Geodesic
Convergence of the successive approximation method in the Cauchy problem for an integro-differential equation with quadratic nonlinearity
Matematičeskie trudy, Tome 21 (2018) no. 2, pp. 136-149 
V. L. Vaskevich; A. I. Shcherbakov. Convergence of the successive approximation method in the Cauchy problem for an integro-differential equation with quadratic nonlinearity. Matematičeskie trudy, Tome 21 (2018) no. 2, pp. 136-149. http://geodesic.mathdoc.fr/item/MT_2018_21_2_a5/
@article{MT_2018_21_2_a5,
     author = {V. L. Vaskevich and A. I. Shcherbakov},
     title = {Convergence of the successive approximation method in the {Cauchy} problem for an integro-differential equation with quadratic nonlinearity},
     journal = {Matemati\v{c}eskie trudy},
     pages = {136--149},
     year = {2018},
     volume = {21},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2018_21_2_a5/}
}
TY  - JOUR
AU  - V. L. Vaskevich
AU  - A. I. Shcherbakov
TI  - Convergence of the successive approximation method in the Cauchy problem for an integro-differential equation with quadratic nonlinearity
JO  - Matematičeskie trudy
PY  - 2018
SP  - 136
EP  - 149
VL  - 21
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/MT_2018_21_2_a5/
LA  - ru
ID  - MT_2018_21_2_a5
ER  - 
%0 Journal Article
%A V. L. Vaskevich
%A A. I. Shcherbakov
%T Convergence of the successive approximation method in the Cauchy problem for an integro-differential equation with quadratic nonlinearity
%J Matematičeskie trudy
%D 2018
%P 136-149
%V 21
%N 2
%U http://geodesic.mathdoc.fr/item/MT_2018_21_2_a5/
%G ru
%F MT_2018_21_2_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

The equations considered in this article have the form in which the time derivative of the unknown function is expressed as a double integral over the space variables of a weighted quadratic expression of the sought function. The domain of integration is unbounded and does not depend on time but depends on the space variable. We study the Cauchy problem in the function classes accompanying the equation with initial data on the positive half-line. In application to this problem, the convergence of the successive approximation method is justified. An estimate is given of the quality of the approximation depending on the number of the iterated solution. It is proved that, on any finite time interval, the posed Cauchy problem has at most one solution in the accompanying function class. An existence theorem is proved in the same class.

[1] Vainberg M. M., “Integro-differentsialnye uravneniya”, Itogi nauki. Matem. analiz. Teoriya veroyatnostei. Regulirovanie 1962, VINITI, M., 1964, 5–37

[2] Vainberg M. M., Trenogin V. A., Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR

[3] Bell N. K., Grebenev V. N., Medvedev S. B., Nazarenko S. V., “Self-similar evolution of Alfven wave turbulence”, J. Phys. A: Math. Theor., 50:43 (2017) | DOI | MR

[4] Galtier S., “Wave turbulence in astrophysics”, Advances in Wave Turbulence, World Scientific Series on Nonlinear Science, Ser. A, 83, World Scientific Publishing, Singapore, 2013, 73–111 | DOI | MR | Zbl

[5] Galtier S., Nazarenko S., Newell A. C., Pouquet A., “A weak turbulence theory for incompressible magnetohydrodynamics”, J. Plasma Physics, 63:5 (2000), 447–488 | DOI

[6] Galtier S., Buchlin E., “Nonlinear diffusion equations for anisotropic magnetohydrodynamic turbulence with cross-helicity”, The Astrophysical J., 722 (2010), 1977–1983 | DOI

[7] Nazarenko S., Wave Turbulence, Lecture Notes in Physics, 825, Springer, Berlin–Heidelberg, 2011 | DOI | MR | Zbl