International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

Quintic B-Spline Technique for Numerical Treatment of Third Order Singular Perturbed Delay Differential Equation

Mandeep Kaur Vaid
Department of Mathematics, Lovely Professional University, Phagwara, Punjab, India.

Geeta Arora
Department of Mathematics, Lovely Professional University, Phagwara, Punjab, India.

DOI https://dx.doi.org/10.33889/IJMEMS.2019.4.6-116

Received on April 14, 2019
  ;
Accepted on August 07, 2019

Abstract

In this paper, a class of third order singularly perturbed delay differential equation with large delay is considered for numerical treatment. The considered equation has discontinuous convection-diffusion coefficient and source term. A quintic trigonometric B-spline collocation technique is used for numerical simulation of the considered singularly perturbed delay differential equation by dividing the domain into the uniform mesh. Further, uniform convergence of the solution is discussed by using the concept of Hall error estimation and the method is found to be of first-order convergent. The existence of the solution is also established. Computation work is carried out to validate the theoretical results.

Keywords- Quintic trigonometric spline, Error estimate, Perturbed equation, Delay.

Citation

Vaid, M. K., & Arora, G. (2019). Quintic B-Spline Technique for Numerical Treatment of Third Order Singular Perturbed Delay Differential Equation. International Journal of Mathematical, Engineering and Management Sciences, 4(6), 1471-1482. https://dx.doi.org/10.33889/IJMEMS.2019.4.6-116.

Conflict of Interest

The authors confirm that this article contents have no conflict of interest.

Acknowledgements

The authors would like to express their sincere thanks to the editor and referee for their valuable suggestions towards the improvement of the paper.

References

Andargie, A., & Reddy, Y. N. (2013). Parameter fitted scheme for singularly perturbed delay differential equations. International Journal of Applied Science and Engineering, 11(4), 361-373.

Cimen, E. (2017). A priori estimates for solution of singularly perturbed boundary value problem with delay in convection term. Journal of Mathematical Analysis, 8(1), 202-211.

Hall, C.A. (1968). On error bounds for spline interpolation. Journal of Approximation Theory, 1(2), 209-218.

Kellogg, R.B., & Tsan, A. (1978). Analysis of some difference approximations for a singular perturbation problem without turning points. Mathematics of Computation, 32(144), 1025-1039.

Kumar, D., & Kadalbajoo, M.K. (2012). Numerical treatment of singularly perturbed delay differential equations using B-Spline collocation method on Shishkin mesh. Journal of Numerical Analysis, Industrial and Applied Mathematics, 7(3-4), 73-90.

Longtin, A., & Milton, J.G. (1988). Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback. Mathematical Biosciences, 90(1-2), 183-199.

Nelson, P.W., & Perelson, A.S. (2002). Mathematical analysis of delay differential equation models of HIV-1 infection. Mathematical Biosciences, 179(1), 73-94.

Nicaise, S., & Xenophontos, C. (2013). Robust approximation of singularly perturbed delay differential equations by the hp finite element method. Computational Methods in Applied Mathematics, 13(1), 21-37.

Rihan, F.A. (2013). Delay differential equations in biosciences: parameter estimation and sensitivity analysis. In Recent Advances in Applied Mathematics and Computational Methods: Proceedings of the 2013 International Conference on Applied Mathematics and Computational Methods (Venice, Italy September 2013) (pp. 50-58).

Stein, R.B. (1965). A theoretical analysis of neuronal variability. Biophysical Journal, 5(2), 173-194.

Stein, R.B. (1967). Some models of neuronal variability. Biophysical Journal, 7(1), 37-68.

Subburayan, V., & Mahendran, R. (2018). An ε-uniform numerical method for third order singularly perturbed delay differential equations with discontinuous convection coefficient and source term. Applied Mathematics and Computation, 331, 404-415.

Swamy, D.K., Phaneendra, K., Babu, A.B., & Reddy, Y.N. (2015). Computational method for singularly perturbed delay differential equations with twin layers or oscillatory behaviour. Ain Shams Engineering Journal, 6(1), 391-398.

Varah, J.M. (1975). A lower bound for the smallest singular value of a matrix. Linear Algebra and its Applications, 11(1), 3-5.

Wilkie, K P., & Hahnfeldt, P. (2013). Mathematical models of immune-induced cancer dormancy and the emergence of immune evasion. Interface Focus, 3(4), 20130010.

Zakaria, N.F., Hassan, N.A., Hamid, N.N.A., Majid, A.A., & Ismail, A.I.M. (2017, April). Solving boussinesq equation using quintic B-spline and quintic trigonometric B-spline interpolation methods. In AIP Conference Proceedings 1830(1), p. 020041. AIP Publishing.

Zhang, Y., Jie, Y., & Meng, X. (2016). The modelling and control of a singular biological economic system in a polluted environment. Discrete Dynamics in Nature and Society, 2016, Article ID 3925386, 7 pages.

Privacy Policy| Terms & Conditions