Integrating ODEs in the complex plane—pole vaulting
Author: George F. Corliss
Journal: Math. Comp. 35 (1980), 1181-1189
MSC: Primary 65L05; Secondary 34A20
MathSciNet review: 583495
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Abstract: Most existing algorithms for solving initial value problems in ordinary differential equations implicitly assume that all variables are real. If the real-valued assumption is removed, the solution can be extended by analytic continuation along a path of integration in the complex plane of the independent variable. This path is chosen to avoid singularities which can make the solution difficult or impossible for standard methods. We restrict our attention to Taylor series methods, although other methods can be suitably modified. Numerical examples are given for (a) singularities on the real axis, (b) singularities in derivatives higher than those involved in the differential equation, and (c) singularities near the real axis. These examples show that the pole vaulting method merits further study for some special problems for which it is competitive with standard methods.
- Constructive and computational methods for differential and integral equations, Lecture Notes in Mathematics, Vol. 430, Springer-Verlag, Berlin-New York, 1974. Symposium, Indiana University, Bloomington, Ind., February 17–20, 1974; Edited by D. L. Colton and R. P. Gilbert. MR 0356437
- Y. F. Chang and G. Corliss, Ratio-like and recurrence relation tests for convergence of series, J. Inst. Math. Appl. 25 (1980), no. 4, 349–359. MR 578082
- Harold T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York, 1962. MR 0181773
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
- Peter Henrici, Applied and computational complex analysis, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros; Pure and Applied Mathematics. MR 0372162
- Keith Miller, Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal. 1 (1970), 52–74. MR 260196, DOI https://doi.org/10.1137/0501006
- Ramon E. Moore, Interval analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0231516
Y. F. CHANG, "Automatic solution of differential equations," Constructive and Computational Methods for Differential and Integral Equations (D. L. Colton and R. P. Gilbert, Eds.), Lecture Notes in Math., Vol. 430, Springer-Verlag, New York, 1974, pp. 61-94.
Y. F. CHANG & G. F. CORLISS, "Ratio-like and recurrence relation tests for convergence of series," J. Inst. Math. Appl. (To appear.)
HAROLD T. DAVIS, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962.
PHILIP HARTMAN, Ordinary Differential Equations, Wiley, New York, 1964.
PETER HENRICI, Applied and Computational Complex Analysis, Vol. 1, Wiley, New York, 1974.
KEITH MILLER, "Least squares methods for ill-posed problems with a prescribed bound," SIAM J. Math. Anal., v. 1, 1970, pp. 52-74.
R. E. MOORE, Interval Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1966.
Keywords: Ordinary differential equations, numerical solutions, Taylor series, numerical analytic continuation, singularities in the complex plane, pole vaulting
Article copyright: © Copyright 1980 American Mathematical Society