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Computational results and analysis for a class of linear and nonlinear singularly perturbed convection delay problems on Shishkin mesh. (English) Zbl 1488.65191

Summary: 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.

MSC:

65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34K26 Singular perturbations of functional-differential equations
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[1] [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. · Zbl 1064.65066
[2] [2] R.E. Bellman and R.E. Kalaba, Quasilinearization and nonlinear boundary value problems, Rand Corporation, 1965. · Zbl 0139.10702
[3] [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. · Zbl 1108.35386
[4] [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. · Zbl 0459.65058
[5] [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. · Zbl 1211.74068
[6] [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. · Zbl 0964.65083
[7] [7] D.D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys. 61 (1), 41, 1989. · Zbl 1129.80300
[8] [8] D.D. Joseph and L. Preziosi, Addendum to the paper heat waves, Rev. Mod. Phys. 62, 375-391, 1990.
[9] [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. · Zbl 1195.65100
[10] [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. · Zbl 1120.65088
[11] [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. · Zbl 1112.65067
[12] [12] A. Lasota and M. Wazewska, Mathematical models of the red blood cell system, Mat. Stos. 6, 25-40, 1976.
[13] [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] [14] M.C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197 (4300), 287-289, 1977. · Zbl 1383.92036
[15] [15] J.J.H. Miller, R.E. ORiordan, and G.I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific, Singapore, 1996. · Zbl 0915.65097
[16] [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. · Zbl 1224.65186
[17] [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. · Zbl 1274.65213
[18] [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. · Zbl 1376.65109
[19] [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] [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. · Zbl 1488.65161
[21] [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. · Zbl 1401.65078
[22] [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. · Zbl 0844.65075
[23] [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. · Zbl 0698.65058
[24] [24] M. Stynes and H.G. Roos, The midpoint upwind scheme, App. Numer. Math. 23 (3), 361-374, 1997. · Zbl 0877.65055
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