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Introduction. The emerging digital society demonstrates the problem of justifying the value, worth and price of digital products and services, which requires consideration of the specificity of value in this society. Theoretical analysis. Following the financial and art market, the boundary between “value” and “worth” is blurred in the digital society. Value, including the value of an individual, becomes variable and is conditioned by the presence of unique attributes and unique experiences that obtain market value. At the same time, under standard digital protocols and technologies, the true uniqueness of an individual experience can only be achieved through the creation of customized digital ecosystems in the form of robotic swarms that support human life activities. A set of digital devices that have sufficient resources to continuously self-renew and form a closed environment around the individual that provides a unique experience can be called a chrysalic swarm. The chrysalic swarm creates an opportunity for the individual to escape from the logic of mutual competition dictated by society. Conclusion. Studying the value problematization in the digital society of swarm intelligence opens a theoretical perspective on the problem of the crisis of social elites in the context of the fulfillment of this scenario of society development.

В данной работе для уравнения в частных производных четвертого порядка в прямоугольной области рассмотрена нелокальная обратная задача по поиску неизвестной правой части, которая зависит от одной переменной. Собственные функции и присоединенные функции соответствующей спектральной задачи, и их биортогональные функции полны и образуют базис Рисса в пространстве L2​(0,1). Установлены критерии единственности и существования решения рассматриваемой нелокальной обратной задачи для уравнения четвертого порядка. Единственность решения нелокальной обратной задачи вытекает из полноты системы биортогональных функций. Решение задачи построено в виде суммы ряда по собственным и присоединенным функциям, соответствующей спектральной задачи. Установлены достаточные условия на граничные функции, которые гарантируют теоремы существования и устойчивости решения рассматриваемой задачи. В замкнутой области показана абсолютная и равномерная сходимость найденного решения обратной задачи в виде ряда, а также рядов, полученных почленным дифференцированием по t и x соответственно два и четыре раза, в зависимости от гладкости функции заданными начальными условиями. При этом возникают малые знаменатели, затрудняющие сходимость этих рядов. Доказано, что в зависимости от размера области, множество ненулевых решений выражения в знаменателе не пусто. А также, показано, что если этот знаменатель равен нулю, то данная задача будет иметь нетривиальное решение при однородных условиях. Доказана также, устойчивость решения обратной задачи по нормам пространств L2​,W2n​ и C(Ω±), относительно изменения входных данных.

This paper investigates the solvability of the inverse problem of finding a solution and an unknown coefficient in a pseudohyperbolic equation known as the Klein-Gordon equation. A distinctive feature of the given problem is that the unknown coefficient is a function that depends only on the time variable. The problem is considered in the cylinder, the conditions of the usual initial-boundary value problem are set. The integral overdetermination condition is used as an additional condition. In this paper, the inverse problem is reduced to an equivalent problem for the loaded nonlinear pseudohyperbolic equation. Such equations belong to the class of partial differential equations, not resolved with respect to the highest time derivative, and they are also called composite type equations. The proof uses the Galerkin method and the compactness method (using the obtained a priori estimates). For the problem under study, the authors prove existence and uniqueness theorems for the solution in appropriate classes.

A nonlinear boundary value problem for a third-order nonlinear equation with multiple characteristics is studied in the article in a curvilinear domain. The unique solvability of the problem is proved. The uniqueness of the solution to the boundary value problem is proved by the energy integral method using some elementary inequalities. An auxiliary problem is considered for the existence of a solution, for which the Green function is constructed. By solving an auxiliary problem, the original problem is reduced to a system of Hammerstein integral equations. The solvability of a nonlinear system is proved by the contracting mapping method.

The fractional differential equations play important roles due to their numerous applications and also for the important role they play not only in mathematics but also in other sciences. In the present research work, we build up the explicit solutions to linear fractional q-differential equations with the q-Caputo fractional derivative of real order a > 0. To speak more precisely, we will achieve our main results we use that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. After that, by using the method of successive approximations is applied to solve the Volterra q-integral equation we construct the the explicit solutions to linear fractional q-differential equations. In the same way we have the more general homogeneous fractional q-differential equation with the Caputo fractional q-derivative of real order a > 0 and we give other The (Mittag-Leffler) q-function. Finally, some examples are presented to illustrate our main results in cases where we can even give concrete formulas for these explicit solutions.

Inverse problems of finding the right-hand side of a partial differential equations arise when an external source is unknown or impossible to measurement, for example, the source is in a high-temperature environment or underground. The partial differential equations with mixed derivatives by time and space variables are usually called pseudo-parabolic equations or Sobolev type equations. Pseudo-parabolic equations occur in mathematical modeling of many physical phenomena such as the motion of non-Newtonian fluids, thermodynamic processes, filtration in a porous medium, unsteady flow of second-order fluids, etc. This paper is devoted to investigate the unique solvability of two inverse problems for a linear one dimensional pseudoparabolic equation. The inverse problems consist of recovering the right hand side of the equation depending on the space variable. An additional information for the first inverse problem is given by an final overdetermination condition and for the second is given by an integral overdetermination condition. Under the suitable conditions on the initial data of the problem, the existence and uniqueness of a classical solution to these inverse problems are established. By using the Fourier method, the explicit formulas of a solutions are presented in the form of a series, which make it possible to perform the necessary numerical calculations with a given accuracy.

In this paper, we study the solvability of some non-local analogs of the second initial-boundary value problem for multidimensional hyperbolic and parabolic equations of the second order. We prove the existence and uniqueness theorems of regular solutions (which have all Sobolev generalized derivatives that are summable with a square and are included in the equation). Some generalization and amplification of the obtained results are also given.

The study of equations of mathematical physics, including inverse problems, is relevant today. This work is devoted to the fundamental problem of studying the solvability and qualitative properties of the solution of the inverse problem for a quasilinear pseudoparabolic equation (also called Sobolev-type equations) with memory term. To date, studies of direct and inverse problems for a pseudoparabolic equations are rapidly developing in connection with the needs of modeling and control of processes in thermal physics, hydrodynamics, and mechanics of a continuous medium. The pseudoparabolic equations similar to those considered in this work arise in the description of heat and mass transfer processes, processes of non-Newtonian fluids motion, wave processes, and in many other areas. The main types of the inverse problems are: boundary, retrospective, coefficient and geometric. The boundary and retrospective inverse problems lead to the study of linear problems. In turn, the statements related to the study of coefficient and geometric types bring to the nonlinear problems. Coefficient inverse problems are divided into two main groups: coefficient inverse problems, where the unknown is a function of one or several variables, and finite-dimensional coefficient inverse problems. In this article the existence and uniqueness of a weak and strong solution of the inverse problem in a bounded domain are proved by the Galerkin method. Also we used Sobolev’s embedding theorems, and obtained a priori estimates for the solution. Moreover, we get local and global theorems on the existence of the solution.

The aim of this paper is to study the solvability of solutions of non-local problems with integral conditions in spatial variables for high-order linear parabolic equations in the classes of regular solutions (which have all the squared derivatives generalized by S. L. Sobolev that are included in the corresponding equation) . Previously, similar problems were studied for high-order parabolic equations, either in the one-dimensional case, or when certain conditions of smallness on the coefficients are met equations. In this paper, we present new results on the solvability of non-local problems with integral spatial variables for high-order parabolic equations a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions. The research method is based on the transition from a problem with non-local integral conditions to a problem with classical homogeneous conditions of the first or second kind on the side boundary for a loaded integro-differential equation. At the end of the paper, some generalizations of the obtained results will be described.

New sufficient conditions for the existence and uniqueness of a periodic solution of a system of differential equations equivalent to the Rayleigh equation are obtained. In contrast to the known results, the existence proof of at least one limit cycle of the system is based on applying curves of the topographic Poincare system. The uniqueness of the limit cycle surrounding a complex unstable focus is proved by the Otrokov method.

В работе рассматривается обратная задача для гиперболического уравнения третьего порядка. Ставится обратная задача, состоящая в определении неизвестного коэффициента, зависящего от времени. В качестве дополнительной информации для решения обратной задачи задаются значения решения задачи во внутренней точке. Доказывается теорема существования и единственности решения обратной задачи. Доказательство основано на выводе нелинейной системы интегральных уравнений типа Вольтерра второго рода и доказательстве его разрешимости.

In this paper, an initial-boundary value problem for the equation of forced vibrations of a cantilever beam is studied. Such a linear differential equation of the fourth order describes bending transverse vibrations of a homogeneous beam under the action of an external force in the absence of rotational motion during bending. The system of eigenfunctions of the one-dimensional spectral problem, which is orthogonal and complete in the space of square-summable functions, is constructed by the method of separation of variables. The uniqueness of the solution to the initial-boundary value problem is proved in two ways: (i) using the energy integral; (ii) relying on the completeness property of the system of eigenfunctions. The solution to the problem was first found in the absence of an external force and homogeneous boundary conditions, and then the general case was considered in the presence of an external force and inhomogeneous boundary conditions. In both cases, the solution of the problem is constructed as the sum of the Fourier series. Estimates of the coefficients of these series and the system of eigenfunctions are obtained. On the basis of the established estimates, sufficient conditions were found for the initial functions, the fulfillment of which ensures the uniform convergence of the constructed series in the class of regular solutions of the beam vibration equation, i.e. existence theorems for the solution of the stated initial-boundary value problem are proved. Based on the solutions obtained, the stability of the solutions of the initial-boundary value problem is established depending on the initial data and the right-hand side of the equation under consideration in the classes of square-summable and continuous functions.

The paper is devoted to the study of the solvability of nonlocal problems with an integral variable $t$ condition for the equations $$u_{tt}+(\alpha\frac{\partial}{\partial t}+\beta)\Delta u=f(x,t)$$($\alpha$, $\beta$ are valid constants, $\Delta$ is Laplace operator by spatial variables). Theorems are proved for the studied problems existence and non-existence, uniqueness and non-uniqueness solutions (having all derivatives generalized by S. L. Sobolev included in the equation).

В начале 2019/20 учебного года исполнилось 60 лет двум первым музыковедческим кафедрам - истории музыки и теории музыки. Деятельность представителей этих кафедр послужила основой для возникновения в Новосибирске авторских школ и различных направлений научно-исследовательской и музыкально-педагогической работы. Наиболее авторитетными фигурами и первопроходцами сибирского музыковедения были А.Н. Котляревский, Ю.Г. Кон, Л.Я. Хинчин, Т.А. Роменская, А.М. Айзенштадт и др. Открытие в Новосибирской консерватории музыковедческих кафедр стало новаторским шагом в процессе подготовки в регионе кадров высшей ступени образования. Успехи в этом деле были обеспечены концептуально значимыми взглядами и идеями на учебно-воспитательный процесс. Поэтому в статье формулируются фундаментальные для воспитания и образования музыкантов-профессионалов установки и постулаты. Они откристаллизовались благодаря суммарному опыту представителей музыковедческих школ различных вузов и регионов - первых педагогов Новосибирской консерватории, которые претворяли их в учебно-педагогической деятельности. Основополагающие идеи сформулированы автором статьи на основе учебного и межличностного общения с первыми музыковедами консерватории. Обращает на себя внимание тот факт, что эти и другие похожие идеи, проверенные временем и органично свойственные педагогике музыкального искусства, никем из них не декларировались, но интуитивно присутствовали и всегда реализовывались на практике. Совершенно очевидно, что педагоги-музыканты намного опередили то время, когда в общей педагогике появились такие прогрессивные термины, как «опережающее обучение», «деятельностный подход» и др. Представленные в статье идеи имеют непреходящее значение, ибо в них заключены вневременные ценностные смыслы, обусловливающие успешность воспитания, образования и развития музыкантов-профессионалов.

The article reflects the current state and the prospects of the Kostroma cattle breed development. The creation and improvement of the breed anticipated the formation and further breed development in the domestic and world zoo technical science. In recent years, the number of Kostroma cattle breed has been unreasonably reducing despite its economic valuable traits. At the same time in the Kostroma region there are prospects for preserving the valuable gene pool of one of the best domestic dairy and meat cattle breeds. As a result of directed selection and breeding work with the use of Brown Swiss bulls, a stud milk breed "Karavaevsky KK-1" has been developed. At present cow-champions of the Kostroma breed give up to 12,000 kg of milk per lactation. The milk of the Kostroma breed cows has a high content of protein and fat. The animals of Kostroma breed have good meat qualities, high resistance to a number of diseases, strong ungulate horn, easy adaptability to intensive technologies, good fertility and high productive longevity. To save the Kostroma breed it is important to put on 8-10 bulls of 8 planned lines and related groups for the validation and accumulation of sperm every year. The restoration of previously created lines with high genetic potential should be foreseen. It is necessary to identify and actively use the stud bulls bred in their own breeding farms; to conduct a linear assessment of bulls and daughter heifers by body type for more competent selection. In addition to the generally accepted forms and methods of selection, it is necessary to continue immunogenetic control of the authenticity of the origin of breeding animals and introduce modern methods of DNA technology to assess the breeding and productive qualities of cattle, which will reveal valuable genetic complexes in the best breeding animals and thus 1.5-2 times accelerate the selection processes.Conflict of interest: the authors stated that there was no conflict of interest.

The article is devoted to the study of the solvability of boundary value problems for third-order Sobolev-type differential equations of the third order with two time variables (such equations are also called composite-type equations or equations not solved for the derivative). The peculiarities of the equations under study are, firstly, that the differential operators acting at the time derivatives are not assumed inverse, and, secondly, that the statements of boundary value problems for them are determined by the coefficients of these differential operators. For the problems proposed in the article, we prove existence and uniqueness theorems for regular solutions (solutions having all weak derivatives in the sense of Sobolev involved in the equation). The technique of proving the existence theorems is based on a special regularization of the equations under study, a priori estimates, and passage to the limit.

In this paper, the intervals of change in the exponent of the degree of degeneration of a mixed-type equation with characteristic degeneration are established. The first boundary problem and the modified boundary problem (analogue of the Keldysh problem) with the conditions of periodicity are correctly set. In the case of the first problem, a criterion for the uniqueness of its solution is manifested. It is shown that the solution of the analogue of the Keldysh problem is unique up to a term of a linear function. Solutions are constructed as the sum of series of eigenfunctions of the corresponding one-dimensional spectral problem. In justifying the convergence of a series representing the solution of the first boundary-value problem, the problem of small denominators of a more complex structure arises in the class of regular solutions of this equation than in previously known works. The estimate on separation from zero is established with the corresponding asymptotic. Based on this estimate, sufficient conditions are found for the boundary functions to substantiate the uniform convergence of the series and their derivatives up to the second order inclusive.

The inverse problems for finding the initial condition and the right-hand side were studied for the heat transfer equation. A solution of the initial boundary value problem for the inhomogeneous heat transfer equation with sufficient conditions for the solvability of the problem was constructed in the first place. On the basis of the solution of the initial boundary value problem, a criterion for the uniqueness of the solution of the inverse problem to determine the initial condition was established. The study of the inverse problem of finding the right-hand side of the component, which depends on time, is equivalent to reducing to the unique solvability of the Volterra integral equation of the second kind. In view of the unique solvability of the given integral equation in the class of continuous functions, we obtained theorems for the unique solvability of the inverse problem. The solution of the inverse problem to determine the factor of the right-hand side, depending on the spatial coordinate, was constructed as a sum of the series in the system of eigenfunctions of the corresponding one-dimensional spectral problem; the criterion of uniqueness was established, and the existence and stability theorems of the solution of the problem were proved.

Background. This work is devoted to the study of sets of functions in which the condition of unique solvability of degenerate polysingular integral equations is satisfied, and to the construction of approximate methods for solving polysingular integral equations in degenerate cases. Nowadays, the study of many sections of singular integral equations can be considered largely completed. Some of the exceptions are singular and polysingular integral equations that vanish on manifolds with measure greater than zero. The theory of singular integral equations in degenerate cases is constructed, from which it follows that degenerate singular integral equations have an infinite number of solutions and for these equations the first and second Noether theorems are not valid. For polysingular integral equations, a similar theory has not yet been constructed. Moreover, there are no specific algorithms and approximate methods for solving polysingular integral equations in degenerate cases. Due to the fact that many processes in physics and technology are modeled by degenerate polysingular integral equations, it becomes necessary to develop approximate methods for their solution. In addition, since in the Hölder space and in the space of functions summable in a square, the degenerate polysingular integral equations have an infinite number of solutions, the actual problem of identifying the sets of uniqueness of solutions of these equations arises. The problem of constructing approximate methods for solving degenerate polysingular integral equations is no less urgent. Materials and methods. To distinguish classes of functions in which degenerate polysingular integral equations have a unique solution, methods of the theory of functions of a complex variable, Riemann boundary value problems and the theory of singular integral equations are used. When constructing approximate methods, iterativeprojection methods are used. Results. Classes of functions are constructed on which solutions of degenerate polysingular integral equations, if they exist, are uniquely determined. In this regard, a new formulation of the problem of solving degenerate polysingular integral equations is proposed. Methods of collocation and mechanical quadratures for solving degenerate polysingular integral equations on the constructed classes of functions are proposed and substantiated. Conclusions. The proposed results can be directly used in solving many problems of physics and technology, in particular, in problems of integral geometry, aerodynamics, hydrodynamics. It is of interest to extend these results to degenerate multidimensional singular integral equations.

In the rectangular domain, the initial-boundary value problem for the Helmholtz equation and its non-local modifications are studied and the inverse problems for finding its right-hand side are studied. The solutions of direct problems with nonlocal boundary conditions and inverse problems are constructed in explicit form as the sums of orthogonal series in the system of eigenfunctions of the one-dimensional Sturm-Liouville spectral problem. The corresponding uniqueness theorems for the solution of all set problems are proved. Sufficient conditions for boundary functions are established, which are guaranteed by the existence and stability theorems for the solution of the proposed new problem statements.

In mathematical modeling of continuous media with memory, we deal with equations that describe a new type of wave motion, something between ordinary wave diffusion and classical wave propagation. There are fractional differential equations, which are the basis for the most mathematical models describing a wide class of physical and chemical processes in the fractal geometry of the Nature. The paper presents a new moisture transfer equation with a fractional Riemann – Liouville derivative that generalize the Aller – Lykov equation. The first boundary value problem for the generalized moisture transfer equation is considered. To prove the uniqueness of a solution we employ the energy inequalities method; an a priori estimate is obtained in terms of the fractional Riemann – Liouville derivative. The existence of the solution for the problem is proved by the Fourier method.

Three-dimensional analogue of the problem of Tricomi with non-characteristic parallel planes of change of types of the equation is investigated for a parabolic-hyperbolic equation. The uniqueness of the solution of the problem is proved by the method of a priori estimates, and the existence of the solution of the problem is enlightened to the existence of a solution of the system of the second type Voltaire integral equation.

In this paper we consider the first boundary value problem in a rectangular area for a mixed-type equation of the second kind with a singular coefficient. The criterion of the uniqueness of the problem solution is determined. The uniqueness of the problem solution is proved on the basis of completeness of the system of eigenfunctions of the corresponding onedimensional spectral problem. The solution of the problem is built explicitly as a sum of Fourier-Bessel. There is the problem of the small denominators that appears when justifying the uniform convergence of the constructed series. In this regard, an evaluation of separateness from zero with a corresponding small denominator asymptotic behavior is found. This estimate has allowed to prove the convergence of the series and its derivatives up to the second order, and the existence theorem for the class of regular solutions of this equation.

We consider a boundary value problem for a hyperbolic equation with Bessel differential operator in a rectangular domain with integral nonlocal boundary value condition of the first kind. The equivalence between boundary value problem with integral nonlocal condition of the first kind and a local boundary value problem with mixed boundary conditions of the first and third kinds is proved. The existence and uniqueness of solution of the equivalent problem are established by means of the spectral method. At the uniqueness proof the completeness of the eigenfunction system of the spectral problem is used . At the existence proof the assessment of coefficients of series, the asymptotic formula for Bessel function of the first kind and asymptotic formula for eigenvalues are used. Sufficient conditions on the functions defining initial data of the problem are received. The solution of the problem is obtained in explicit form. The solution is obtained in the form of the Fourier-Bessel series. Its convergence is proved in the class of regular solutions.

This writing attempts to capture mind and matter to rational description by focusing on disparity between scientific method as it extracts concepts and discards particulars and ordinary cognitive functioning that extends from proximal experience. Discourse underlined with the concept of strict uniqueness of world entities is intended to examine critically paradox and solution efforts in the mathematical and cosmological sciences as problems of zero, infinity: vast and small numbers and discreteness are addressed.

A two-point boundary value problem on the interval [0, 1] is considered, where the highest-order derivative is a Caputo fractional derivative of order 2 − δ with 0 < δ < 1. A necessary and sufficient condition for existence and uniqueness of a solution u is derived. For this solution the derivative uŐ is absolutely continuous on [0, 1]. It is shown that if one assumes more regularity — that u lies in C2[0, 1] — then this places a subtle restriction on the data of the problem.

Processes and tendencies that became substantial and determinative when forming integrity of a human being in the modern world are analyzed in the article. The "mounting" of identity by means of life arrangement has a major value, i.e. those forms of daily routine, in particular its artistic working-through, due to which the unique complex of a man is manifested. It is significant that the artistic working-through of some segments of the "habitat", their account and mass production not necessarily lead to total unification and standardization, but, on the opposite, can be only a stage when forming unique individuality of a man.

The inverse problem of finding the initial distribution has been studied on the basis of formulas for the solution of the first initial-boundary value problem for the inhomogeneous two-dimentional heat equation. The uniqueness of the solution of the direct initial-boundary value problem has proved with the completeness of the eigenfunctions of the corresponding homogeneous Dirichlet problem for the Laplace operator. The existence theorem for solving direct initial boundary value problem has been proved. Inverse problem has been investigated on the basis of the solution of direct problem, a criterion for the uniqueness of the inverse problem of finding the initial distribution has been proved. The existence of the inverse problem solution has been equivalently reduced to Fredholm integral equation of the first kind.

We study the problem of determining the kernel of the integral term in the one-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of the kernel. Next, assuming that there are two solutions $ k_1 (x, t) $ and $ k_2 (x, t), $ integro-differential equations, Cauchy and additional conditions for the difference of solutions of the Cauchy problem corresponding to the functions $ k_1 (x, t), $ $ k_2 (x, t)$ are obtained. Further research is being conducted for the difference $k_1 (x, t) - k_2 (x, t) $ of solutions of the problem and using the techniques of integral equations estimates it is shown that if the unknown kernel $ k (x, t) $ can be represented as $ k_j (x, t) = \sum_ {i = 0} ^ N a_i (x) b_i (t)$, $ j = 1, 2, $ then $ k_1 (x, t ) \equiv k_2 (x, t). $ Thus, the theorem on the uniqueness of the solution of the problem is proved.

We study the first boundary value problem for the elliptic-hyperbolic type equation with two perpendicular lines of change of type and spectral parameter. We prove the existence and uniqueness of the solution. In the proof of the uniqueness of solution we use the completeness of biorthogonal system in space $L_2$ . When building a solution as the sum of a series there is a problem of small denominators. We obtained estimates of the denominators of the separation from zero.

In this paper we study the initial problem for the equation of a beam with clamped ends. Uniqueness, existence and stability theorems are proved for the problem in the classes of regular and generalized solutions. Solution of the initial-boundary value problem is constructed in the form of a series in the system of eigenfunctions of one-dimensional spectral problem. We found the spectral problem eigenvalues as roots of the transcendental equation and the corresponding system of eigenfunctions. It is shown that the system of eigenfunctions is orthogonal and complete in L 2. On the basis of the completeness of the eigenfunctions the uniqueness theorem for the initial-boundary value problem for the equation of the beam is obtained. The generalized solution is defined as the limit of a sequence of regular solutions of the mean-square norm on the space variable.

We consider the Dirichlet spectral problem with the homogeneous boundary conditions in a cylindrical domain of Euclidean space for multidimensional hyperbolic equation with wave operator. We construct the solution as an expansion in multidimensional spherical functions; prove the existence and uniqueness theorems. The obtained conditions of the problem unique solvability essentially depend on the “height” of the cylinder.

In this paper, the existence and the uniqueness of solution of the Frankl type boundary value problem for degenerating equation of the mixed type are proved.

The problem for the equation of the mixed elliptic-hyperbolic type with nonlocal boundary conditions is viewed. This problem is reduced to the inverse problem for elliptichyperbolic equation with unknown right-hand parts. The criterion of the uniqueness is established. The explicit solution is constructed as the sum of orthogonal trigonometric series of the one-dimensional spectral problem eigenfunctions. The argumentation of the series convergence under some restrictions is given. The stability of the solution by the boundary functions is proved.

The multidimensional inverse problem of determining spatial part of integral member kernel in integro-differential wave equation is considered. Herein, the direct problem is represented by the initial-boundary problem for this with zero initial data and Neyman’s boundary condition as Dirac’s delta-function concentrated on the boundary of the domain $(x, t) \in \mathbb R^{n+1}$, $z > 0$. As information in order to solve the inverse problem on the boundary of the considered domain the traces of direct problem solution are given. The significant moment of the problem setup is such a circumstance that all given functions are real analytical functions of variables $x \in \mathbb R^n$. The main result of the work is concluded in obtaining the local unique solvability of the inverse problem in the class of continuous functions on variable $z$ and analytical on other spatial variables. For this, by means of singularity separation method, the inverse problem is replaced by the initial-boundary problem for the regular part of the solution of this problem. Further, direct and inverse problems are reduced to the solution of equivalent system of Volterra type integro-differential equations. For the solution of the latter, the method of Banach space scale of real analytical functions is used.

Non-local boundary value problem for loaded equation of parabolic type with the sign- variable characteristic form is solved. Common representation of solution is constructed. The theorems of common representation, existence and uniqueness of solution for boundary value problems set in rectangular domain are proved.

In work necessary and sufficient conditions of uniqueness of the decision of a regional problem for the loaded equation mixed parabolic hyperbolic type in rectangular area are established. The problem decision is constructed in the form of the number sum on own functions of a corresponding one-dimensional problem on own values.

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