Boundary Value Problems For Equations Research Articles

Boundary value problems of nonlinear Maxwell equations arising in mesoscopic electromagnetism

Boundary value problems of nonlinear Maxwell equations arising in mesoscopic electromagnetism

Determination of the Thermal Conductivity and Volumetric Heat Capacity of Substance from Heat Flux

The study of nonlinear problems related to heat transfer in a substance is of great practical important. Earlier, this paper’s authors proposed an effective algorithm for determining the volumetric heat capacity and thermal conductivity of a substance based on experimental observations of the dynamics of the temperature field in the object. In this paper, the problem of simultaneous identification of temperature-dependent volumetric heat capacity and thermal conductivity of the substance under study from the heat flux at the boundary of the domain is investigated. The consideration is based on the first boundary value problem for a one-dimensional unsteady heat equation. The coefficient inverse problem under consideration is reduced to a variational problem, which is solved by gradient methods based on the application of fast automatic differentiation. The uniqueness of the solution of the inverse problem is investigated.

Qualitative analysis of fourth-order hyperbolic equations

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

Review of Some Modified Generalized Korteweg–de Vries–Kuramoto–Sivashinsky Equations (Part II)

In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form N(ux), where ux denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto–Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in Rn. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case.

Existence of Nontrivial Solutions for Boundary Value Problems of Fourth-Order Differential Equations

This article investigates the solvability problem of fourth-order differential equations with two-point boundary conditions; specifically, conclusions regarding sign-changing solutions are obtained. The methods used in this article are fixed-point theorems on lattices. Firstly, under some sublinear conditions, the existence of three nontrivial solutions is demonstrated, including a sign-changing solution, a negative solution and a positive solution. Secondly, under some unilaterally asymptotically linear and superlinear conditions, the existence of at least one sign-changing solution is proved. Finally, this article provides several specific examples to illustrate the obtained conclusions.

Review of Some Modified Generalized Korteweg–De Vries–Kuramoto–Sivashinsky (mgKdV-KS) Equations

This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg–De Vries equation, the Kuramoto–Sivashinsky equation, the generalized Korteweg–De Vries–Kuramoto–Sivashinsky equation and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane.

Domain sampling methods for an inverse boundary value problem of the heat equation

Abstract We investigate the reconstruction of an unknown cavity inside a heat conductor with only single boundary measurement, which is a typical non-destructive testing of defects in material science using thermography. Mathematically, it can be formulated as an inverse boundary value problem for the heat equation. Two domain sampling methods, i.e., the range test (RT) and no-response test (NRT), are developed to realize the reconstruction. We first justify the convergence of the RT, which is based on the analytical extension property of solutions to the heat equation. Then we prove the duality of the RT and NRT by looking at the relation between their indicator functions. Consequently, the unknown cavity can be reconstructed by using either the RT or NRT if the solution associated to the single measurement cannot be analytically extended across its boundary. Next, we design two efficient algorithms for the RT and NRT to reconstruct the unknown cavity numerically. Each algorithm consists of two steps. The first step is to determine an approximation of the unknown cavity by implementing the sampling methods in a coarse grid, while the second step is to improve the reconstruction by using a fine grid in the approximate domain. The numerical results suggest that both algorithms can generate some rough information on the cavity with only one boundary measurement. Finally, by using this information as an initial guess for Newton's method, we improve the accuracy of the mentioned reconstruction results.

A Study on Positive Solutions to Nonlinear Fractional Differential Equations

The purpose of this article concerns a type of nonlinear boundary value problem of fractional differential equations with the Caputo derivative. Using the fixed point index theory of condensing mapping in cones, the existence results of positive solutions for such a problem are obtained.

Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions.

Numerical solution of the mixed boundary value problem for the heat equation in two-dimensional domains of complex shape

A finite-difference computational algorithm is proposed for solving a mixed boundary value problem for heat equation given in a two-dimensional domains of complex shape. To solve the problem, generalized curvilinear coordinates are used. The physical domain is mapped to the computational domain (unit square) in the space of generalized coordinates. The original problem is written in curvilinear coordinates and approximated on a uniform grid in the computational domain. The obtained results are mapped on a non-uniform boundary-fitted difference grid in the physical domain. The second-order approximations of mixed Neumann – Dirichlet boundary conditions are constructed. The results of computational experiments are presented. The second order of accuracy of the presented computational algorithm is confirmed.

Boundary Value and Control Problems for the Stationary Magnetic Hydrodynamic Equations of Heat Conducting Fluid with Variable Coefficients

Boundary Value and Control Problems for the Stationary Magnetic Hydrodynamic Equations of Heat Conducting Fluid with Variable Coefficients

ON A BOUNDARY VALUE PROBLEM FOR HIGH-ORDER HYPERBOLIC EQUATION WITH IMPULSE DISCRETE MEMORY

The investigation focuses on a boundary value problem for a high-order hyperbolic equation with impulse discrete memory in a rectangular domain. By introducing new functions, the problem is transformed into a set of boundary value problems for a first-order differential equation with impulse discrete memory, which depends on unknown functions and integral relations. D.S. Dzhumabaev's parametrization method is applied to this equivalent problem. The domain is divided according to the time variable, and functional parameters representing discrete memory values are introduced within the interior domains. As a result, the family of boundary value problems for the first-order differential equation with impulse discrete memory and unknown functions is converted into an equivalent family of integral-multipoint boundary value problems involving functional parameters and unknown functions. These equivalent problems include initial value problems for first-order differential equations related to the new functions. The solutions to the initial value problems are expressed using Volterra integral equations. By substituting these solutions into the boundary and impulse conditions, a system of linear functional equations concerning the functional parameters is derived. An algorithm is developed to solve the equivalent problem, and sufficient conditions for the unique solvability of the family of integral-multipoint boundary value problems with functional parameters and unknown functions are provided. Additionally, sufficient conditions for the unique solvability of the original boundary value problem for the high-order hyperbolic equation with impulse discrete memory are established based on the initial data.

Heat equation for Sturm–Liouville operator with singular propagation and potential

Abstract This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm–Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm–Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

NUMERICAL METHOD FOR SOLVING THE BOUNDARY VALUE PROBLEM FOR THE PARABOLIC EQUATION

A linear boundary value problem for a parabolic equation is considered in a closed domain. Based on the broken line method, the boundary value problem for a parabolic equation is replaced by a two-point boundary value problem for a system of linear ordinary differential equations by discretizing the unknown function u(t,x) with respect to the variable x. The obtained two-point boundary value problem is investigated by the parameterization method of Professor Dzhumabaev. Based on this method, an algorithm for finding a numerical solution to the two-point boundary value problem for a system of linear ordinary differential equations is constructed. The constructed algorithm is realized by applying known numerical methods. The constructiveness and efficiency of the parameterization method also allows us to construct a numerical solution of the considered linear boundary value problem for the parabolic equation. One numerical example is given to verify and illustrate the proposed algorithm.

Chaos of the initial and boundary value problems for the reaction-diffusion equations

This paper investigates an initial and boundary value problem for the reaction-diffusion equations, which can be considered as a linearized form of the advective Fisher-KPP equations. It is demonstrated that the initial and boundary value problem is chaotic when the three parameters of the reaction-diffusion equation vary above a specific surface. However, stable solutions are obtained both on and below this surface within a particular subset of initial values. The chaos and stability of the nonhomogeneous initial boundary value problem are further studied. Finally, some numerical examples are provided to illustrate the validity of the obtained results.

Singular Abreu equations and linearized Monge–Ampère equations with drifts

We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher-dimensional case by transforming singular Abreu equations into linearized Monge–Ampère equations with drifts. We establish global Hölder estimates for linearized Monge–Ampère equations with drifts under suitable hypotheses, and then apply them to prove the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side are also discussed.

Дослідження процесів самозаймання у насипі з прямокутним перерізом методами Роте та двобічних наближень

Self-ignition of a stockpile of material (coal, peat, grain) occurs as a result of the accumulation of heat released by an exothermic reaction of oxidation, which makes it possible to consider the stockpile as a body with an internal heat source. The research of self-ignition processes using mathematical modeling methods leads to the need to find a solution to the initial boundary value problem for a two-dimensional semilinear heat conduction equation. This cannot always be done analytically, so it makes sense to use numerical analysis methods. The aim of this article is to apply Rothe’s method in combination with the method of two-sided approximations based on the use of the Green's function to find the solution of the initial boundary value problem for the two-dimensional semilinear heat conduction equation that arises during the mathematical modeling of self-ignition processes of an stockpile of bulk material of cylindrical shape with a rectangular base. To achieve the goal, after the discretization of the heat conduction equation by the time variable, a sequence of boundary value problems was obtained by the Rothe’s method, each of which was reduced to the Hammerstein equation. For this nonlinear operator equation, an iterative process of the two-way method was constructed with its stopping condition obtained through a posteriori error estimation. The power of the internal heat source was approximated using an exponential dependence. As a result of the computational experiment, a sequence of approximate solutions was obtained. The graphs of heat maps constructed for them made it possible to examine over time the course of the self-ignition process in the cross-section of a stockpile of cylindrical shape with a rectangular base and to identify areas of heat accumulation

The boundary value problem for an ordinary linear half-order differential equation

This study is devoted to the study of the solution of a boundary value problem for an ordinary linear differential equation of half order with constant coefficients. Using of the fundamental solution of the main part of the considered equation, we obtained the principal relations, from which we obtain the necessary conditions for the Fredholm property of the original problem. Further, using the Mittag-Leffler function, a general solution of the homogeneous equation is obtained. Finally, the problem under consideration is reduced to an integral Fredholm equation of the second kind with a non-singular kernel, i.e., the Fredholm property of the stated problem is proved.

Застосування методу двобічних наближень до знаходження додатних аксіально-симетричних розв’язків крайових задач з монотонними нелінійностями

This paper considers the application of the method of two-sided approximations for finding positive solutions to boundary value problems for nonlinear elliptic differential equations with axial symmetry. The case of Dirichlet boundary conditions (first kind) is considered, with a power-type monotone nonlinearity, where the exponent ranges from 0 to 1. By transitioning to polar coordinates in the boundary value problem for the elliptic equation, due to the axial symmetry of the solution, the problem is reduced to a boundary value problem for an ordinary differential equation on an interval with respect to a function that depends only on the polar radius, thus eliminating the dependence on the angle. The pole of the polar coordinate system becomes a singular point, where a boundedness condition must be imposed. For the boundary value problem, a Green's function is constructed to further reduce the problem to a Hammerstein integral equation. The integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on the interval, partially ordered by a cone of non-negative continuous functions on this interval. The operator is examined for properties such as monotonicity (isotonicity), positivity, boundedness, and concavity. Next, the initial approximation is found as the endpoints of the invariant conical segment for the isotone operator in such a way as to ensure the highest convergence rate of the iterative process. The following iterative sequences of two-sided approximations are constructed: the first sequence, which is non-decreasing with respect to the cone, and the second sequence, which is non-increasing with respect to the cone. At each iteration, the arithmetic mean of the upper and lower approximations is taken as the next approximation. The iterative process is terminated when the error estimate of the solution satisfies the specified accuracy. The theoretical results obtained in this work were validated through a computational experiment. The dependence of the solution on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs

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.