Computational results and analysis for a class oflinear and nonlinear singularly perturbedconvection delay problems on Shishkin mesh

Computational results and analysis for a class oflinear and nonlinear singularly perturbedconvection delay problems on Shishkin mesh

This article presents a hybrid numerical scheme for a class of linear and nonlinear sin-gularly perturbed convection delay problems on piecewise uniform. The proposed hybridnumerical scheme comprises with the tension spline scheme in the boundary layer regionand the midpoint approximation in the outer region on piecewise uniform mesh. Erroranalysis of the proposed scheme is discussed and is shownε-uniformly convergent. Nu-merical experiments for linear and nonlinear are performed to confirm the theoreticalanalysis.

___

  • [1]T. Aziz, A. Khan, I. Khan, and M. Stojanovic,A variable-mesh approximation methodfor singularly perturbed boundary-value problems using cubic spline in tension, Int. J.Comput. Math.81(12), 1513–1518, 2004.
  • [2]R.E. Bellman and R.E. Kalaba,Quasilinearization and nonlinear boundary valueproblems, Rand Corporation, 1965.
  • [3]M. Bestehorn and E.V. Grigorieva, E.V.Formation and propagation of localized statesin extended systems, Ann. Phys.13(7-8), 423–431, 2004.
  • [4]E.P. Doolan, J.J.H. Miller, and W.H.A. Schilders,Uniform numerical methods forproblems with initial and boundary layers, Boole Press, Dublin, 1980.
  • [5]M.A. Ezzat, M.I. Othman, and A.M. El-Karamany, ıState space approach to two-dimensional generalized thermo-viscoelasticity with two relaxation times, Int. J. Eng.Sci.40(11), 1251–1274, 2002.
  • [6]P.A. Farrell, A.F. Hegarty, J.J.H. Miller, R.E. O’Riordan, and G.I. Shishkin,Robustcomputational techniques for boundary layers, CRC Press, New York, 2000.
  • [7]D.D. Joseph and L. Preziosi,Heat waves, Rev. Mod. Phys.61(1), 41, 1989.
  • [8]D.D. Joseph and L. Preziosi,Addendum to the paper heat waves, Rev. Mod. Phys.62, 375–391, 1990.
  • [9]M.K. Kadalbajoo and D. Kumar,A computational method for singularly perturbednonlinear differential-difference equations with small shift, Appl. Math. Model.34(9), 2584–2596, 2010.
  • [10]M.K. Kadalbajoo and V.P. Ramesh,Hybrid method for numerical solution of singu-larly perturbed delay differential equations, Appl. Math. Comput.187(2), 797–814,2007.
  • [11]M.K. Kadalbajoo and K.K. Sharma,Parameter-uniform fitted mesh method for sin-gularly perturbed delay differential equations with layer behavior, Electron. T. Numer.Ana.23, 180–201, 2006.
  • [12]A. Lasota and M. Wazewska,Mathematical models of the red blood cell system, Mat.Stos.6, 25–40, 1976.
  • [13]Q. Liu, X. Wang, and D. De Kee,Mass transport through swelling membranes, Int.J. Eng. Sci.43(19-20), 1464–1470, 2005.
  • [14]M.C. Mackey and L. Glass,Oscillation and chaos in physiological control systems,Science197(4300), 287–289, 1977.
  • [15]J.J.H. Miller, R.E. ORiordan, and G.I. Shishkin,Fitted numerical methods for singu-lar perturbation problems, World Scientific, Singapore, 1996.
  • [16]J. Mohapatra and S.Natesan,Uniform convergence analysis of finite difference schemefor singularly perturbed delay differential equation on an adaptively generated grid,Numer. Math.: Theory, Methods and Appl.3(1), 1–22, 2010.
  • [17]R.N. Rao and P.P. Chakravarthy,A finite difference method for singularly perturbeddifferential-difference equations with layer and oscillatory behavior, Appl. Math.Model.37(8), 5743–5755, 2013.
  • [18]A.S.V. Ravi Kanth and P. Murali Mohan Kumar,A numerical approach for solvingsingularly perturbed convection delay problems via exponentially fitted spline method,Calcolo54(3), 943–961, 2017.
  • [19]A.S.V. Ravi Kanth and P. Murali Mohan Kumar,Numerical treatment for a singularlyperturbed convection delayed dominated diffusion equation via tension spliens, Int. J.Pure Appl. Math.113(6), 110–118, 2017.
  • [20]A.S.V. Ravi Kanth and P. Murali Mohan Kumar,A numerical technique for solvingnonlinear singularly perturbed delay differential equations, Math. Model. Anal.23(1),64–78, 2018.
  • [21]A.S.V. Ravi Kanth and P. Murali Mohan Kumar,Numerical method for a class of non-linear singularly perturbed delay differential equations using parametric cubic spline,Int. J. Nonlin. Sci. Numer. Simul.19(3-4), 357–365, 2018.
  • [22]H.G. Roos, M. Stynes, and L. Tobiska, em Numerical methods for singularly per-turbed differential equations, convection-diffusion and flow problems, Springer-Verlag,Berlin Heidelberg, 1996.
  • [23]G.I. Shishkin,A difference scheme for a singularly perturbed equation of parabolic typewith discontinuous boundary conditions, Comput. Math. Math. Phys.28(6), 32–41,1988.
  • [24]M. Stynes and H.G. Roos,The midpoint upwind scheme, App. Numer. Math.23(3),361–374, 1997.