A.s.v. Ravi KANTH, P. Murali Mohan KUMAR

Computational results and analysis for a class of linear and nonlinear singularly perturbed convection delay problems on Shishkin mesh

This article presents a hybrid numerical scheme for a class of linear and nonlinear singularly perturbed convection delay problems on piecewise uniform. The proposed hybrid numerical scheme comprises with the tension spline scheme in the boundary layer region and the midpoint approximation in the outer region on piecewise uniform mesh. Error analysis of the proposed scheme is discussed and is shown $\varepsilon$-uniformly convergent. Numerical experiments for linear and nonlinear are performed to confirm the theoretical analysis.

Keywords:

Nonlinear delay problems, differential-difference equations, singular perturbation problems tension spline, Shishkin mesh,

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[1] T. Aziz, A. Khan, I. Khan, and M. Stojanovic, A variable-mesh approximation method for 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 value problems, Rand Corporation, 1965.
[3] M. Bestehorn and E.V. Grigorieva, E.V. Formation and propagation of localized states in 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 for problems 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 twodimensional 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, Robust computational 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 perturbed nonlinear 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 singularly 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 singularly 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, Science 197 (4300), 287–289, 1977.
[15] J.J.H. Miller, R.E. ORiordan, and G.I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, 1996.
[16] J. Mohapatra and S.Natesan, Uniform convergence analysis of finite difference scheme for 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 perturbed differential-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 solving singularly perturbed convection delay problems via exponentially fitted spline method, Calcolo 54 (3), 943–961, 2017.
[19] A.S.V. Ravi Kanth and P. Murali Mohan Kumar, Numerical treatment for a singularly perturbed 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 solving nonlinear 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 nonlinear 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 perturbed 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 type with 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.