 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4760 |
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| L. D. Akulenko and S. V. Nesterov, "An effective numerical-analytical method for solving one-dimensional variational problems of mechanics," Mech. Solids. 36 (6), 64-75 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
6 |
Pages |
64-75 |
| Title |
An effective numerical-analytical method for solving one-dimensional variational problems of mechanics |
| Author(s) |
L. D. Akulenko (Moscow)
S. V. Nesterov (Moscow) |
| Abstract |
A high-precision effective numerical-analytical method is presented
for solving nonlinear boundary-value problems for the Euler-Lagrange
equation subject to various boundary conditions. The computational
algorithm is based on a non-standard approach employing the dependence
of the missing initial (or terminal) value of the variable on the length
of the interval and introducing a small variable of the boundary-value
problem in an original way. A recurrence procedure involving
numerical solution of initial value problems (Cauchy problems)
is suggested for successive
refinement of this value and the desired solution. This procedure
has a quadratic convergence with respect to the small parameter
characterizing the relative residual of the approximate solution with
respect to the argument. The method is developed for boundary conditions
of the first and second kinds, as well as for nonlinear boundary
conditions of the third kind. A modification of this algorithm is
suggested utilizing the residual of the solution of the Cauchy problem at
the terminal point. The algorithm is tested by solving and analyzing
essentially nonlinear case problems from magnetic hydrodynamics and
propagation of rays in nonhomogeneous media satisfying Fermat's principle. |
| References |
| 1. | L. S. Polak (Editor), Variational Principles of Mechanics [in
Russian], Fizmatgiz, Moscow, 1959. |
| 2. | R. Courant and D. Hilbert, Methoden der mathematischen Physik.
Volume 1, Springer, Berlin, 1968. |
| 3. | N. I. Akhiezer, Lectures on the Calculus of Variations [in Russian],
Gostekhizdat, Moscow, 1955. |
| 4. | L. E. El'sgol'ts, Differential Equations and Calculus of Variations
[in Russian], Nauka, Moscow, 1965. |
| 5. | L. C. Young, Lectures on the Calculus of Variations and Optimal
Control Theory, W. B. Sounders Co., Philadelphia, 1969. |
| 6. | L. Collatz, Funktionalanalysis and numerische Mathematik,
Springer-Verlag, Berlin, 1969. |
| 7. | R. E. Bellman and R. E. Kalaba, Quasilinearization
and Nonlinear Boundary-value Problems, American Elsiver Publ. Co.,
INC, New York, 1965. |
| 8. | F. L. Chernous'ko and N. V. Banichuk, Variational Problems
of Mechanics and Control [in Russian], Nauka, Moscow, 1973. |
| 9. | K. I. Babenko, Fundamentals of the Numerical Analysis [in Russian],
Nauka, Moscow, 1986. |
| 10. | L. D. Akulenko and S. V. Nesterov, "An effective method
for investigating vibrations of essentially nonhomogeneous distributed
systems," PMM [Applied Mathematics and Mechanics], Vol. 61, No. 3,
pp. 466-478, 1997. |
| 11. | L. D. Akulenko and S. V. Nesterov, "An effective solution
of the generalized Sturm-Liouville problem,"
Doklady AN, Vol. 363, No. 3, pp. 323-326, 1997. |
| 12. | L. M. Brekhovskikh, Waves in Laminated Media [in Russian],
Izd-vo AN SSSR, Moscow, 1957. |
| 13. | I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations
[in Russian], Gostekhizdat, Moscow, 1956. |
| 14. | H. Poincaré, Les Methodes Nouvelles de la Mechanique Celeste,
3 Vols, Paris, 1892, 1893, 1899. |
| 15. | L. D. Akulenko and S. V. Nesterov, "An effective
numerical-analytical solution of variational problems of mechanics,"
Doklady AN, Vol. 374, No. 5, pp. 624-627, 2000. |
| 16. | L. D. Akulenko, D. V. Georgievskii, S. A. Kumakshev,
and S. V. Nesterov, "Numerical-analytical investigation of the steady-state
flow of a viscous fluid in a plane convergent channel,"
Doklady AN, Vol. 374, No. 1, pp. 44-48, 2000. |
| 17. | D. M. Klimov, S. V. Nesterov, L. D. Akulenko,
D. V. Georgievskii, and S. A. Kumakshev, "The flow
of a viscoplastic medium with small yield limit in a plane
convergent channel,"
Doklady AN, Vol. 375, No. 1, pp. 37-41, 2000. |
|
| Received |
18 October 1999 |
| << Previous article | Volume 36, Issue 6 / 2001 | Next article >> |
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