Wang, Xiao-Yun; Jiang, Yao-Lin A general method for solving singular perturbed impulsive differential equations with two-point boundary conditions. (English) Zbl 1090.65094 Appl. Math. Comput. 171, No. 2, 775-806 (2005). Summary: A general method is presented for solving singularly perturbed impulsive two-point boundary value problems with boundary layer at one end (left or right) point. For the method, we divide the defined interval into a series of subintervals and obtain the solution on each subinterval. Namely, on the subinterval of boundary layer, we yield an expansion by using matching techniques. Then, by using impulsive and boundary conditions, we obtain the solutions on other subintervals. Finally, several linear and nonlinear singular perturbed impulsive problems are provided to be concretely solved according to the new approach. Cited in 4 Documents 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 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:singularly perturbation; impulsive differential equations; method of small intervals; left-end boundary layer; right-end boundary layer; numerical examples; two-point boundary value problems PDFBibTeX XMLCite \textit{X.-Y. Wang} and \textit{Y.-L. Jiang}, Appl. Math. Comput. 171, No. 2, 775--806 (2005; Zbl 1090.65094) Full Text: DOI References: [1] Guo, D. J., Existence of solutions of boundary value problems for nonlinear second order impulsive differential equations in Banach spaces, J. Math. Anal. Appl., 181, 407-421 (1994) · Zbl 0807.34076 [2] Guo, D. J.; Sun, J. X.; Liu, Z. J., Functional methods of nonlinear ordinary differential equations (1995), Shandong Scientific Publishing: Shandong Scientific Publishing Jinan, (in Chinese) [3] Zhang, X., Singular perturbation of the nonlinear boundary value problems for impulsive differential equations, Acta. Math. Sci., 14, 419-423 (1994) · Zbl 0900.34072 [4] Valarmathi, S.; Ramanujam, N., Computational methods for solving two-parameter singularly perturbed boundary value problems for second-order ordinary differential equations, Appl. Math. Comput., 136, 415-441 (2003) · Zbl 1025.65046 [5] Reddy, Y. N.; Chakravarthy, P. P., Numerical patching method for singularly perturbed two-point boundary value problems using cubic splines, Appl. Math. Comput., 149, 441-468 (2004) · Zbl 1038.65068 [6] Reddy, Y. N.; Chakravarthy, P. P., Method of reduction of order for solving singularly perturbed two-point boundary value problems, Appl. Math. Comput., 136, 27-45 (2003) · Zbl 1026.34063 [7] Bainov, D.; Covachev, V., Impulsive differential equations with a small parameter (1994), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 0828.34001 [8] O’Malley, R. E., Give your ODEs a singular perturbation, J. Math. Anal. Appli., 251, 433-451 (2000) · Zbl 0964.34046 [9] O’Malley, R. E., Introduction to singular perturbations (1974), Academic Press: Academic Press New York · Zbl 0287.34062 [10] Khadalbajoo, M. K.; Reddy, Y. N., Approximations for the numerical solution of singular perturbation problems, Appl. Math. Comput., 21, 185-199 (1987) · Zbl 0626.65075 [11] O’Malley, R. E., Singular perturbation methods for ordinary differential equations (1990), Springer-Verlag: Springer-Verlag Berlin [12] Lakshmikantham, V.; Bainov, D. D.; Simenov, P. S., Theory of impulsive differential equations (1989), World Scientific Publishing: World Scientific Publishing Singapore [13] 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 [14] Holmes, M. H., Introduction to perturbation methods (1999), World Scientific Publishing: World Scientific Publishing Singapore [15] Nafeh, A. H., Introduction to perturbation techniques (1981), John Wiley & Sons: John Wiley & Sons New York 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.