Boundary Value Problems For Equations Research Articles

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.

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.

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.

Numerical solution of source identification multi-point problem of parabolic partial differential equation with Neumann type boundary condition

We study a source identification boundary value problem for a parabolic partial differential equation with multi-point Neumann type boundary condition. Stability estimates for the solution of the overdetermined mixed BVP for multi-dimensional parabolic equation were established. The first and second order of accuracy difference schemes for the approximate solution of this problem were proposed. Stability estimates for both difference schemes were obtained. The result of numerical illustration in test example was given.

Inverse coefficient problem for a partial differential equation with multi-term orders fractional Riemann–Liouville derivatives

This work studies direct initial boundary value and inverse coefficient determination problems for a one-dimensional partial differential equation with multi-term orders fractional Riemann–Liouville derivatives. The unique solvability of the direct problem is investigated and a priori estimates for its solution are obtained in weighted spaces, which will be used for studying the inverse problem. Then, the inverse problem is equivalently reduced to a nonlinear integral equation. The fixed-point principle is used to prove the unique solvability of this equation.

On the existence of a positive solution to a boundary value problem for a third-order nonlinear functional differential equation with an integral boundary condition at one of the ends of the segment

The article studies the existence of positive solutions on the segment [0,1] of a two-point boundary value problem for one nonlinear third-order functional differential equation with an integral boundary condition at one of the ends of the segment. Using the Go–Krasnoselsky fixed point theorem and some properties of the Green's function of the corresponding differential operator, sufficient conditions for the existence of at least one positive solution to the problem under consideration are obtained. An example is given to illustrate the results obtained.

Solving the boundary value problem of the first-order measure differential equations

This article is to develop a method to solve the boundary value problems of the first-order measure differential equations in the space of bound-ed variation functions. Firstly, we obtain the solution and Green’s function by applying the integration by parts. Secondly, the criterion for the existence of solution is given by using fixed point theorem and regularization theory. Finally, an example is provided to validate these conclusions.

Existence result for a nonlinear mixed boundary value problem for the heat equation

In this paper we study the existence of solutions in parabolic Schauder space of a nonlinear mixed boundary value problem for the heat equation in a perforated domain. From a given regular open set Ω⊆Rn we remove a cavity ω⊆Ω. On the exterior boundary of Ω∖ω‾ we prescribe a Neumann boundary condition, while on the interior boundary we set a nonlinear Robin-type condition. Under suitable assumptions on the data and by means of Leray Schauder Fixed-Point Theorem, we prove the existence of (at least) one solution u∈C01+α2;1+α([0,T]×(Ω‾∖ω)).

On differential equations with exponential nonlinearities

Two-point boundary value problems for the second order nonlinear ordinary differential equations, arising in the heat conductivity theory, are considered. Multiplicity and existence results are established. The properties of solutions are studied. Estimates of the number of solutions are obtained. A bifurcation analysis was made and the bifurcation curves were presented. The analytical technique together with the phase plane analysis is used to obtain the results.

Existence of Positive Solutions for Hadamard-Type Fractional Boundary Value Problems at Resonance on an Infinite Interval

This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.

The overdetermined Cauchy problem for the hyperbolic Gellerstedt equation

Abstract Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions.

Improved growth estimate for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity

This paper provides an improved exponential growth estimate, surpassing the growth rate given in the previous work. This finding elucidates the impact of the power index k in the logarithmic nonlinearity uln|u|k on the growth behavior of the solution to the initial boundary value problem for the one-dimensional sixth-order nonlinear Boussinesq equation with logarithmic nonlinearity.

Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework

We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics equations in the half-plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform L∞ estimates of the error functions comes from the unboundedness of vorticity of strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential component of magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl boundary layer ansatz in L∞ sense in Sobolev framework. Compared with the homogeneous incompressible case considered in [33], there exists a strong boundary layer of density. Consequently, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.