System Of Eigenfunctions Research Articles

Solution of the heat equation with a discontinuous coefficient with nonlocal boundary conditions by the Fourier method

This paper substantiates the solution by the method of separation of variables of the initial-boundary value problem for the heat equation with a discontinuous coefficient, under periodic or anti-periodic boundary conditions. Using the Fourier method, this problem is reduced to the corresponding spectral problem. The eigenvalues and eigenfunctions of this spectral problem are found. It is shown that the spectral problem is non-self-adjoint and a conjugate spectral problem of this original spectral problem is constructed. Further, it is proved that the system of eigenfunctions forms a Riesz basis. For this purpose, a self-adjoint spectral problem is constructed and its eigenvalues and eigenfunctions are found. In conclusion, using biorthogonality, the main theorem on the existence and uniqueness of a classical solution to the problem is proven.

The basis property of generalized eigenfunctions for one boundary value problem with discontinuities at two interior points

In this study, we consider a spectral problem for one boundary value problem with discontinuities at two interior points. The boundary conditions involve a spectral parameter. We consider some compact, positive, self-adjoint operators to reduce the spectral problem to an operator-pencil equation. Then, it was proven that this operator-pencil is positive definite, the spectrum is discrete, and the system of weak eigenfunctions forms a Riesz basis of the appropriate Sobolev space.

Galerkin-Type Solution of the Föppl–von Kármán Equations for Square Plates

The solution of the non-linear Föppl–von Kármán equations for square plates in the form of expansion over a system of eigenfunctions, generated by a linear self-adjoint operator, is obtained. The coefficients of the expansion are determined via the reduction method from the infinite-dimensional system of cubic equations. This allows the proposed solution to be considered as a non-linear generalization of the classical Galerkin approach. The novelty of the study is in the strict formulation of the auxiliary boundary problem, which makes it possible to take into account a rigid fixation against any displacements along the boundary. To verify the proposed solution, it is compared with experimental data. The latter is obtained by the holographic interferometry of small deflection increments superimposed on the large deflection caused by initial pressure. Experiment and theory show a good agreement.

INITIAL BOUNDARY VALUE PROBLEM FOR THE NONLINEAR MODIFIED BOUSSINESQ EQUATION

The problem in Sobolev spaces is investigated for a modified Boussinesq equation with a homogeneous Neumann boundary condition and with classical initial conditions. Based on the compactness method, it is shown that the approximate analytical solution, constructed in the form of Galerkin’s sum over the system of eigenfunctions of the Neumann boundary value problem, *-weakly converges to the exact solution.

Solution of boundary value problems for the heat equation with a piecewise constant coefficient using the Fourier method

This paper considers some initial boundary value problems for the heat equation in a bounded segment with a piecewise constant coefficient. Using the method of separation of variables, the problem under consideration is reduced to a spectral problem and eigenvalues and eigenfunctions of the resulting spectral problem are found. It is shown that the system of eigenfunctions forms a Riesz basis. Next, we prove a theorem on the existence and uniqueness of solutions to the initial-boundary value problems under consideration.

Two-phase heat conduction problems with Sturm-type boundary conditions

In this work, the solution is justified using the method of separation of variables for initial-boundary value problems of the heat conduction equation with piecewise constant thermal conductivity coefficient under Sturm-type boundary conditions (separated boundary conditions). Various cases are considered. A system of eigenfunctions of the corresponding spectral problem is constructed and it is proved that this system of eigenfunctions forms a Riesz basis. The theorem of the existence of a unique classical solution to the problem is proved.

On the quadratic closeness of eigenfunctions of «unperturbed» and «perturbed» differentiation operators on an interval

The article considers the eigenvalue problem of the differentiation operator, when the spectral param- eter is also present in the boundary condition with an integral perturbation, where the integrand has the property of limited variation and has a value of unity at the ends of the segment [-1, 1]. The derivatives with respect to the time variable included in the boundary condition naturally arise when solving (by the Fourier method) initial boundary value problems for evolution equations. The conjugate operator is constructed. It is shown that the spectral questions of the conjugate operator have a similar structure. The characteristic determinant of the original direct spectral problem with an integral perturbation of the boundary condition and for the eigenvalue problem of a loaded first-order differential equation on a segment with a periodic boundary condition, which is an entire analytical function of the spectral parameter, is constructed. Based on the formula of the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the original «perturbed» differentiation operator and the loaded first-order differential equation on the segment. A special feature of the operator under consideration is the non-self-adjointness of the operator in L2(–1,1). The quadratic proximity of the systems of eigen functions of the «unperturbed» and «perturbed» differentiation operators and the Riesz basis property of these systems are proved. In this case, the system of eigenfunctions of the «perturbed» operator is not orthonormal.

ON A PROBLEM FOR A THREE-DIMENSIONAL ELLIPTIC EQUATIONWITH BESSEL OPERATORS

The Keldysh problem for a three-dimensional elliptic equation with three singular coefficients in a rectangular parallelepiped is studied. Based on the property of completeness of systems of eigenfunctions of two one-dimensional spectral problems, a uniqueness theorem is proved. The solution to the problem posed is constructed as the sum of a double Fourier Bessel series.

Properties of eigenfunctions of a boundary value problem for ordinary differential equations of fourth-order with boundary conditions depending on the spectral parameter

In this paper, we consider an eigenvalue problem for fourth-order ordinary differential equations with a spectral parameter contained in all boundary conditions. We characterize the location of eigenvalues on the real axis, find their multiplicities, and study the oscillatory properties of eigenfunctions. Moreover, we obtain refined asymptotic formulas for the eigenvalues, as well as for the values at the endpoints of the interval of eigenfunctions and their derivatives. Then, using these results and the operator-theoretic formulation, we establish sufficient conditions for the system of eigenfunctions of this problem to form a basis in the space Lp,1<p<∞, after removing four functions.

An inverse time-dependent source problem for a time-fractional diffusion equation with nonlocal boundary conditions

In this paper, we investigate an inverse time-dependent source problem for a time-fractional diffusion equation with nonlocal boundary and integral over-determination conditions. The fractional derivative is described in the generalized Caputo sense. The nonlocal boundary conditions are regular but not strongly regular. The special thing about this problem is: the system of eigenfunctions is not complete, but the system of eigen-and associated functions forming a basis in L2 (0,1). Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown by using the generalized Fourier method, the estimates of Mittag-Leffler function and Banach’s contraction mapping principle.

Neumann Problem for Second-Order Differential Equation with Fractional Derivative in the Analysis and Modeling of Structures Made of Viscoelastic Elements

This article addresses a second-order differential equation containing a Gerasimov-Kaputo fractional differentiation operator of order less than two. The Neumann problem is formulated for this equation. A system of eigenfunctions and eigenvalues for the considered homogeneous boundary problem of the second kind is found. A conjugate boundary problem for the Gerasimov-Kaputo fractional derivative is introduced. A biorthogonal system is obtained that is orthogonal to the found system of eigenfunctions. Visualizations of the eigenfunction system, biorthogonal system, and an example of eigenvalue distribution on the real axis are provided.

Uniform asymptotics for eigenvalues of model Schro&amp;#x0308;dinger operator with small translation

We consider a model Schro&amp;#x0308;dinger operator with a constant coefficient on the unit segment and the Dirichlet and Neumann condition on opposite ends with a small translation in the free term. The value of the translation is small parameter, which can be both positive and negative. The main result is the spectral asymptotics for the eigenvalues and eigenfunctions with an estimate for the error term, which is uniform in the small parameter. For finitely many first eigenvalues and associated eigenfunctions we provide asymptotics in the small parameter. We prove that each eigenvalue is simple, and the system of eigenfunctions forms a basis in the space $L_2(0, 1).$

On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

&lt;abstract&gt;&lt;p&gt;In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.&lt;/p&gt;&lt;/abstract&gt;

Basis Property of the System of Eigenfunctions Corresponding to a Problem with a Spectral Parameter in the Boundary Conditio

Basis Property of the System of Eigenfunctions Corresponding to a Problem with a Spectral Parameter in the Boundary Conditio

On Self-Adjoint Boundary Value Problems for a Functional Equation of Even Order

Investigating a functional differential equation of even order, we explore a class of self-adjoint boundary value problems with two-point conditions. We establish the basis property of the system of eigenfunctions and compare the eigenvalues.

Gellerstedt Problem with a Nonlocal Oddness Boundary Condition for the Lavrent’ev–Bitsadze Equation

We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with the oddness boundary condition on the boundary of the ellipticity domain. All eigenvalues and eigenfunctions are obtained in closed form. It is proved that the system of eigenfunctions is complete in the elliptic part of the domain and incomplete in the entire domain. The unique solvability of the problem is also proved; the solution is written in the form of a series if the spectral parameter is not equal to an eigenvalue. For the spectral parameter coinciding with an eigenvalue, solvability conditions are obtained under which the family of solutions is found in the form of a series. A condition for the solvability of the problem depending on the eigenvalues is obtained. The constructed analytical solutions can be used efficiently in numerical modeling of transonic gas dynamics problems.

On the Multiple Spectrum of a Problem for the Bessel Equation of an Integer Order with Squared Spectral Parameter in the Boundary Condition

The problem for the Bessel equation of an integer order with complex physical and spectral parameters in the boundary condition is considered. The spectral parameter enters the boundary condition quadratically. The question of the basis property of the system of eigenfunctions in the case of the appearance of a multiple eigenvalue is studied

On Symplectic Self-Adjointness of Hamiltonian Operator Matrices

The symmetry of the spectrum and the completeness of the eigenfunction system of the Hamiltonian operator matrix have important applications in the symplectic Fourier expansion method in elasticity. However, the symplectic self-adjointness of Hamiltonian operator matrices is important to the characterization of the symmetry of the point spectrum. Therefore, in this paper, the symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied by using the spectral method of unbounded block operator matrices, and some sufficient conditions of the symplectic self-adjointness of infinite dimensional Hamiltonian operators are obtained. In addition, the necessary and sufficient conditions are also investigated for some special infinite dimensional Hamiltonian operators.

On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property

This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function px, which belongs to the space L20,1. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when px≡0) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions px, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in L20,1, is dense within the space L20,1. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.

On eigenfunctions and eigenvalues of some boundary value problems for a nonlocal biharmonic operator

In this note, the concept of a nonlocal biharmonic operator is introduced. When introducing this operator, mappings of the type of involution are used. Namely, in the differential expression of this operator, in addition to the variables 12( , ,..., )nxx xx, transformed arguments with mappings of the form 111,...,,,,...,,1jjjjjnS xxxpx xxj n and their multiplication also involved. Spectral problems with Dirichlet and Neumann-type boundary conditions are considered in an n-dimensional parallelepiped for a given nonlocal biharmonic operator. The eigenfunctions and eigenvalues of the problems under consideration are explicitly constructed. When constructing these elements, eigenfunctions and eigenvalues of the classical biharmonic operator with Dirichlet and Neumann type boundary conditions are essentially used. Theorems on the orthonomization and completeness of the systems of eigenfunctions of the problems under consideration are proved. Examples of the corresponding parameters for special cases involved in the problems under consideration are given. In addition, in the two-dimensional case for the corresponding nonlocal biharmonic operator, spectral issues of boundary value problems of the Samarsky-Ionkin type are also investigated. The proper and attached functions of the problem under consideration are found and theorems on the completeness ofthese systems are proved.