# zbMATH — the first resource for mathematics

Solving second-order singularly perturbed ODE by the collocation method based on energetic Robin boundary functions. (English) Zbl 07144733
Summary: For a second-order singularly perturbed ordinary differential equation (ODE) under the Robin type boundary conditions, we develop an energetic Robin boundary functions method (ERBFM) to find the solution, which automatically satisfies the Robin boundary conditions. For the non-singular ODE the Robin boundary functions consist of polynomials, while the normalized exponential trial functions are used for the singularly perturbed ODE. The ERBFM is also designed to preserve the energy, which can quickly find accurate numerical solutions for the highly singularly perturbed problems by a simple collocation technique.

##### MSC:
 34B60 Applications of boundary value problems involving ordinary differential equations
Full Text:
##### References:
 [1] Ascher, U. M.; Mattheij, R. M. M.; Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Classics in Applied Mathematics 13, SIAM, Society for Industrial and Applied Mathematics, Philadelphia (1995) [2] Awoke, A.; Reddy, Y. N., An exponentially fitted special second-order finite difference method for solving singular perturbation problems, Appl. Math. Comput. 190 (2007), 1767-1782 [3] Bender, C. M.; Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers, International Series in Pure and Applied Mathematics, McGraw-Hill Book, New York (1978) [4] Cash, J. R., Numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections. I. A survey and comparison of some one-step formulae, Comput. Math. Appl., Part A 12 (1986), 1029-1048 [5] Cash, J. R., On the numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections. II. The development and analysis of highly stable deferred correction formulae, SIAM J. Numer. Anal. 25 (1988), 862-882 [6] Cash, J. R.; Wright, R. W., Continuous extensions of deferred correction schemes for the numerical solution of nonlinear two-point boundary value problems, Appl. Numer. Math. 28 (1998), 227-244 [7] Doğan, N.; Ertürk, V. S.; Akı{n}, Ö., Numerical treatment of singularly perturbed two-point boundary value problems by using differential transformation method, Discrete Dyn. Nat. Soc. 2012 (2012), Article ID 579431, 10 pages [8] Ilicasu, F. O.; Schultz, D. H., High-order finite-difference techniques for linear singular perturbation boundary value problems, Comput. Math. Appl. 47 (2004), 391-417 [9] Kadalbajoo, M. K.; Aggarwal, V. K., Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems, Appl. Math. Comput. 161 (2005), 973-987 [10] Keller, H. B., Numerical Methods for Two-Point Boundary Value Problems, Blaisdell Publishing Company, Waltham (1968) [11] Khuri, S. A.; Sayfy, A., Self-adjoint singularly perturbed boundary value problems: an adaptive variational approach, Math. Methods Appl. Sci. 36 (2013), 1070-1079 [12] Lin, T.-C.; Schultz, D. H.; Zhang, W., Numerical solutions of linear and nonlinear singular perturbation problems, Comput. Math. Appl. 55 (2008), 2574-2592 [13] Liu, C.-S., Efficient shooting methods for the second-order ordinary differential equations, CMES, Comput. Model. Eng. Sci. 15 (2006), 69-86 [14] Liu, C.-S., The Lie-group shooting method for singularly perturbed two-point boundary value problems, CMES, Comput. Model. Eng. Sci. 15 (2006), 179-196 [15] Liu, C.-S., An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation, Eng. Anal. Bound. Elem. 36 (2012), 1235-1245 [16] Liu, C.-S., The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1506-1521 [17] Liu, C.-S.; Li, B., Reconstructing a second-order Sturm-Liouville operator by an energetic boundary function iterative method, Appl. Math. Lett. 73 (2017), 49-55 [18] Liu, C.-S.; Liu, D.; Jhao, W.-S., Solving a singular beam equation by using a weak-form integral equation method, Appl. Math. Lett. 64 (2017), 51-58 [19] Patidar, K. C., High order parameter uniform numerical method for singular perturbation problems, Appl. Math. Comput. 188 (2007), 720-733 [20] Reddy, Y. N.; Chakravarthy, P. Pramod, An initial-value approach for solving singularly perturbed two-point boundary value problems, Appl. Math. Comput. 155 (2004), 95-110 [21] Varner, T. N.; Choudhury, S. R., Non-standard difference schemes for singular perturbation problems revisited, Appl. Math. Comput. 92 (1998), 101-123 [22] Vigo-Aguiar, J.; Natesan, S., An efficient numerical method for singular perturbation problems, J. Comput. Appl. Math. 192 (2006), 132-141
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.