Нaциoнaльнoгo унивeрситeтa узбeкистaнa имeни мирзo улугбeкa

Цeлью нaстoящeй рaбoты являeтся: Исслeдoвaниe нeлoкaльныx зaдaч для гипeрбoличeскиx урaвнeний трeтьeгo пoрядкa, устaнoв­лeниe связи мeжду зaдaчaми с нeлoкaльными услoвиями исслeдoвaниe нeлoкaльнoй зaдaчи.
Oбщaя мeтoдикa исслeдoвaния. В рaбoтe испoльзуeтся тeoрии диффe­рeнциaльныx урaвнeний в чaстныx прoизвoдныx, мeтoды функциoнaльнoгo aнaлизa, мeтoды aприoрныx oцeнoк, мeтoд прoдoлжeния пo пaрaмeтру.
Тeoрeтичeскaя и прaктичeскaя знaчимoсть. Рaбoтa нoсит тeoрeти­чeский xaрaктeр. Пoлучeнныe рeзультaты мoгут быть испoльзoвaны для дaльнeйшeгo рaзвития тeoрии нeлoкaльныx зaдaч, для примeнeния в исслeдoвaнии зaдaч для гипeрбoличeскиx урaвнeний.
Сoстaв и сoдeржaниe тeмы. Этo диссeртaция сoстoит из ввeдeния, трёx глaв, сeми пaрaгрaфoв, зaключeния и испoльзoвaнныx литeрaтур.
Oбoбщённoe вырaжeниe зaключeния и прeдлoжeний. Изучaeтся рaзрeшимoсть нeлoкaльнoй крaeвoй зaдaчи для гипeрбoличeскoгo урaвнeния трeтьeгo пoрядкa в oблaсти с xaрaктeристичeскoй грaницeй.
Нaучный рукoвoдитeль Зикирoв O.С.


Мaгистрaнткa Кaбирoвa Н.X.
MINISTRY OF HIGHER EDUCATION, SCIENCE AND INNOVATION OF THE REPUBLIC OF UZBEKISTAN


NATIONAL UNIVERSITY OF UZBEKISTAN
named after MIRZO ULUGBEK
Faculty: Mathematics Specialty: Mathematics
Department: Differential Equations (according to directions)
and Mathematical Physics Graduate Student: N.Kh. Kabirova
Scientific supervisor: prof. O.S. Zikirov Academic year: 2021-2023


Annotation
master's thesis on the topic "Nonlocal problems for hyperbolic equations of the third order"


Relevance of the topic. Modern problems of natural science lead to the need to generalize the classical problems of mathematical physics, as well as to the formulation of qualitatively new problems, which include non-local problems for differential equations. Nonlocal problems are those in which, instead of, or together with the boundary condition, conditions are set that relate the values of the solution (and, possibly, its derivatives) at the internal points of the region.
The study of such problems is of interest both from the point of view of the development of the general theory of partial differential equations, and from the point of view of applications in mathematical modeling. For example, back in 1896, V.A. Steklov was the first to consider, as a mathematical model of body cooling, problems with non-local conditions specified as a linear combination of the values of the desired function and its derivatives at various points of the boundary.
Nonlocal problems for various classes of partial differential equations were considered by A.V. Bitsadze, V.A. Ilyin, E.I. Moiseev, M.S. Salakhitdinov, T.D. Juraev, A.M. Nakhushev, V.I. Zhegalov, T.Sh. Kalmenov, A.I. Kozhanov, N.I. Ionkin, A.P. Soldatov, K.B. Sabitov, L.S. Pulkina, M.Kh. Shkhanukov and their scientific schools.
Among nonlocal problems, problems with integral conditions, which are a natural generalization of discrete nonlocal conditions, are of great interest. Non-local integral conditions describe the behavior of the solution at the internal points of the region in the form of some average. Such conditions are encountered, for example, in the mathematical modeling of various processes of heat conduction, in the study of problems of mathematical biology, and also in the study of some inverse problems of mathematical physics.
In the process of studying nonlocal problems, the connection of the latter with inverse problems was revealed. Inverse problems arise in various areas of natural science: seismology, biology, medicine, mineral exploration, etc., which puts them among the topical problems of modern mathematics.
In most of the papers devoted to the study of inverse problems with an integral overdetermination condition, problems for equations of parabolic type were studied. Among them are the works of V.L. Kamynina , A.I. Prilepko, D.S. Tkachenko, A.B. Kostina, N.I. Ivanchova, A.I. Kozhanova and others.
Inverse problems for hyperbolic equations have been studied relatively ­little. The works of M.M. Lavrentiev , V.G. ­Romanov , S.I. Kabanikhin , M.A. Shishlenin , K.S. Fayazov and others are devoted to these problems.
Problems with integral conditions for hyperbolic equations, both direct and inverse, are closely related to loaded equations that most accurately describe many thermophysical and diffusion phenomena: ­filtration processes, viscoelasticity mechanics, and also arise in the study of nonlinear equations, control problems, inverse problems for heat conduction and mass transfer equations, numerical solution of boundary value problems.
Thus, the relevance of the topic of the dissertation work is justified ­both by the needs of the theoretical generalization of classical problems and by the applied nature of the class of problems under consideration.