# ИССЛЕДОВАНИЕ СУЩЕСТВОВАНИЯ И АНАЛИЗ РЕШЕНИЯ ЗАДАЧИ КОШИ ДЛЯ ВОЗМУЩЕННОГО УРАВНЕНИЯ КЛЕЙНА – ГОРДОНА

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• 3 (27), 2013 - .

Physicsal and mathematical sciences. Mathematics 25

517.95 . .

. . - , , . , - - . , - , - . - - . . - . - . , . : , , - , .

E. A. Budylina

THE STUDY OF THE EXISTENCE AND THE ANALYSIS OF THE CAUCHY PROBLEM SOLUTION

FOR THE PERTURBED KLEIN-GORDON EQUATION Abstract. Background. At present time different approximate models for the descrip-tion of the gas-liquid mix move are used, and the Klein Gordon equation is one of them. Studying acoustic properties of liquids with bubbles of gas, as well as waves with the finite length in mixes with sufficiently large bubbles is based on these models. Besides, there are a number of mathematical models describing nonlinear seismic effects in geophysical environments, for instance, the sine-Gordon equation and its modifications. The objective of the paper is to study the existence and the Cauchy problem solution for the perturbed Klein Gordon equation and to deter-mine the relative error of the solution of Cauchy problem when replacing the per-turbed Klein Gordon equation with the non-perturbed one. Results and conclu-sions. The existence of the Cauchy problem solution for the perturbed Klein-Gordon equation has been proved. The relative error of the solution of the Cauchy problem for the perturbed Klein Gordon equation when replacing the perturbed Klein Gordon equation with non-perturbed has been determined. The results of the re-search make it possible to find borders at which it is admissible to replace small constant coefficients with zero. Key words: the Klein Gordon's equation, the problem of existence, the relative er-ror of the approximate solution, applications.

• .

University proceedings. Volga region 26

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• 3 (27), 2013 - .

Physicsal and mathematical sciences. Mathematics 27

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• .

University proceedings. Volga region 28

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• 3 (27), 2013 - .

Physicsal and mathematical sciences. Mathematics 29

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• .

University proceedings. Volga region 30

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Chichester, New York, Brisbane, Toronto, Signature, 1998.

References 1. Danilova E. A. Nekotorye voprosy, svyazannye s modifikatsiyami uravneniya sinus-

Gordona: dis. kand. fiz.-mat. nauk [Some questions relating to modifications of sine-Gordon equations: dissertation to apply for the degree of the candidate of physical and mathematical sciences]. Moscow, 2012, 71 p.

2. Yakushevich L. V. Nonlinear physics of DNA. Wiley, Chichester, New York, Brisbane, Toronto, Signature, 1998.

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Budylina Evgeniya Aleksandrovna Candidate of physical and mathematical sciences, senior lecturer, sub-department of information systems and distance technology, Moscow State University of mechanical engineering (MAMI) (38 Bol'shaya Semenovskaya street, Moscow, Russia)

E-mail: bud-ea@yandex.ru

517.95

, . .

/ . . // . . - . 2013. 3 (27). . 2530.