Numerical integration of a class of singular perturbation problems. (English) Zbl 0579.65081
We discuss an approximate method for the numerical integration of a class of linear, singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This method requires a minimum of problem preparation and can be implemented easily on a computer. We replace the original singular perturbation problem by an approximate first-order differential equation with a small deviating argument. Then, we use the trapezoidal formula to obtain the three-term recurrence relationship. Discrete invariant imbedding algorithm is used to solve a tridiagonal algebraic system. The stability of this algorithm is investigated. The proposed method is iterative on the deviating argument. Several numerical experiments are included to demonstrate the efficiency of the method.
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 34B05 | Linear boundary value problems for ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
Keywords:
boundary layer; trapezoidal formula; discrete invariant imbedding algorithm; numerical experimentsReferences:
| [1] | Pearson, C. E.,On a Differential Equation of Boundary Layer Type, Journal of Mathematics and Physics, Vol. 47, pp. 134-154, 1968. · Zbl 0167.15801 |
| [2] | Hemker, P. W., andMiller, J. J. H.,Numerical Analysis of Singular Perturbation Problems, Academic Press, New York, New York, 1979. |
| [3] | Axelsson, O., Frank, L. S., andVan der Sluis, A.,Analytical and Numerical Approaches to Asymptotic Problems in Analysis, North-Holland Publishing Company, Amsterdam, Holland, 1981. · Zbl 0441.00011 |
| [4] | Cole, J. D., andKevorkian, J.,Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, New York, 1981. · Zbl 0456.34001 |
| [5] | Bender, C. M., andOrszag, S. A.,Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, New York, 1978. · Zbl 0417.34001 |
| [6] | O’Malley, R. E.,Introduction to Singular Perturbations, Academic Press, New York, New York, 1974. |
| [7] | Nayfeh, A. H.,Perturbation Methods, Wiley, New York, New York, 1973. · Zbl 0265.35002 |
| [8] | Bellman, R.,Perturbation Techniques in Mathematics, Physics, and Engineering, Holt, Rinehart, and Winston, New York, New York, 1964. · Zbl 0133.24107 |
| [9] | Elsgolts, L. E., andNorkin, S. B.,Introduction to Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, New York, 1973. |
| [10] | Elsgolts, L. E.,Qualitative Methods in Mathematical Analysis, American Mathematical Society, Providence, Rhode Island, 1964. |
| [11] | Bellman, R., andCooke, K. L.,Differential-Difference Equations, Academic Press, New York, New York, 1963. · Zbl 0105.06402 |
| [12] | Bellman, R., Buell, J. D., andKalaba, R.,Numerical Integration of a Differential-Difference Equation with a Decreasing Time-Lag, Communications of the ACM, Vol. 8, pp. 227-228, 1965. · Zbl 0135.17902 · doi:10.1145/363831.364879 |
| [13] | Driver, R. D.,Ordinary and Delay Differential Equations, Springer-Verlag, New York, New York, 1977. · Zbl 0374.34001 |
| [14] | Schmitt, K.,Delay and Functional Differential Equations and Their Applications, Academic Press, New York, New York, 1972. · Zbl 0259.00008 |
| [15] | Norkin, S. B.,Differential Equations of the Second Order with Retarded Argument, American Mathematical Society, Providence, Rhode Island, 1972. · Zbl 0234.34080 |
| [16] | Angel, E., andBellman, R.,Dynamic Programming and Partial Differential Equations, Academic Press, New York, New York, 1972. · Zbl 0312.49011 |
| [17] | Reinhardt, H. J.,Singular Perturbations of Difference Methods for Linear Ordinary Differential Equations, Applicable Analysis, Vol. 10, pp. 53-70, 1980. · doi:10.1080/00036818008839286 |
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.
