PERIODIC SOLUTIONS OF A NON-SMALL AMPLITUDE OF THE QUASILINEAR EQUATION FOR OSCILLATIONS OF AN I-BEAM
Journal Title: Современные информационные технологии и ИТ-образование - Year 2018, Vol 14, Issue 3
Abstract
The problem of time-periodic solutions of the quasilinear equation of forced oscillations of an I-beam with hinged ends is investigated. The nonlinear summand and the right side of the equation are time periodic functions. The paper studies the case when the time period is commensurate with the length of the beam. The solution is sought in the form of a Fourier series. To prove the convergence of the Fourier series and their derivatives, we study the eigenvalues of the differential operator corresponding to the linear part of the equation. Conditions are obtained under which the kernel of a differential operator is finite-dimensional and the inverse operator is completely continuous on the complement of the kernel. A lemma on the existence and regularity of solutions of the corresponding linear problem is proved. The proof is based on the properties of the sums of the Fourier series. A theorem on the existence and regularity of a periodic solution is proved if the nonlinear term satisfies the non-resonance condition at infinity. The non-resonance condition implies the fact that, for large values of the argument, the graph of the non-linear term does not intersect the straight lines whose slope is an eigenvalue of the linear part of the equation. In the proof of the theorem, an a priori estimate is made of the solutions of the corresponding operator equation and the Leray-Schauder principle of a fixed point is applied. Additional conditions are obtained under which the periodic solution found in the main theorem is unique.
Authors and Affiliations
Igor Rudakov, Elena Romanenco
Keywords
Related Articles
- EP ID EP521612
- DOI 10.25559/SITITO.14.201803.647-653
- Views 104
- Downloads 0
THE IMPLEMENTATION OF PARALLEL DATA PROCESSING IN CLOUD SYSTEMS
The methods of using cloud computing technology for the development of software that implements parallel processing of distributed data are considered. The method of parallel implementation of the JOIN operation based on...
ON WORLD MARKET OF AUTONOMOUS (DRIVERLESS) CARS
This article is devoted to the review of the world market of unmanned vehicles. It is noted that the global automotive market is now showing two key trends. This is electric mobility (replacement of batteries in electric...
PREPARATION OF QUALIFIED SPECIALISTS FOR THE FIELD OF IT FROM D-LINK
The article is devoted to the consideration of D-Link's experience in the development and implementation of educational programs for the training of qualified specialists for the field of IT. To remain competitive, a spe...
USING THE VARIABLE-LENGTH MANTISSA FOR MODELS’ PARAMETERS ESTIMATION
The article is dedicated to the problem of computing process design for computing processes used for systems parameters estimation based on measurements using maximum likelihood method, using variable length mantissa cal...
DYNAMIC ONTOLOGY BASED RETRIEVAL AND ANALYSIS OF INFORMATION ON SCIENCE AND ENGINEERING
RDF ontologies are proposed as modelling pattern for associations of innovation cycles being performed by science-intensive technical products. An appropriate ontology is applied as framework of personalized archives of...