# zbMATH — the first resource for mathematics

The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems. (English) Zbl 1244.65113
Summary: A new computational method for solving a second-order nonlinear singularly perturbed boundary value problems is provided. In order to overcome a highly singular behavior very near to the boundary as being not easy to treat by numerical method, we adopt a coordinate transformation from an $$x$$-domain to a $$t$$-domain via a rescaling technique, which can reduce the singularity within the boundary layer. Then, we construct a Lie-group shooting method to search a missing initial condition through the finding of a suitable value of a parameter $$r\in [0,1]$$. Moreover, we derive a closed-form formula to express the initial condition in terms of $$r$$, which can be determined properly by an accurate matching to the right-boundary condition. Numerical examples are examined, showing that the present approach is highly efficient and accurate.

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34E15 Singular perturbations for ordinary differential equations
Full Text:
##### References:
 [1] Iserles, A.; Zanna, A., Preserving algebraic invariants with runge – kutta methods, J comput appl math, 125, 69-81, (2000) · Zbl 0969.65069 [2] Iserles, A.; Munthe-Kass, H.Z.; Norsett, S.P.; Zanna, A., Lie-group methods, Acta numer, 9, 215-365, (2000) · Zbl 1064.65147 [3] Roos, H.G.; Stynes, M.; Tobiska, L., Numerical methods for singularly perturbed differential equations, (1996), Spring-Verlag Berlin [4] Ou, C.H.; Wong, R., Shooting method for nonlinear singularly perturbed boundary-value problems, Stud appl math, 112, 161-200, (2004) · Zbl 1141.34328 [5] Bender, C.M.; Orszag, S.A., Advanced mathematical methods for scientists and engineers, (1978), McGraw-Hill New York · Zbl 0417.34001 [6] Shampine, L.F.; Gear, C.W., A user’s view of solving stiff ordinary differential equations, SIAM rev, 21, 1-17, (1979) · Zbl 0415.65038 [7] Nayfeh, A.H., Introduction to perturbation techniques, (1981), Wiley New York · Zbl 0449.34001 [8] Kevorkian, J.; Cole, J.D., Perturbation methods in applied mathematics, (1981), Springer-Verlag New York · Zbl 0456.34001 [9] Kevorkian, J.; Cole, J.D., Multiple scale and singular perturbation methods, (1996), Springer-Verlag New York · Zbl 0846.34001 [10] O’Malley, R.E., Singular perturbation methods for ordinary differential equations, (1991), Springer-Verlag New York · Zbl 0743.34059 [11] De Jager, E.M.; Jiang, F.R., The theory of singular perturbation, (1996), Noth-Holland Amsterdam [12] Kadalbajoo, M.K.; Patidar, K.C., A survey of numerical techniques for solving singularly perturbed ordinary differential equations, Appl math comput, 130, 457-510, (2002) · Zbl 1026.65059 [13] Kadalbajoo, M.K.; Reddy, Y.N., Initial-value technique for a class of nonlinear singular perturbation problems, J optim theory appl, 53, 395-406, (1986) · Zbl 0594.34017 [14] Gasparo, M.G.; Macconi, M., New initial-value method for singularly perturbed boundary value problems, J optim theory appl, 63, 213-224, (1989) · Zbl 0664.65083 [15] Gasparo, M.G.; Macconi, M., Initial-value methods for second-order singularly perturbed boundary value problems, J optim theory appl, 66, 197-210, (1990) · Zbl 0681.34018 [16] Natesan, S.; Ramanujam, M., Initial-value technique for singularly-perturbed turning-point problems exhibiting twin boundary layers, J optim theory appl, 99, 37-52, (1998) · Zbl 0983.34050 [17] Reddy, Y.N.; Chakravarthy, P.P., An initial-value approach for solving singularly perturbed two-point boundary value problems, Appl math comput, 155, 95-110, (2004) · Zbl 1058.65079 [18] Li, Z.; Wang, W., Mechanization for solving SPP by reducing order method, Appl math comput, 169, 1028-1037, (2005) · Zbl 1119.65361 [19] Awoke, A.; Reddy, Y.N., An exponentially fitted special second-order finite difference method for solving singular perturbation problems, Appl math comput, 190, 1767-1782, (2007) · Zbl 1122.65377 [20] Patidar, K.C., High order parameter uniform numerical method for singular perturbation problems, Appl math comput, 188, 720-733, (2007) · Zbl 1119.65070 [21] Vigo-Aguiar, J.; Natesan, S., An efficient numerical method for singular perturbation problems, J comput appl math, 192, 132-141, (2006) · Zbl 1095.65068 [22] Lin, T.C.; Schultza, D.H.; Zhang, W., Numerical solutions of linear and nonlinear singular perturbation problems, Comput math appl, 55, 2574-2592, (2008) · Zbl 1142.65306 [23] Kubicek, M.; Hlavacek, V., Numerical solution of nonlinear boundary value problems with applications, (1983), Prentice-Hall New York · Zbl 1140.65340 [24] Keller, H.B., Numerical methods for two-point boundary value problems, (1992), Dover New York [25] Ascher, U.M.; Mattheij, R.M.M.; Russell, R.D., Numerical solution of boundary value problems for ordinary differential equations, (1995), SIAM Philadelphia [26] Liu, C.-S., Cone of non-linear dynamical system and group preserving schemes, Int J non linear mech, 36, 1047-1068, (2001) · Zbl 1243.65084 [27] Liu, C.-S., The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions, CMES: comput model eng sci, 13, 149-163, (2006) · Zbl 1232.65108 [28] Liu, C.-S., Efficient shooting methods for the second order ordinary differential equations, CMES: comput model eng sci, 15, 69-86, (2006) · Zbl 1152.65453 [29] Liu, C.-S., The Lie-group shooting method for singularly perturbed two-point boundary value problems, CMES: comput model eng sci, 15, 179-196, (2006) · Zbl 1152.65452 [30] Liu, C.-S., The Lie-group shooting method for solving multi-dimensional nonlinear boundary value problems, J optim theory appl, (2011) [31] Liu, C.-S., A Lie-group shooting method for computing eigenvalues and eigenfunctions of sturm – liouville problems, CMES: comput model eng sci, 26, 157-168, (2008) · Zbl 1232.65110 [32] Liu, C.-S., The Lie-group shooting method for computing the generalized sturm – liouville problems, CMES: comput model eng sci, 56, 85-112, (2010) · Zbl 1231.65125 [33] Abbasbandy, S.; Hashemi, M.S.; Liu, C.-S., The Lie-group shooting method for solving the bratu equation, Commun nonlinear sci numer simulat, 16, 4238-4249, (2011) · Zbl 1222.65067 [34] Liu, C.-S., A group preserving scheme for Burgers equation with very large Reynolds number, CMES: comput model eng sci, 12, 197-211, (2006) · Zbl 1232.76012 [35] Varner, T.N.; Choudhury, S.R., Non-standard difference schemes for singular perturbation problems revisited, Appl math comput, 92, 101-123, (1998) · Zbl 0942.65081 [36] Ilicasu, F.O.; Schultz, D.H., High-order finite-difference techniques for linear singular perturbation boundary value problems, Comput math appl, 47, 391-417, (2004) · Zbl 1168.76343
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.