Computational methods for time-periodic solutions of singular semilinear parabolic problems. (English) Zbl 0732.65087
Using the method of alternating bounds and the Picard method, existence and uniqueness of time-periodic solutions of a class of singular semilinear parabolic problems are established. The authors’ technique allows one to determine, for these two methods, the number of iterations and the number of terms retained (after truncating the infinite series) in each iterate such that the approximate solution is within the degree of accuracy desired. Under additional assumptions, the authors are able to improve the rate of convergence so that the number of terms in each iterate may be reduced without diminishing the degree of accuracy.
Reviewer: V.P.Tyagi (Bombay)
MSC:
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
Keywords:
method of alternating bounds; Picard method; time-periodic solutions; singular semilinear parabolic problems; number of iterations; degree of accuracy; rate of convergenceReferences:
| [1] | (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (1972), National Bureau of Standards: National Bureau of Standards Washington), 362, Applied Mathematics Series 55 · Zbl 0543.33001 |
| [2] | (Beyer, W. H., CRC Standard Mathematical Tables (1984), CRC Press: CRC Press Boca Raton, Fla.), 54 |
| [3] | Chan, C. Y.; Wong, B. M., Periodic solutions of singular linear and semilinear parabolic problems, Quart. Appl. Math., 47, 405-428 (1989) · Zbl 0693.35096 |
| [4] | C.Y. Chan and B.M. Wong, Computational error analysis of periodic solutions for singular semilinear parabolic problems, Appl. Math. Comput., to appear.; C.Y. Chan and B.M. Wong, Computational error analysis of periodic solutions for singular semilinear parabolic problems, Appl. Math. Comput., to appear. · Zbl 0767.65070 |
| [5] | Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman Advanced Publishing Program: Pitman Advanced Publishing Program Boston · Zbl 0658.35003 |
| [6] | Liu, B. P.; Pao, C. V., Periodic solutions of coupled semilinear parabolic boundary value problems, Nonlinear Anal., 6, 237-252 (1982) · Zbl 0499.35012 |
| [7] | McLachlan, N. W., (Bessel Functions for Engineers (1955), Clarendon: Clarendon London), 191-192 |
| [8] | Tolstov, G. P., (Fourier Series (1976), Dover: Dover New York), 231-233 · Zbl 0358.42001 |
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