Reduction of a Mixed Characteristic Problem to the Initial Cauchy Problem for Nonlinear Oscillation Equations

Volume 6, Issue 1, February 2021     |     PP. 1-17      |     PDF (221 K)    |     Pub. Date: March 8, 2021
DOI: 10.54647/mathematics11242    184 Downloads     5873 Views  

Author(s)

Rusudan Bitsadze, Department of Mathematics, Georgian Technical University, Tbilisi, Georgia

Abstract
For the well-known nonlinear oscillation equation, we consider a nonlinear mixed characteristic problem, which is a nonlinear analogue of the Darboux problem and consists in the simultaneous definition of a solution and its regular propagation domain. The question of solvability of the formulated problem is solved by the method of characteristics and reduction to the Cauchy problem.

Keywords
characteristics, general integral, definition domain, initial problem

Cite this paper
Rusudan Bitsadze, Reduction of a Mixed Characteristic Problem to the Initial Cauchy Problem for Nonlinear Oscillation Equations , SCIREA Journal of Mathematics. Volume 6, Issue 1, February 2021 | PP. 1-17. 10.54647/mathematics11242

References

[ 1 ] Goursat, E., Lecons sur l’intégration des équations aux dérivées partielles du second ordre à deux variables indépendantes. Tome II: La méthode de Laplace. Les systèmes en involutions. La methode de M. Darboux. Les équations de la première classe. Transformations des équations du second ordre. Généralisations diverses, 2 Notes A. Hermann, Paris, 1898.
[ 2 ] Bers, L., Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Surveys in Applied Mathematics, vol. 3 John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958.
[ 3 ] Gvazava, J., “On one nonlinear version of the characteristic problem with a free support of data”, Proc. A. Razmadze Math. Inst., 140. 91-107. 2006.
[ 4 ] Gvazava, J., “The mean value property for nonstrictly hyperbolic second order quasilinear equations and the nonlocal problems”, Proc. A. Razmadze Math. Inst. 135. 79-96. 2004.
[ 5 ] Klebanskaya, M., “Some nonlinear versions of Darboux and Goursat problems for a hyperbolic equation with parabolic degeneracy”, in International Symposium on Differential Equations and Mathematical Physics dedicated to the 90th birthday anniversary of Academician I. Vekua, Tbilisi, Georgia, June 21-25. 1997.
[ 6 ] Gvazava, J.K., “Second-order nonlinear equations with complete characteristic systems and characteristic problems for them”. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 87. 45-53. 1987.
[ 7 ] Bitsadze, R., “On one version of the initial-characteristic Darboux problem for one equation of nonlinear oscillation”, Reports of an Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 8 (1). 4-6. 1993.
[ 8 ] Kumei S. and Bluman G.W., “When nonlinear differential equations are equivalent to linear differential equations”, SIAM J. Appl. Math. 42(5). 1157-1173. 1982.
[ 9 ] Olver P., Applications of Lie Groups to Differential Equations, Mir, Moscow, 1989.
[ 10 ] Tricomi F., Lectures on Partial Equations, Moscow, 1957.
[ 11 ] Bitsadze R.G., “General representation of solutions of a quasilinear equation of a nonlinear oscillations problem”, Soobshch. Akad. Nauk Gruzin. SSR 128 (3). 493-496. 1988 (in Russian).
[ 12 ] Bitsadze R., “Mixed characteristic problem for a nonlinear oscillation equation”, (Russian) Sovrem. Mat. Prilozh. No. 89 (2013); translation in J. Math. Sci. (N.Y.) 206 (4). 329-340. 2015.
[ 13 ] Goursat E., Course of Mathematical Analysis, vol. 3, part I, Moscow, 1933.