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 115
- Downloads 0
HASH ALGORITHM WITH THE CONTROLLING TREE-LIKE STRUCTURE AND THE METHOD OF ITS IMPLEMENTATION ON PARALLEL ARCHITECTURES
The article is devoted to research and development of new hashing methods that have the ability to adapt to increased requirements for cryptographic stability, and also support parallelization of computations, which is e...
PROGRAMS FOR ATRIBUTION AND FOR VIEWING PHOTOCOPIES OF A COLLECTION OF PORTRAITS
The article deals with the problem of digitization and attribution of portraits in archive collections. The principal example is the collection of portraits from the Moscow Society of Naturalists archives. Special softwa...
INNOVATIVE DIDACTIC ELECTRONIC RESOURCES AND TEACHER'S PRODUCTS IN IT-EDUCATION
The article describes the design and implementation of computer support for the teacher’s activities, which combines such opportunities as data accounting and analysis, the use of pedagogical technology, and the situatio...
TIME EFFICIENT ALGORITHM FOR THE GENERALIZED0 ENTROPYCALCULATION OF TWO-DIMENSIONS WORDS BY THE METHOD OF A SLIDING WINDOW
In the article we consider methods for calculating the entropy of two-dimensional words of finite length over in finite alphabet, with reference to the problem of self-organization of rod-like particles on a torus. Monit...
NEURAL NETWORK MODEL OF PREDICTING THE RISK GROUP FOR THE ACCESSION OF STUDENTS OF THE FIRST COURSE
Many Russian universities face the problem when applicants who successfully passed a single state examination in core disciplines become outsiders as a result of the first academic period. Especially it concerns the spec...