Integral Representation Of Solution Research Articles

ON AN INVERSE SCATTERING PROBLEM FOR THE QUADRATIC PENCIL OF THE STURM-LIOUVILLE OPERATOR WITH A PIECE-WISE CONSTANT COEFFICIENT

In this work, direct and inverse scattering problems on the real axis for the quadratic pencil of the Sturm-Liouville operator with piece-wise constant coefficient are studied. The new integral representations for solutions are given, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proven, the necessary and sufficient conditions for recovering of the potentials are examined.

THE PROPERTIES OF THE VOLUME POTENTIAL FOR ONE PARABOLIC EQUATION WITH GROWING LOWEST COEFFICIENTS

The class of equations considered in the paper is a combination of two classes of equations: a degenerate parabolic equation of the Kolmogorov type and a parabolic equation with increasing coefficients in the group of younger members. Such a combination occurs in the problems of the theory of stochastic processes where, in the case of a normal Markov process, the Kolmogorov-Fokker-Planck equation has a similar form. The coefficients of this equations are constant in the group of principal terms and ones are increasing functions in the group of lowest terms. The article is devoted to the study of the properties of the volume potential, the kernel of which is the fundamental solution of the Cauchy problem for such an equation. Estimates of the fundamental solution of the Cauchy problem have a more complex structure than in the case of the classical Kolmogorov equation. These properties concern the existence of the derivatives included in the equation. They are used to establish theorems on the integral representations of solutions of the Cauchy problem and theorems on the correct solvability of the Cauchy problem in the corresponding classes of functional spaces. Such studies are carried out in this work. The obtained results are new and published for the first time.

On the Solvability of Some Boundary Value Problems for the Nonlocal Poisson Equation with Boundary Operators of Fractional Order

In this paper, in the class of smooth functions, integration and differentiation operators connected with fractional conformable derivatives are introduced. The mutual reversibility of these operators is proved, and the properties of these operators in the class of smooth functions are studied. Using transformations generalizing involutive transformations, a nonlocal analogue of the Laplace operator is introduced. For the corresponding nonlocal analogue of the Poisson equation, the solvability of some boundary value problems with fractional conformable derivatives is studied. For the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Necessary and sufficient conditions for solvability of the studied problems are obtained, and integral representations of solutions are given.

Four Boundary Value Problems for a Nonlocal Biharmonic Equation in the Unit Ball

Solvability issues of four boundary value problems for a nonlocal biharmonic equation in the unit ball are investigated. Dirichlet, Neumann, Navier and Riquier–Neumann boundary value problems are studied. For the problems under consideration, existence and uniqueness theorems are proved. Necessary and sufficient conditions for the solvability of all problems are obtained and an integral representations of solutions are given in terms of the corresponding Green’s functions.

Solvability of nonlocal Dirichlet problem for generalized Helmholtz equation in a unit ball

In this paper, we study the solvability of a new class of nonlocal boundary value problems for the generalized Helmholtz equation in the unit ball. These problems are a generalization of the classical Dirichlet boundary value problem to the Helmholtz equation. For the considered problems, existence and uniqueness theorems are proved. Integral representation of solutions is established. Corresponding spectral questions are also investigated, namely, eigenfunctions and eigenvalues of the problem are found.

ON PROBLEMS FOR EIDELMAN TYPE EQUATIONS AND SYSTEM OF EQUATIONS

The problems for Eidelman type equations and systems of equations are considered in this paper. They were the large part of scientific interests for Prof. Ivasyshen S.D. The results of investigations of Cauchy problem, initial-boundary and the inverse problems for this type of equations in bounded or unbounded domains are given. The results are represented as the estimates of the solutions, the integral representations of solutions, theorems of the existence, uniqueness and stability of solutions.

Properties of Fundamental Solutions, Theorems on Integral Representations of Solutions and Correct Solvability of the Cauchy Problem for Ultraparabolic Kolmogorov-Type Equations with Two Groups of Space Variables of Degeneration

The property of normality and the convolution formula are established for the classical fundamental solution of the Cauchy problem for a homogeneous ultraparabolic Kolmogorov-type equation with two groups of space variables of degeneration. Theorems on the integral representations of solutions and correct solvability of the Cauchy problem are presented for the classes of functions exponentially increasing as |x| → ∞ .

Green's functions of some boundary value problems for the biharmonic equation

In this paper the Green's functions for three boundary value problems for the biharmonic equation are investigated. First, an integral representation of solutions to the inhomogeneous biharmonic equation is given. Then the Green's function of the Dirichlet problem is found and an integral representation of the solution to the Dirichlet problem in terms of the Green's function is given. After that, the Green's function of the Navier problem and the integral representation of the solution to the Navier problem are presented. To study the Neumann-2 problem, the Green's function of the Neumann problem for the Poisson equation is discussed and on its basis the Green's function of the Neumann-2 problem is constructed. To illustrate the results obtained, solutions of the three considered homogeneous problems for the polynomial right-hand side of the equation are found.

A Single-Step Correction Scheme of Crank–Nicolson Convolution Quadrature for the Subdiffusion Equation

We develop a new correction scheme for time discretization of the subdiffusion equation based on the fractional Crank–Nicolson convolution quadrature. Due to the weak singularity of solution near time $$t=0$$ , a single-step initial correction of the scheme is proposed with rigorous analysis to render the time discretization of second-order accuracy. Optimal error estimates of the numerical schemes are proved for $$L^2$$ initial data based on the integral representations of solutions and resolvent estimates of elliptic operator, with regularity assumptions only on the source term. Numerical examples are presented to demonstrate the performance of the proposed method and the consistency with the theoretical analysis.

ASSESSMENT OF GEOECOLOGICAL CONSEQUENCES OF SUBDUCTION PROCESSES

t. The aim of the work is to develop a numerical-analytical method for assessing the geoecological consequences of volcanic activity accompanying the subduction process. The boundary problem was formulated, including the three-dimensional transport and diffusion equation and boundary conditions at the bottom and surface of the water. For research, a block structure with quasi-homogeneous layers is introduced. In each block, a differential factorization method is implemented and integral representations of solutions are obtained. Calculations for a model problem are carried out, conclusions are formulated about the features of the behavior of the heavy and light fractions of volcanic ejections depending on the speed of currents, intensity and duration of the eruption.

On Solvability of One Class of Quasielliptic Systems

We study the class of systems of differential equations defined by one class of matrix quasielliptic operators and establish solvability conditions for the systems and boundary value problems on $ {𝕉}^{n}_{+} $ in the special scales of weighted Sobolev spaces $ W^{l}_{p,\sigma} $ . We construct the integral representations of solutions and obtain estimates for the solutions.

On Mellin transforms of solutions of differential equation $$\chi ^{(n)}(x)+\gamma _{n}x\chi (x)=0$$

In this paper, for $$ n=2,3,\ldots ,$$ we consider the differential equation $$\begin{aligned} \chi ^{(n)}(x)+\gamma _{n}x\chi (x)=0,\quad \left\{ \begin{array}{ll} \gamma _{n}=(-1)^{k},&{}\quad n=2k,\\ \gamma _{n}=-1,&{}\quad n=2k+1, \end{array}\right. \end{aligned}$$ and find the linear independent solutions in terms of the higher-order Airy functions ( $$n=2k$$ ) and the higher-order Levy stable functions ( $$n=2k+1 $$ ). The integral representations of solutions are presented and their Mellin transforms are also given.

On the Maximum Principle for Solutions of Second Order Elliptic Equations

In this paper, we found sufficient conditions, under fulfillment of which the maximum principle for the solution of a second order partial differential elliptic equation in the unit circle meets maximum principle. It is proved that if a quasiconformality coefficient of such function satisfies certain boundary conditions, then this function meets maximum principle. While proving the main result, we use integral representations of solutions of this equation and properties of the Cauchy type integral and functions of Hardy and Smirnoff classes.

Integral representation of solutions using Green function for fractional Hardy equations

Our main aim is to study Green function for the fractional Hardy operator P:=(−Δ)s−θ|x|2s in RN, where 0<θ<ΛN,s and ΛN,s is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds.

On solvability of some nonlocal boundary value problems for biharmonic equation

Abstract In this paper a new class of well-posed boundary value problems for the biharmonic equation is studied. The considered problems are nonlocal boundary value problems of Bitsadze- -Samarskii type. These problems are solved by reducing them to Dirichlet and Neumann type problems. Theorems on existence and uniqueness of the solution are proved and exact solvability conditions of the considered problems are found. In addition, the integral representations of solutions are obtained.

MULTIPERIODIC SOLUTIONS OF SECOND-ORDER SYSTEMS WITH DIFFERENTIATION OPERATOR ON THE LYAPUNOV'S VECTOR FIELD

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

GREEN TENSOR OF MOTION EQUATIONS OF TWO COMPONENTS BIOT’S MEDIUM BY STATIONARY VIBRATIONS

Here processes of wave propagation in a two-component Biot’s medium are considered which are generated by periodic forces actions. By use Fourier transformation of generalized functions, the Green tensor - a fundamental solutions of oscillation equations of this medium has been constructed. This tensor describes the process of propagation of harmonic waves of a fixed frequency in spaces of dimension N = 1,2,3 under the action of power sources concentrated at the coordinates origin, described by a singular delta -function. Based on it, generalized solutions of these equations are constructed under the action of various sources of periodic perturbations, which are described by both regular and singular generalized functions. For regular acting forces, integral representations of solutions are given that can be used to calculate the stress-strain state of a porous water-saturated medium.

Far Fields of Internal Waves Excited by a Pulsing Source in a Stratified Medium with Shear Flows

A problem of the far field of internal gravity waves excited by an oscillating point source of perturbations in a stratified medium with a shear flow is solved. A model distribution of the shear flow velocity by depth is considered and an analytical solution to this problem is obtained in the form of the characteristic Green function expressed in terms of the modified Bessel functions of the imaginary index. Expressions for dispersion relations are obtained and integral representations of solutions are constructed. The dependences of the wave characteristics of the excited fields on the main parameters of the used stratification models, flows, and generation regimes are investigated.

On a step method and a propagation of discontinuity

In this paper we analyze how to compute discontinuous solutions for functional differential equations, looking at an approach which allows to study simultaneously continuous and discontinuous solutions. We focus our attention on the integral representation of solutions and we justify the applicability of such an approach. In particular, we improve the step method in such a way to solve a problem of vanishing discontinuity points. Our solutions are considered as regulated functions.

Analytical Solutions of the Internal Gravity Wave Equation in a Stratified Medium with Shear Flows

The problem of constructing internal gravity wave fields generated by an oscillating localized point source of disturbances in a stratified medium with an average shear flow is considered. A model distribution of shear flow over depth is considered, and an analytical solution of the problem is obtained in the form of a characteristic Green function expressed in terms of modified Bessel functions of imaginary index. Analytical expressions for the dispersion relations are obtained using Debye asymptotics of the modified Bessel functions. Integral representations of solutions are constructed. The wave characteristics of the excited fields are investigated depending on the basic parameters of the used stratification models, shear flows, and generation modes.