Solution Of Boundary Value Problem Research Articles

On weak solutions of boundary value problems for some general differential equations

Изучаются обобщенные постановки задачи Дирихле, Неймана, других граничных задач для уравнений и систем вида $\mathcal{L}^+ A\mathcal{L}u=f$ с общей, вообще говоря, матричной дифференциальной операцией $\mathcal{L}$ и некоторым линейным или нелинейным оператором $A$, действующим в векторных пространствах $L^k_2(\Omega)$. Получены утверждения о существовании и единственности слабого решения и корректности поставленных граничных задач. В качестве оператора $A$ рассмотрены случаи операторов Немыцкого, а также интегральных операторов. Рассмотрены случаи вхождения младших производных. Библиография: 19 наименований.

Complex dynamics of the system of nonlinear difference equations in the Hilbert space

In the given article the necessary and sufficient conditions of the existence of solutions of boundary value problem for the nonlinear system in the Hilbert spaces are obtained. Examples of such systems like a Lotka–Volterra are considered. Bifurcation and branching conditions of solutions are obtained.

Approximation of solutions of boundary value problems for integro-differential equations of the neutral type using a spline function method

Boundary value problems for nonlinear integro-differential equations of the neutral type are investigated. A scheme for approximating the boundary value problem solution using cubic splines of defect two is proposed and substantiated. A model example illustrating the proposed approximation scheme is considered.

On weak solutions of boundary value problems for some general differential equations

We study general settings of the Dirichlet problem, the Neumann problem, and other boundary value problems for equations and systems of the form $\mathcal{L}^+ A\mathcal{L}u=f$ with general (matrix, generally speaking) differential operation $\mathcal{L}$ and some linear or non-linear operator $A$ acting in $L^k_2(\Omega)$-spaces. For these boundary value problems, results on well-posedness, existence and uniqueness of a weak solution are obtained. As an operator $A$, we consider Nemytskii and integral operators. The case of operators involving lower-order derivatives is also studied.

Approximate Solution of Boundary Value Problem for Heat Equation after Represented by Volterra Integral Equation of the First Kind

In this work, we study the regularization method for solving the Boundary Value Problem (BVP) for heat equation. The discretization method applied with two variables on Volterra integral equation in order to covert the problem into a linear operator equation after applied the separation of variables method to solve the partial differential equation. The regularization way used to obtain the estimate solution by using the Lavrentiev regularization method.

Temperature stresses in a rectangular two-layer plate under the action of a locally distributed temperature field

A rectangular isotropic two-layer plate of an irregular structure is considered, the edges of which are freely supported, and a constant temperature is maintained on them. Two-dimensional Kirchhoff-type thermoelasticity equations and two-dimensional heat equations written for an inhomogeneous material were used to study the temperature stresses in the plate. Using the method of double trigonometric series in spatial variables and the Laplace integral transformation over time, the general solutions of boundary value problems of thermoelasticity and heat conductivity for this plate under the action of a locally distributed temperature field specified at the initial moment of time are written down. The normal stresses in the layers of the plate are numerically analyzed depending on the geometric parameters, heat transfer coefficient, and time.

The method of penalty functions in the analysis of optimal control problems of Navier — Stokes evolutionary systems with a spatial variable in a network-like domain

The article considers the Navier — Stokes evolutionary differential system used in the mathematical description of the evolutionary processes of transportation of various types of liquids through network or main pipelines. The Navier—Stokes system is considered in Sobolev spaces, the elements of which are functions with carriers on n-dimensional networklike domains. These domains are a totality of a finite number of mutually non-intersecting subdomains connected to each other by parts of the surfaces of their boundaries like a graph (in applications these are the places of branching of pipelines). Two main questions of analysis are discussed: the weak solvability of the initial boundary value problem of the Navier — Stokes system and the optimal control of this system. The main method of research of weak solutions is the semidigitization of the input system by a time variable, that is the reduction of a differential system to a differential-difference system, and using a priori estimates for weak solutions of boundary value problems to prove the theorem of the existence of a solution of the input differential system. For the optimal control problem a minimizing functional (the penalty function) and a family of the approximate functional with parameters that characterize the “penalty” for failure to fulfill the equations of state of the system are introduced. At the same time, a special Hilbert space is created, the elements of which are pairs of functions that describe the state of the system and controlling actions. The convergence of the sequence of such functions to the optimal state of the system and its corresponding optimal control is proved. The latter essentially widen the possibilities of analysis of stationary and nonstationary network-like processes of hydrodynamics and optimal control of these processesd.

Numerical solution of boundary value problem for the Bagley–Torvik equation using Hermite collocation method

Numerical solution of boundary value problem for the Bagley–Torvik equation using Hermite collocation method

Geometric singularities and regularity of solution of the Stokes system in nonsmooth domains

This paper deals with the geometrical singularities of the weak solution of the mixed boundary value problem governed by the stationary Stokes system in two-dimensional nonsmooth domains with corner points and points at which the type of boundary conditions changes. The presence of these points on the boundary generally generates local singularities in the solution. We will see the impact of the geometrical singularities of the boundary or the mixed boundary conditions on the qualitative properties of the solution including its regularity. Moreover, the asymptotic singular representations for the solution which inherently depend on the zeros of certain transcendental functions are presented.

EXISTENCE RESULTS FOR MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS INVOLVING ATANGANA–BALEANU DERIVATIVE

In this paper, the existence results for the solutions of the multi-term ABC-fractional differential boundary value problem (BVP) [Formula: see text] of order [Formula: see text] with nonlocal boundary conditions have been derived by using Krasnoselskii’s fixed point theorem. The uniqueness of the solution is obtained with the help of Banach contraction principle. Examples are provided to confirm our obtained results.

The existence of positive solutions for a Caputo-Hadamard boundary value problem with an integral boundary condition

The existence of positive solutions for a Caputo-Hadamard boundary value problem with an integral boundary condition

Existence of solutions for first order impulsive periodic boundary value problems on time scales

In this paper we study a class of first order impulsive periodic boundary value problems on time scales. We give conditions under which the considered problem has at least one and at least two solutions. The arguments are based upon recent fixed point index theory in cones of Banach spaces for a k-set contraction perturbed by an expansive operator. An example is given to illustrate the obtained result.

Existence of solutions for a delay singular high order fractional boundary value problem with sign-changing nonlinearity

This paper consider the existence of at least one positive solution of a Riemann-Liouville fractional delay singular boundary value problem with sign-changing nonlinerty. To establish sufficient conditions we use the Guo-Krasnosel?skii fixed point theorem.

Effective Boundary Conditions Arising from the Heat Equation with Three-dimensional Interior Inclusion

We study the initial boundary value problem for a heat equation in a domain containing a thin layer. The thermal conductivity of the layer is drastically different from that of the bulk of the domain; moreover, the layer is anisotropic and 'optimally aligned' in the sense that the normal direction in the layer is always an eigenvector of the thermal tensor. To reveal the effects of the layer, we regard it as a thickless surface on which 'effective boundary conditions' (EBCs) are satisfied by the limit of solutions of the initial boundary value problem as the thickness of the layer shrinks to zero. These EBCs are rich in variety and type, including some nonstandard ones such as the Dirichlet-to-Neumann mapping and the fractional Laplacian.

CALCULATED RELATIONS OF THE SEMI-ANALYTICAL FINITE ELEMENT METHOD OF PRISMATIC BODIES FOR A FINITE ELEMENT BASED ON THE REPRESENTATION OF DIS-PLACEMENTS BY POLYNOMIALS

In article [8, 10], a variant of the semi-analytical finite element method for the calculation of prismatic bodies was developed using the Fourier series function as a coordinate system. The use of trigonometric series ensures maximum efficiency of the semi-analytical finite element method, however, at the ends of the body it is possible to satisfy only the boundary conditions corresponding to the support of the object on an absolutely rigid in its plane and flexible diaphragm. As a result of the performed researches the basis of representation of movements by polynomials is received that allows to expand considerably a range of boundary conditions on end faces of a body. In this case, it is not possible to reduce the solution of the original spatial boundary value problem to a sequence of two-dimensional problems, so a reasonable choice of appropriate polynomials becomes especially important. Their correct choice depends on the conditionality of the matrix of the system of separate equations and, consequently, the convergence of integration algorithms for its solution, and the universality of the approach to the possibility of satisfying different variants of boundary conditions at the ends of the body.

EXACT SOLUTION OF CONTACT PROBLEMS IN A FINITE-WIDTH BAND ON A MULTILAYER MEDIUM

In this paper, for the first time, an exact solution of contact problems for rigid or deformable stamps in a strip of finite width is obtained. It is assumed that the strip is located on a multilayer base of finite thickness. The universal modeling method previously developed by the authors is used. With its help, the solutions of complex boundary value problems for systems of partial differential equations are reduced, using the Galerkin transform, to solving individual differential equations, among which the Helmholtz equations are the simplest. In an earlier work of the authors published, when studying the problem for a deformable stamp in a strip, we had to limit ourselves to an asymptotic solution that is valid only for strips of large relative width. The solution for a strip of any finite size was constrained by the impossibility of constructing an exact solution to the contact problem for a rigid stamp in a strip of any finite width. As a result of the exact solution of the Wiener-Hopf integral equation in a finite-width band for the case of a multilayer medium, this contact problem was solved. The approach applied to the solution consists in constructing an exact operator equation of an infinite system of algebraic equations for large-width bands, using the operator formula of functions from matrices and investigating the constructed solution in the range of small relative bandwidth. The solution obtained in this way coincides with the solution obtained another method, namely, the singular integral method for the case of a small relative bandwidth. The constructed solution brings closer to the problems of research for materials of complex rheologies of contact problems with a deformable stamp, the description of cracks of a new type in limited bodies, modeling of nano particles, the study of tectonic plates of limited dimensions.

Fiber pull-out in a flexoelectric material

This work examines the fiber pull-out problem in the context of flexoelectricity. As a first approach we examine the deformation of an infinitely long prism due to the action of a concentrated out-of-plane line force. We assume that the material of the prism responds in the context of flexoelectricity and anti-plane conditions prevail. We incorporate the approach proposed by Giannakopoulos and Zisis [1] according to which the gradient of the polarization is incorporated into the strain energy density function. Accordingly, the field equations decouple in terms of the primary variables of the problem, that is the out-of-plane displacement and the polarization. The concentrated out-of-plane line force is materialized in the form of an out-of-plane body force. For the proposed problem we develop analytical solutions with respect to the out-of-plane displacement and polarization under static and dynamic conditions. Next, we propose a finite element (FE) model based on a structural plate analogue, for the solution of anti-plane flexoelectric boundary value problems. The proposed FE model is applied to the predefined problem and the numerical and the analytical results regarding the out-of-plane displacement are compared and found to be in excellent agreement. Next, we propose yet another structural analogue (membrane analogue) for the numerical treatment of the polarization equation and again the results are compared with the analytical solutions with respect to the polarization of the predefined problem and excellent agreement is attained. Finally, for the predefined problem the electromagnetic response is investigated and analytical solutions are proposed.

A boundary value problem of heat transfer within DBD-based plasma jet setups.

We claim an analytical solution for the thermal boundary value problem that arises in DBD-based plasma jet systems as a preliminary and consistent approach to a simplified geometry. This approach involves the outline of a coaxial plasma jet reactor and the consideration of the heat transfer to the reactor solids, namely, the dielectric barrier and the grounded electrode. The non-homogeneous initial and boundary value thermal problem is solved analytically, while a simple cut-off technique is applied to deal with the appearance of infinite series relationships, being the outcome of merging dual expressions. The results are also implemented numerically, supporting the analytical solution, while a Finite Integration Technique (FIT) is used for the validation. Both the analytical and numerical data reveal the temperature pattern at the cross-section of the solids in perfect agreement. This analytical approach could be of importance for the optimization of plasma jet systems employed in tailored applications where temperature-sensitive materials are involved, like in plasma biomedicine.

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

A method for an approximate numerical solution of two-point boundary value problems: nonstandard finite difference method on a semi-open interval

Qualitative properties of weighted fourth order elliptic problems

Qualitative properties of nonnegative solutions of weighted fourth order elliptic problems arising from the equations on singular manifolds with conical metrics are studied. The weights may be singular in the domains. We obtain the uniqueness of positive radial solution of the Navier boundary value problem in the ball $ B $ and its exact regularity at the singular point $ x = 0 $, which determines the exact regularity of the unique positive radial solution in $ B $. We also establish the Liouville type results for the equation on $ \mathbb{R}^N \backslash \{0\} $.