DOI QR코드

DOI QR Code

FINITE DIFFERENCE SCHEME FOR SINGULARLY PERTURBED SYSTEM OF DELAY DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

  • SEKAR, E. (DEPARTMENT OF MATHEMATICS, BHARATHIDASAN UNIVERSITY TIRUCHIRAPPALLI) ;
  • TAMILSELVAN, A. (DEPARTMENT OF MATHEMATICS, BHARATHIDASAN UNIVERSITY TIRUCHIRAPPALLI)
  • Received : 2018.05.21
  • Accepted : 2018.08.28
  • Published : 2018.09.25

Abstract

In this paper we consider a class of singularly perturbed system of delay differential equations of convection diffusion type with integral boundary conditions. A finite difference scheme on an appropriate piecewise Shishkin type mesh is suggested to solve the problem. We prove that the method is of almost first order convergent. An error estimate is derived in the discrete maximum norm. Numerical experiments support our theoretical results.

References

  1. G. M. Amiraliyev, I. G. Amiraliyev, Mustafa Kudu,A numerical treatment for singularly perturbed differential equations with integral boundary condition, Applied mathematics and computation 185, 574-582,(2007). https://doi.org/10.1016/j.amc.2006.07.060
  2. D. Bahuguna, S.Abbas and J. Dabas, Partial functional differential equation with an integral condition and applications to population dynamics, Nonlinear Analysis 69 (2008) 2623-2635 https://doi.org/10.1016/j.na.2007.08.041
  3. D. Bahuguna and J. Dabas, Existence and Uniqueness of a Solution to a Semilinear Partial Delay Differential Equation with an Integral Condition, Nonlinear Dynamics and Systems Theory, 8 (1) (2008), 7-19.
  4. A. Boucherif, Second order boundary value problems with integral boundary condition, Nonlinear analysis, 70(1), 368-379, (2009).
  5. M. Cakir and G. M. Amiraliyev, A finite difference method for the singularly perturbed problem with nonlocal boundary condition, Applied mathematics and computation 160, 539-549,(2005). https://doi.org/10.1016/j.amc.2003.11.035
  6. J.R. Cannon , The solution of the heat equation subject to the specification of energy, Qart Appl Math 21(1963),155-160. https://doi.org/10.1090/qam/160437
  7. Y. S. Choi and Kwono-Yu Chan, A Parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Analysis Theory, Methods and Applications, Vol.18,No.4, pp.317-331,1992. https://doi.org/10.1016/0362-546X(92)90148-8
  8. W. A. Day, Parabolic equations and thermodynamics, Quart Appl Math 50(1992), 523-533. https://doi.org/10.1090/qam/1178432
  9. Hongyu Li and Fei Sun, Existence of solutions for integral boundary value prob- lems of second order ordinary differential equations, Li and Sun boundary value problems, (2012).
  10. M.K. Kadalbajoo, K.K. Sharma, Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations, Comput. Appl. Math. 24(2), 151-172 (2005).
  11. M.K. Kadalbajoo,K.K. Sharma,Parameter-Uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior. Electron. Trans. Numer. Anal. 23, 180-201 (2006).
  12. M.K. Kadalbajoo,D. Kumar, Fitted mesh B-spline collocation method for singularly perturbed differential equations with small delay, Appl. Math. Comput. 204, 90-98 (2008).
  13. C.G. Lange,R.M. Miura,Singularly perturbation analysis of boundary-value problems for differential-difference equations, SIAM J. Appl. Math. 42(3), 502-530 (1982). https://doi.org/10.1137/0142036
  14. Meigiang Feng, Dehong Ji, and Weigao Ge Positive solutions for a class of boundary value problem with integral boundary conditions in banach spaces, Journal of computational and applied mathematics 222, 351-363, (2008). https://doi.org/10.1016/j.cam.2007.11.003
  15. J.J.H. Miller,E. ORiordan,G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems,World Scientific Publishing Co., Singapore, New Jersey, London, Hong Kong (1996).
  16. Mustfa kudu and Gabil Amiraliyev, Finite difference method for a singularly perturbed differential equations with integral boundary condition, International journal of mathematics and computation Vol (26),(2015).
  17. S.Nicaise, C.Xenophontos, Robust approximation of singularly perturbed delay differential equations by the hp finite element method. Comput. Meth. Appl. Math. 13(1), 21-37 (2013).
  18. Z.Q. Tang, F.Z. Geng, Fitted reproducing kernel method for singularly perturbed delay initial value problems, Applied Mathematics and Computation 284 (2016) 169-174. https://doi.org/10.1016/j.amc.2016.03.006
  19. H. Zarin, On discontinuous Galerkin finite element method for singularly perturbed delay differential equations, Applied Mathematics Letters 38 (2014) 27-32. https://doi.org/10.1016/j.aml.2014.06.013
  20. Zhang Lian and Xie Feng, Singularly perturbed first order differential equations with integral boundary condition, J. Shanghai Univ (Eng), 20-22, (2009).
  21. Zhongdi Cen and Xin Cai, A second order upwind difference scheme for a sin- gularly perturbed problem with integral boundary condition in netural network , Springer verlag berlin heidelberg, 175-181, 2007.