Numerical Simulation of Singularly Perturbed Delay Differential Equations With Large Delay Using an Exponential Spline

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-20-2022-63
Published: Nov 25, 2022
Cite as:
Ramavath Omkar, K. Phaneendra, Numerical Simulation of Singularly Perturbed Delay Differential Equations With Large Delay Using an Exponential Spline, Int. J. Anal. Appl., 20 (2022), 63.

Main Article Content

Ramavath Omkar, K. Phaneendra

Abstract

In this study, numerical solution of a differential-difference equation with a boundary layer at one end of the domain is suggested using an exponential spline. The numerical scheme is developed using an exponential spline with a special type of mesh. A fitting parameter is inserted in the scheme to improve the accuracy and to control the oscillations in the solution due to large delay. Convergence of the method is examined. The error profiles are represented by tabulating the maximum absolute errors in the solution. Graphs are being used to show that how the fitting parameter influence the layer structure.

Article Details

References

  1. R. Bellman, K.L. Cooke, Differential-Difference Equations, Academic Press, New York, USA, 1963.
  2. E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, et al. Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
  3. R.D. Driver, Ordinary and Delay Differential Equations, Springer, Belin-Heidelberg, New York, 1977.
  4. G.M. Amiraliyev, E. Cimen, Numerical Method for a Singularly Perturbed Convection–diffusion Problem With Delay, Appl. Math. Comput. 216 (2010), 2351–2359. https://doi.org/10.1016/j.amc.2010.03.080.
  5. V.Y. Glizer, Asymptotic Solution of a Boundary-Value Problem for Linear Singularly-Perturbed Functional Differential Equations Arising in Optimal Control Theory, J. Optim. Theory Appl. 106 (2000), 309–335. https://doi.org/10.1023/a:1004651430364.
  6. M.K. Kadalbajoo, K.K. Sharma, A Numerical Method Based on Finite Difference for Boundary Value Problems for Singularly Perturbed Delay Differential Equations, Appl. Math. Comput. 197 (2008), 692–707. https://doi.org/10.1016/j.amc.2007.08.089.
  7. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
  8. D. Kumara Swamy, K. Phaneendra, A. Benerji Babu, Y.N. Reddy, Computational Method for Singularly Perturbed Delay Differential Equations With Twin Layers or Oscillatory Behaviour, Ain Shams Eng. J. 6 (2015), 391–398. https://doi.org/10.1016/j.asej.2014.10.004.
  9. M. Lalu, K. Phaneendra, S.P. Emineni, Numerical Approach for Differential-Difference Equations Having Layer Behaviour With Small or Large Delay Using Non-Polynomial Spline, Soft Comput. 25 (2021), 13709-13722. https://doi.org/10.1007/s00500-021-06032-5.
  10. L. Sirisha, K. Phaneendra, Y.N. Reddy, Mixed Finite Difference Method for Singularly Perturbed Differential Difference Equations With Mixed Shifts via Domain Decomposition, Ain Shams Eng. J. 9 (2018), 647–654. https://doi.org/10.1016/j.asej.2016.03.009.
  11. C.G. Lange, R.M. Miura, Singular Perturbation Analysis of Boundary Value Problems for Differential-Difference Equations. V. Small Shifts with Layer Behavior, SIAM J. Appl. Math. 54 (1994), 249–272. https://doi.org/10.1137/s0036139992228120.
  12. C.G. Lange, R.M. Miura, Singular Perturbation Analysis of Boundary-Value Problems for Differential-Difference Equations. VI. Small Shifts with Rapid Oscillations, SIAM J. Appl. Math. 54 (1994), 273–283. https://doi.org/10.1137/s0036139992228119.
  13. A. Lasota, M. Walewska, Mathematical Models of the Red Blood Cell System, Mat. Stos. 6 (1976), 25–40.
  14. M.C. Mackey, L. Glass, Oscillation and Chaos in Physiological Control Systems, Science. 197 (1977), 287–289. https://doi.org/10.1126/science.267326.
  15. A. Martin, S. Ruan, Predator-Prey Models With Delay and Prey Harvesting, J. Math. Biol. 43 (2001), 247–267. https://doi.org/10.1007/s002850100095.
  16. K. Phaneendra, M. Lalu, Numerical Solution of Singularly Perturbed Delay Differential Equations Using Gaussion Quadrature Method, J. Phys.: Conf. Ser. 1344 (2019), 012013. https://doi.org/10.1088/1742-6596/1344/1/012013.
  17. P. Rai, K.K. Sharma, Numerical Analysis of Singularly Perturbed Delay Differential Turning Point Problem, Appl. Math. Comput. 218 (2011), 3483–3498. https://doi.org/10.1016/j.amc.2011.08.095.
  18. P. Rai, K.K. Sharma, Numerical Approximation for a Class of Singularly Perturbed Delay Differential Equations With Boundary and Interior Layer(s), Numer Algor. 85 (2019), 305–328. https://doi.org/10.1007/s11075-019-00815-6.
  19. A.S.V. Ravikanth, P. Murali, Numerical Treatment for a Singularly Perturbed Convection Delayed Dominated Diffusion Equation via Tension Splines, Int. J. Pure Appl. Math. 113 (2017), 1314-3395.
  20. R.K. Adivi Sri Venkata, M.M.K. Palli, A numerical approach for solving singularly perturbed convection delay problems via exponentially fitted spline method, Calcolo. 54 (2017), 943-961. https://doi.org/10.1007/s10092-017-0215-6.
  21. Y.N. Reddy, GBSL Soujanya, K. Phaneendra, Numerical Integration Method for Singularly Perturbed Delay Differential Equations, Int. J. Appl. Sci. Eng. 10 (2012), 249-261.
  22. R.B. Stein, Some Models of Neuronal Variability, Biophys. J. 7 (1967), 37-68. https://doi.org/10.1016/s0006-3495(67)86574-3.
  23. W.K. Zahra, A.M. El Mhlawy, Numerical Solution of Two-Parameter Singularly Perturbed Boundary Value Problems via Exponential Spline, J. King Saud Univ. - Sci. 25 (2013), 201-208. https://doi.org/10.1016/j.jksus.2013.01.003.