Boundary Value Problems For Equations Research Articles

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian

The fractional order p-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with p-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space.

On boundary value problems for the Boussinesq-type equation with dynamic and non-dynamic boundary conditions

The work studies boundary value problems with non-dynamic and dynamic boundary conditions for one- and two-dimensional Boussinesq-type equations in domains representing a trapezoid, triangle, "curvilinear" trapezoid, "curvilinear" triangle, truncated cone, cone, truncated "curvilinear" cone, and "curvilinear" cone. Combining the methods of the theory of monotone operators and a priori estimates, in Sobolev classes, we have established theorems on the unique weak solvability of the boundary value problems under study.

A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity

The present paper is devoted to the study of a class of nonlinear Helmholtz equations that is essentially used in mathematical models for the theoretical and numerical analysis of scattering and radiation effects. While well-known works relate to geometrically simple domains (supporting the nonlinearity) and selected nonlinearities, the new aspects lie in the transition to more generally shaped, two- or three-dimensional objects, to more general nonlinearities (including saturation), and in the possibility of an efficient numerical approximation of the electromagnetic fields and derived quantities (such as energy, transmission coefficient, etc.).The paper describes and investigates an approach that consists in transforming the original full-space transmission problem for a nonlinear Helmholtz equation into an equivalent boundary-value problem on a bounded domain by means of a nonlocal Dirichlet-to-Neumann (DtN) operator. It is shown that the transformed nonlinear problem is equivalent to the original one and is uniquely solvable under appropriate conditions. In addition, the effect of truncation of the DtN operator on the resulting solution is investigated. It is shown that the corresponding sesquilinear form satisfies a parameter-uniform inf–sup condition, that the modified nonlinear problem has a unique solution, and that the solution error caused by the truncated DtN operator can be estimated in dependence on the truncation parameter.

Numerical modeling of the boundary value problem of an ordinary differential equation with a small parameter at the highest derivative by Chebyshev polynomials of the second kind

In the present work the application of the spectral method with Chebyshev polynomials of the second kind is considered for solving a boundary value problem of an ordinary differential equation with a small parameter at the highest derivative. Numerical and graphical results illustrate the effectiveness of the applied method for various values of polynomials.

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

On the Stabilization of the Solution to the Initial Boundary Value Problem for One-Dimensional Isothermal Equations of Viscous Compressible Multicomponent Media Dynamics

An initial boundary value problem is considered for one-dimensional isothermal equations of the dynamics of viscous compressible multicomponent media, which are a generalization of the Navier–Stokes equations. The stabilization of the solution to the initial boundary value problem is proven while the time tends to infinity, without simplifying assumptions for the structure of the viscosity matrix, except for the standard physical requirements of symmetry and positive definiteness. It is shown that the stabilization of the solution is exponential.

Local null controllability for the thermistor problem

In this paper, we deal with the local null controllability of an initial boundary value problem for a thermistor equation. The control is distributed, locally in space. The main ingredients of the proof are suitable Carleman estimates for an adjoint system and Liusternik’s Inverse Mapping Theorem in Hilbert spaces. As a complement to the theoretical results, we present formulations and numerical simulations for the controllability problem, the results were obtained through the finite element method and a quasi-Newton method, simulated in the software Freefem++. Finally, large time null controllability, stabilization and others possible extensions and open problems concerning other systems are presented.

Global existence and asymptotic stability of the free boundary problem of the primitive equations with heat insulation

In this paper, we deal with the free boundary value problem of the three-dimensional full primitive equations for the large-scale ocean with heat insulation in a bounded domain. The domain is periodic in the horizontal direction and finite in the vertical direction, and its upper boundary is a free surface driven by the fluid and bottom boundary is fixed. We show that the unique strong solution of the free boundary value problem for the three-dimensional full primitive equations exists globally in time and converges to the equilibrium state with exponential decay rate, when the initial data is a small perturbation of the steady state in regular Sobolev space.

On a Method for Solving a Mixed Boundary Value Problem for a Parabolic Equation Using Modified Sinc-Approximation Operators

A new method is proposed for obtaining a generalized solution of a mixed boundary value problem for a parabolic equation with boundary conditions of the third kind and a continuous initial condition. Generalized functions are understood in the sense of the sequential approach. As an intermediate approximation, a modified sinc-approximation operator is used. The solution is obtained in the form of a series uniformly converging inside the solution domain.

Problem of the Boundary Control of Oscillations of a Sample of a Layered Two-Phase Composite Material

We consider the problem of the boundary control of one-dimensional oscillations of an effective (averaged) medium corresponding to a two-phase medium consisting of periodically alternating layers of elastic and viscoelastic materials with long-term memory or various viscoelastic materials with Kelvin–Voigt friction and long-term memory. The averaged model is described by a boundary value problem for an integrodifferential equation. It is shown that for this model, it is impossible to bring oscillations to a state of rest in finite time (in contrast to the equation of string oscillations) by a force acting at one end of the band. A hypothesis is formulated on the possibility of bringing the specified object to a state of rest with the help of force effects distributed along the entire length of the object.

On a general discrete boundary value problem for an elliptic pseudo-differential equation in a quadrant

On a general discrete boundary value problem for an elliptic pseudo-differential equation in a quadrant

Analytical solutions to the isothermal Euler equations with time-dependent damping

In this paper, we construct a class of analytical solutions to the one dimensional compressible isothermal Euler equations with time-dependent damping. By introducing a special density function ρ(x, t) = e c(t)x+d(t), we obtain a family of analytical solutions. A sufficient condition for the solution to blow up in finite time is given. On the basis of the conclusion, the analytical solutions to the initial boundary value problem of the pressureless Euler equations are obtained.

On one approximate solution of a nonlocal boundary value problem for the Benjamin-Bona-Mahony equation

The paper investigates a non-local boundary value problem for the Benjamin-Bona-Mahony equation. This equation is a nonlinear pseudoparabolic equation of the third order with a mixed derivative. To find a solution to this problem, an algorithm for finding an approximate solution is proposed. Sufficient conditions for the feasibility and convergence of the proposed algorithm are established, as well as the existence of an isolated solution of a non-local boundary value problem for a nonlinear equation. Estimates are obtained between the exact and approximate solution of this problem.

On Some Non-local Boundary Value and Internal Boundary Value Problems for the String Oscillation Equation

The work is devoted to the problem of setting new boundary and internal boundary value problems for hyperbolic equations. The consideration of these settings is given on the example of a wave equation. The research involves the d’Alembert method, the mean value theorem and the method of successive approximations. The paper formulates and studies a number of non-local problems summarizing the classical Goursat and Dardu tasks. Some of them are marginal, and the other part is internal-marginal, and in both cases both characteristic and uncharacteristic displacements are considered. It should also be noted that a number of problems discussed below arose as a special case in the construction of the theory of correct problems for the model loaded equation of string oscillation.

Chebyshev collocation method for solving second order ODEs using integration matrices

The spectral collocation method for solving two-point boundary value problems for second order differential equations is implemented, based on representing the solution as an expansion in Chebyshev polynomials. The approach allows a stable calculation of both the spectral representation of the solution and its pointwise representation on any required grid in the definition domain of the equation and additional conditions of the multipoint problem. For the effective construction of SLAE, the solution of which gives the desired coefficients, the Chebyshev matrices of spectral integration are actively used. The proposed algorithms have a high accuracy for moderate-dimension systems of linear algebraic equations. The matrix of the system remains well-conditioned and, with an increase in the number of collocation points, allows finding solutions with ever-increasing accuracy.

Boundary value problems of integrodifferential equations under boundary conditions taking into account physical nonlinearity

When solving integrodifferential equations under boundary conditions, taking into account physical nonlinearity, a broad class of boundary-value problems of oscillations arises associated with various boundary conditions at the edges of a flat element. When taking into account non-stationary external influences, the main parameters is the frequency of natural vibrations of a flat component, taking into account temperature, prestressing, and other factors. The study of such problems, taking into account complicating factors, reduces to solving rather complex problems. The difficulty of solving these problems is due to both the type of equations and the variety. We analyze the results of previous works on the boundary problems of vibrations of plane elements. Possible boundary conditions at the edges of a flat element and the necessary initial conditions for solving particular problems of self-oscillation and forced vibrations, and other problems are considered. The set of equations, boundaries, and initial conditions make it possible to formulate and solve various boundary value problems of vibrations for a flat element. The oscillation equations for a flat element in the form of a plate given in this paper contain viscoelastic operators that describe the viscous behavior of the materials of a flat component. In studying oscillations and wave processes, it is advisable to take the kernels of viscoelastic operators regularly, since only such operators describe instantaneous elasticity and then viscous flow.

APPLICATION OF CALCULATION METHODS FOR SOLVING APPLIED PROBLEMS OF HEAT AND MASS EXCHANGE IN COMPLEX SYSTEMS

The accuracy of the calculation and optimization of the objective function and its parameters when solving applied engineering problems depends on the accuracy of the formulation of the main optimization problem, calculation and applied optimization problems, as well as the accuracy of computational methods for their implementation. An increase in the considered features of applied optimization problems will complicate the setting and methods of implementing boundary value problems. Thus, for the implementation of modernized boundary value problems, it will be necessary to apply several computational methods that will create a computational structure. The main condition for constructing a physically based boundary value problem is to find and justify the conditions for the existence of a unique solution. To increase the efficiency of the use of methods of calculation and optimization of technical parameters, it is necessary to increase the number of considered features of calculation and applied optimization mathematical models for heat and mass transfer in technical systems. Along with the construction of boundary value problems, it is important to define and justify the conditions for the existence of a single solution.
 The research article deals with some aspects of solving applied problems of heat and mass transfer in technical systems. Nonlocal boundary value problems for inhomogeneous and homogeneous pseudodifferential equations in partial derivatives with integral boundary conditions are considered, methods of solving a nonlocal inhomogeneous boundary value problem are proposed, and the correctness conditions of this problem in the class of infinitely differentiable generalized functions of power growth are defined and proven. Proved conditions for the existence of a correct problem for pseudo-differential equations with an integral boundary condition. The research of this article should be applied for controlling possible risks when solving applied problems in technical systems, biotechnology and veterinary medicine.

Вычисление коэффициентов определяющих соотношений нелинейно-упругих материалов с использованием эволюционных алгоритмов

Abstract. The possibility of parameter identification for constitutive relations of materials based on uniaxial stretching of a nonlinear-elastic prismatic sample and on inflating a hollow cylinder has been studied. The experiments are modeled using the semi-inverse method of nonlinear theory of elasticity. To describe the mechanical properties of the studied objects, the potentials of compressible nonlinearly elastic media - Blatz and Ko material, and Murnaghan model are used. In the first stage, a series of direct problems are solved. Analytical expressions for the load-elongation dependence are used for the stretching problem. The three-dimensional problem of a cylinder inflation is reduced to a nonlinear boundary value problem for an ordinary second-order differential equation for the function of the radial displacement of cylinder points. In all cases, load diagrams are constructed based on the specified parameters of the nonlinearly elastic models, which are subsequently considered as initial data for the inverse problem of parameter identification. The problem is reduced to finding the minimum of the objective function which is defined as a least squares function. Two evolutionary algorithms were used for analysis: the Squirrel search algorithm (SSA), based on the model of the foraging behaviour of southern flying squirrel population, and the differential evolution algorithm (DE) from the SciPy library. Reliable identification of material parameters of models was shown using evolutionary algorithms based on load diagrams of samples, including condition of artificial noise of "experimental" data.

Central part interpolation schemes for fractional differential equations

We consider a two-point boundary value problem for fractional integro-differential equations with weakly singular kernels. Using an integral equation reformulation of the underlying problem, a collocation method based on the central part interpolation by continuous piecewise polynomials on the uniform grid is constructed and analysed. Optimal convergence order of the proposed method is established and confirmed by numerical experiments.

On a Certain Functional Equation and Its Application to the Schwarz Problem

The Schwarz problem for J-analytic functions in an ellipse is considered. In this case, the matrix J is assumed to be two-dimensional with different eigenvalues located above the real axis. The Schwarz problem is reduced to an equivalent boundary value problem for the scalar functional equation depending on the real parameter l. This parameter is determined by the Jordan basis of the matrix J. An analysis of the functional equation was performed. It is shown that for l∈[0,1], the solution of the Schwarz problem with matrix J exists uniquely in the Hölder classes in an arbitrary ellipse.