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A computational method for solving third-order singularly perturbed boundary-value problems. (English) Zbl 1138.65060

Summary: A new computational method is presented for solving a class of third-order singularly perturbed boundary-value problems with a boundary layer at the left of the underlying interval. In this method, first the given third-order singularly perturbed boundary-value problem is transformed into a system of two ordinary differential equations subject to suitable initial and boundary conditions and a zeroth-order asymptotic expansion for the solution of the given problem is constructed. Then the reduced terminal value problem is solved analytically in the reproducing kernel space.
This method is effective and easy to implement. A numerical example is studied to demonstrate the accuracy of the present method. Results obtained by the method are found to be in good agreement with the exact solution not only in the boundary layer, but also away from the layer.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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References:

[1] Roos, H.G.; Stynes, M.; Tobiska, L., Numerical methods for singularly perturbed differential equations, (1996), Springer-Verlag
[2] Howes, F.A., Singular perturbations and differential inequalities, (1976), Memoirs of the American Mathematical Society Providence, Rhode Island, p. 168 · Zbl 0285.34038
[3] Ascher, V.; Weiss, R., Collocation for singular-perturbation problems, III: nonlinear problems without turning points, SIAM journal on scientific and statistical computing, 5, 811-829, (1984) · Zbl 0558.65060
[4] Kadalbajoo, Mohank; Aggarwal, Vivek K., Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems, Applied mathematics and computation, 161, 973-987, (2005) · Zbl 1073.65062
[5] Roos, H.G., A second-order monotone upwind scheme, Computing, 36, 57-67, (1986) · Zbl 0572.65063
[6] Ilicasu, F.O.; Schultz, D.H., High-order finite-difference techniques for linear singular perturbation boundary value problems, Computers and mathematics with applications, 47, 391-417, (2004) · Zbl 1168.76343
[7] Stynes, M.; O’Riordan, E., A uniformly accurate finite-element method for a singular-perturbation problem in conservative form, SIAM journal on numerical analysis, 23, 369-375, (1986) · Zbl 0595.65091
[8] Vigo-Aguiar, J.; Natesan, S., A parallel boundary value techniques for singularly perturbed two-point boundary value problems, The journal of supercomputing, 27, 195-206, (2004) · Zbl 1070.65066
[9] Reddy, Y.N.; Chakravarthy, P. Pramod, An initial-value approach for singularly perturbed two-point boundary value problems, Applied mathematics and computation, 155, 95-110, (2004) · Zbl 1058.65079
[10] Valanarasu, T.; Ramanujam, N., Asymptotic initial-value method for singularly-perturbed boundary-value problems for second-order ordinary differential equations, Journal of optimization theory and applications, 116, 167-182, (2003) · Zbl 1043.34060
[11] Bawa, Rajesh K., Spline based computational technique for linear singularly perturbed boundary value problems, Applied mathematics and computation, 167, 225-236, (2005) · Zbl 1083.65513
[12] Aziz, Tariq; khan, Arshad, A spline method for second-order singularly perturbed boundary-value problems, Journal of computational and applied mathematics, 147, 445-452, (2002) · Zbl 1034.65059
[13] Howes, F.A., Differential inequalities of higher order and the asymptotic solution of nonlinear boundary value problems, SIAM journal on mathematical analysis, 131, 61-80, (1982) · Zbl 0487.34066
[14] Howes, F.A., The asympototic solution of a class of third-order boundary problem arising in the theory of thin film flow, SIAM journal of applied mathematics, 43, 5, 993-1004, (1983) · Zbl 0532.76042
[15] Weili, Z., Singular perturbations of boundary value problems for a class of third order nonlinear ordinary differential equations, Journal of differential equations, 88, 2, 265-278, (1990) · Zbl 0718.34010
[16] Roberts, S.M., Further examples of the boundary value technique in singular perturbation problems, Journal of mathematical analysis and applications, 133, 411-436, (1988) · Zbl 0658.65076
[17] Valarmathi, S.; Ramanujam, N., A computational method for solving boundary value problems for third-order singularly perturbed ordinary differential equations, Applied mathematics and computation, 129, 345-373, (2002) · Zbl 1025.65045
[18] Nayfeh, A.H., Introduction to perturbation method, (1981), Wiley New York
[19] Natesan, S.; Ramanujam, N., A computational method for solving singularly perturbed Turing point problems exhibiting twin boundary layer, Applied mathematics and computation, 93, 259-275, (1998) · Zbl 0943.65086
[20] Li, Chunli; Cui, Minggen, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied mathematics and computation, 143, 2-3, 393-399, (2003) · Zbl 1034.47030
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