scholar