Homogeneous Boundary Value Problem Research Articles

The Self-Consistent Problem of Oscillations and Waves Associated with Sources

It is shown that the source function on the right-hand side of the Helmholtz equation, which is a solution of a homogeneous boundary value problem for this equation, provides the excitation of an associated wave (oscillation). Since this wave (oscillation) exists only in the presence of a source of the specified type, it is proposed to classify them (wave and oscillation) as associated with the source, and to consider the problem “generating” them as self-consistent, because the excited wave (oscillation) is in synchronism with the source in all wave numbers and energy balance.

An inverse problem of reconstructing option drift rate from market observation data

Drift rate is a very important parameter in the evolution of stock price, which has significant impact on the corresponding option pricing. This paper deals with an inverse problem of recovering the drift function by current market prices of options. Different from the usual inverse volatility problem, our mathematical model does not tend to zero at infinity, which may bring great trouble to theoretical analysis and numerical calculation. To overcome this difficulty, we use an artificial boundary and homogenization technique to transform the original problem into a homogeneous initial boundary value problem on a bounded domain. Then, based on the optimal control framework, we construct the corresponding optimization problem and strictly prove the well-posedness of the minimizer. Finally, we design an iterative algorithm to obtain the numerical solution. We give some typical examples to verify the validity of our method.

Complex Resonance As a Self-Consistent Electromagnetic Process

The phenomenon, which is classified as a complex resonance and takes place in electromagnetic structures described by nonself-adjoint boundary value problems, is considered. It is shown that the oscillation adjoint to a source is formed, when complex waves that are complex conjugate by wave numbers and amplitudes are excited in couples. This oscillation is described by the self-consistent boundary value problem for the adjoint Helmholtz equation, i.e., for the equation, whose right-hand side is the solution of the homogeneous boundary value problem. The distinctive feature of the complex resonance is in the fact that it exists in the whole range of complex waves, when the source is necessarily present. Theoretical and experimental results of the investigation of the considered phenomenon are presented.

On the quasi-static approximation in the initial boundary value problem of linearised elastodynamics

Continuous data dependence estimates are employed to rigorously derive conditions that validate the quasi-static approximation for the initial homogeneous boundary value problem in the theory of small elastic deformations superposed upon large elastic deformations. This theory imposes no sign-definite assumptions on the linearised elastic moduli and in consequence the requisite estimates are established using methods principally motivated by known Lagrange identity arguments.

Related thermoelastic elastomeric task for cylinder

A new closed solution of the coupled non-stationary problem of thermoelectroelasticity for a long piezoceramic radially polarized cylinder is constructed when the boundary conditions of the 1st kind of thermal conductivity are satisfied on its front surfaces. The case is considered when the rate of change of the temperature field does not affect the inertial characteristics of the elastic system, which makes it possible to include in the initial calculated relations of the problem under consideration the linear equations of equilibrium, electrostatics and thermal conductivity with respect to the radial component of the displacement vector, electric potential, as well as the function of changing the temperature field. The calculations use the classical Fourier law of thermal conductivity. To solve the problem, the mathematical apparatus of incomplete separation of variables is used in the form of a generalized biorthogonal finite integral transformation (KIP), based on a multicomponent relation of eigenvector-functions of two homogeneous boundary value problems. An important point in the procedure of the structural algorithm of this method is the selection of the adjoint operator, without which it is impossible to solve non-self-adjoint linear problems of mathematical physics.

Mathematical formulations for elastic magneto-electrically coupled soft materials at finite strains: Time-independent processes

Recently, among other smart and multifunctional materials, magneto-electric soft materials are expected to open a new horizon with myriad of potential applications such as wireless energy harvesting, spintronics and nonvolatile memories, magneto-electric random access memory, to mention a few. Magneto-electric coupling can be defined as the ability of a material to electrically polarize upon the application of a magnetic field and conversely, to magnetize under the application of an electric field. In contrast to traditional multi-ferroic hard materials, magneto-electric soft materials are of largely deformable where electric and magnetic fields and mechanical deformations are intricately coupled at finite strians. In this contribution, we will emphasis to formulate generalised mathematical frameworks of finitely deformed magneto-electric soft materials. After elaborating fundamental and governing equations, some homogeneous and non-homogeneous classical boundary value problems are studied under magneto-electrically coupled loads.

Equation Including Local Fractional Derivative and Neumann Boundary Conditions

The aim of this study to discuss the construction of the solution of fractional partial differential equations (FPDEs) with initial and boundary conditions. Since the homogenous initial boundary value problem involves local fractional-order derivative, it has classical initial and boundary conditions. By means of the separation of variables method (SVM) and the inner product on L^2\left[0,l\right], we construct the solution in this series form in terms of eigenfunctions of related Sturm-Liouville problem. An illustrative example presents the applicability and influence of the separation of variables method on fractional mathematical problems.

Inverting the variable fractional order in a variable-order space-fractional diffusion equation with variable diffusivity: analysis and simulation

Abstract Variable-order space-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena, including anomalously superdiffusive transport of solutes in heterogeneous porous media, long-range spatial interactions and other applications, as well as eliminating the nonphysical boundary layers of the solutions to their constant-order analogues. In this paper, we prove the uniqueness of determining the variable fractional order of the homogeneous Dirichlet boundary-value problem of the one-sided linear variable-order space-fractional diffusion equation with some observed values of the unknown solutions near the boundary of the spatial domain. We base on the analysis to develop a spectral-Galerkin Levenberg–Marquardt method and a finite difference Levenberg–Marquardt method to numerically invert the variable order. We carry out numerical experiments to investigate the numerical performance of these methods.

Variable-order space-fractional diffusion equations and a variable-order modification of constant-order fractional problems

ABSTRACT Fractional diffusion equations (FDEs) were shown to provide very competitive descriptions of challenging phenomena of anomalously diffusive transport or long-range interactions. However, FDEs introduce mathematical issues that are not common in the context of integer-order diffusion equations. For instance, the homogeneous Dirichlet boundary-value problems of linear elliptic FDEs with smooth data in one space dimension may generate solutions with singularities that do not seem physically relevant, which are in sharp contrast to their integer-order analogues do. We prove the wellposedness of the Dirichlet boundary-value problem of one dimensional variable-order linear space-fractional diffusion equations (sFDEs). We further prove that their solutions have the similar regularities as their integer-order analogues if the order has an integer limit at the boundary or have the same singularity near the boundary as their constant-order sFDE analogues if the order has a non-integer limit at the boundary. In particular, we prove that constant-order sFDEs with a variable-order modification indeed generate solutions with significantly improved regularities.

A spring-damping regularization of the Fourier sine series solution to the inverse Cauchy problem for a 3D sideways heat equation

For solving the inverse Cauchy problem of a three-dimensional (3D) non-homogeneous sideways heat equation in a cuboid, we first transform it into a partial homogeneous boundary value problem, using a time-dependent two-dimensional (2D) homogenization technique. After a time step-by-step approach of the inverse Cauchy problem, it becomes an inverse Cauchy problem of the 3D modified Helmholtz equation. A sequence of 2D sine functions and a set of second-order ordinary differential equations are derived to determine the expansion coefficients, which can be solved in closed-form. In order to stably solve the inverse Cauchy problem, we simply introduce a spring-damping regularization technique to modify the coefficients, which results in a regularized 2D Fourier sine series solution of the unknown Dirichlet data on the top of the cuboid. The novel method can find the solution in the whole domain, which is quite accurate with small errors, even a large noise up to is imposed on the given data. The present method together with the analyses of regularization parameter and the stability of solution is novel and shows promise in the 3D sideways problems of the heat equation.

Растяжение и раздувание нелинейно-упругого неоднородного полого цилиндра

The influence of the inhomogeneity of material properties on the process of three-dimensional stability loss of a hollow cylinder stretched by axial force and loaded by uniform pressure on the outer or inner side surface is investigated. We used two standard models describing the compressible nonlinearly elastic material's mechanical properties, namely the three-constant Blatz and Ko model, as well as the five-constant Mournaghan model. Usage of the semi-inverse method allows the reduction of a three-dimensional cylinder equilibrium problem to the study of a non-linear boundary-value problem for an ordinary second-order differential equation. Stability analysis was carried out based on the linearization of the equilibrium equations in the vicinity of the constructed solution. The value of a de-formation characteristic for which there were nontrivial solutions of a homogeneous boundary-value problem for the equations of neutral equilibrium obtained in the linearization process was identified with the critical value of the loading parameter, i.e., the value at which the system loses stability. The coefficients of the cylinder's stretching or radial expansion and the dimensionless characteristic of the applied pressure served as such parameters. On the plane of the loading parameters, stability regions are determined. The influence of heterogeneity on the size and shape of these regions is analyzed.

Intensity factors of homogeneous boundary value problem in angular area of the domain

A relevant task in terms of ensuring reliable functioning of structures and constructions is studying structural areas with complex boundary shapes. Areas with geometrically non-linear shape of boundaries – angular cut-outs, cuts, and element connections with rupturing forced deformations – are characterized by singularities of the stress-strain state. In experimental research of areas with complex boundary shapes one must take into account the difference between stress intensity factors and deformation intensity factors. This paper describes a procedure for obtaining the stress-strain state in neighborhood of the angular cutout apex on the domain boundary using stress intensity factors and deformation intensity factors for a variety of homogeneous boundary conditions. In the angular cut-out area on the domain boundary, the stress-strain state is characterized by limit values of stresses and deformations, similar to stress intensity factors used with force criteria in fracture mechanics. Taking into account the ratios of stress and deformation intensity factors in experimental analysis of the stress-strain state in angular areas of structures is practically relevant for determining critical values of stress and deformation intensity factors.

Equilibrium stability of nonlinear elastic sphere with distributed dislocations

An exact formulation and a solution of the stability problem for a three-dimensional elastic body containing distributed dislocations are given. The buckling phenomenon for a hollow nonlinear elastic sphere of the semi-linear (harmonic) material with edge dislocations is studied. The study is carried out within the framework of the continuum theory of continuously distributed dislocations using the bifurcation method of buckling analysis. The bifurcation method is to find the equilibrium positions of an elastic body, which differ little from the subcritical (unperturbed) state. The perturbed equilibrium state is described by a linearized boundary value problem. By solving a homogeneous linear boundary value problem, the minimum critical value of the external pressure at which the sphere loses stability is found. The influence of dislocations on the buckling of both thin and thick spherical shells is analyzed.

A fast method for variable-order space-fractional diffusion equations

We develop a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical approximation does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires $O(kN\log^2 N)$ memory and $O(k N\log^3 N)$ computational complexity with $N$ and $k$ being the numbers of unknowns and the approximants, respectively. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

Nonstationary torsion problems for the finite cylinder partly coupled with rigid base

The axisymmetric dynamic problems of determining the stress state in the vicinity of the delamination in a finite cylinder partly coupled with the rigid base were solved. In contrast to the traditional solution methods based on the use of the integral Laplace transform, the proposed method consists in the difference approximation only of the time derivative. As a result, the original problem is replaced by a sequence of homogeneous boundary value problems for the Helmholtz equation that reduce to the Fredholm integral equation of the second kind. The numerical solution found made it possible to obtain an approximate formula for calculating the stress intensity factor.

Связанная нестационарная задача термоэлектроупругости для длинного полого цилиндра

Построено новое замкнутое решение связанной нестационарной задачи термоэлектроупругости для длинного пьезокерамического радиально поляризованного цилиндра при удовлетворении на его лицевых поверхностях граничных условий теплопроводности 1-го и 3-го рода. Рассматривается случай, когда скорость изменения температурного поля не оказывает влияние на инерционные характеристики упругой системы, что позволяет включить в исходные расчетные соотношения рассматриваемой задачи линейные уравнения равновесия, электростатики и теплопроводности относительно радиальной компоненты вектора перемещений, электрического потенциала, а также функции изменения температурного поля. В расчетах применяется классический закон теплопроводности Фурье. Для решения задачи используется математический аппарат неполного разделения переменных в виде обобщенного биортогонального конечного интегрального преобразования, основанного на многокомпонентном соотношении собственных вектор-функций двух однородных краевых задач. Важным моментом в процедуре структурного алгоритма данного метода является выделение сопряженного оператора, без которого невозможно осуществить решение несамосопряженных линейных задач математической физики. Построенные расчетные соотношения дают возможность определить напряженно-деформированное состояние, температурное и электрическое поля, индуцируемые в пьезокерамическом элементе при произвольном температурном внешнем воздействии. Анализ численных результатов позволяет определить толщину стенки цилиндра, при которой электрическое поле приводит к перераспределению температурного поля. Установлено, что скорость изменения объема пьезокерамического тела при внешнем температурном воздействии не оказывает существенного влияния на температурное поле. Разработанный алгоритм расчета находит свое применение при проектировании нерезонансных пьезоэлектрических датчиков температуры.

Априорные оценки локального разрывного метода Галеркина на разнесенных сетках для решения уравнения параболического типа в рамках однородной задачи Дирихле

Представлены априорные оценки точности решения однородной краевой задачи для параболического уравнения с помощью локального метода Галеркина с разрывными базисными функциями на разнесенных сетках. Дискретизация по пространству строится с помощью обращения к смешанной конечно-элементной формулировке. Производные второго порядка не могут быть согласованы напрямую в слабой вариационной формулировке, используя пространство разрывных функций. Для понижения порядка компоненты вектора потока рассматриваются как вспомогательные неизвестные искомого уравнения второго порядка. Аппроксимация строится на разнесенных сетках. Основная сетка состоит из треугольников, двойственная сетка состоит из медианных контрольных объемов вокруг узлов треугольной сетки. Аппроксимация искомой функции строится на ячейках основной сетки, в то время как аппроксимация вспомогательных неизвестных строится на ячейках двойственной сетки. Для вычисления потоков на границе между элементами используется стабилизирующий параметр. При этом поток искомой функции не зависит от вспомогательных функций, в то время как поток вспомогательных величин зависит от искомой функции. Для решения поставленной задачи в работе формулируются и доказываются необходимые леммы. В результате сформулирована и доказана основная теорема, результатом которой являются априорные оценки при решении параболического уравнения с помощью метода Галеркина с разрывными базисными функциями. Основную роль при анализе сходимости играет оценка для отрицательной нормы градиента. В работе для стабилизирующего параметра порядка $1$ показано, что порядок сходимости будет $k+{1}/{2}$, а в случае использования стабилизирующего параметра порядка $h^{-1}$ порядок сходимости увеличивается до $k+1$, когда в качестве базиса используются полиномы степени не выше $k$.

An exact solution of the dynamical problem for the infinite elastic layer with a cylindrical cavity

The wave field of an infinite elastic layer weakened by a cylindrical cavity is constructed in this paper. The ideal contact conditions are given on the upper and bottom faces of the layer. The normal dynamic tensile load is applied to a cylindrical cavity's surface at the initial moment of time. The Laplace and finite $sin-$ and $cos-$ Fourier integral transforms are applied successively directly to axisymmetric equations of motion and to the boundary conditions, on the contrary to the traditional approaches, when integral transforms are applied to solutions' representation through harmonic and biharmonic functions. This operation leads to a one-dimensional vector homogeneous boundary value problem with respect to unknown transformations of displacements. The problem is solved using matrix differential calculus. The field of initial displacements is derived after application of inverse integral transforms. The case of the steady-state oscillations was investigated. The normal stress on the faces of the elastic layer are constructed and investigated depending on the mechanical and dynamic parameters.

Non-stationary coupled axisymmetric thermoelasticity problem for a rigidly fixed round plate

A new closed solution is constructed for the axisymmetric dynamic problem of the classical (CTE) theory of thermoelasticity for a rigidly fixed circular isotropic plate in the case of a temperature change on its face surfaces (boundary conditions of the first kind). The mathematical formulation of the problem under consideration includes linear equations of thermal conductivity and equilibrium in a spatial setting, assuming that their inertial elastic characteristics can be neglected in the structures under study. In constructing a general solution of related non-self-conjugate equations, we use the mathematical apparatus of separation of variables in the form of finite integral transformations i.e. Hankel along the radial coordinate and biorthogonal transformation (FIT) with respect to the axial variable. At each stage of the investigation, a procedure is performed to reduce the boundary conditions to a form that allows the corresponding transformation to be applied. A particular feature of this solution is the application of a FIT based on a multicomponent relation of the eigenvector functions of two homogeneous boundary value problems. An important point in the procedure of the structural algorithm is the separation of the adjoint operator, without which it is impossible to solve non-self-adjoint linear problems of mathematical physics. This transformation is the most effective method for studying similar boundary value problems. The calculated design relationships make it possible to determine the stress-strain state and the character of the distribution of the temperature field in a rigidly fixed circular isotropic plate for an arbitrary external temperature effect with respect to time. Numerical analysis of the strength characteristics of the concrete structure shows that during the period of the unsteady load the maximum values of mechanical stresses are observed. Later, at a constant temperature regime, as a result of heating the entire plate, the displacements increase and the stresses fall.

Excitation of Spatially Developed Lamb Waves by a System of Body and Surface Loads (Part 1)

Principle and method for designing electroacoustic transducers operating in the mode of elastic wave excitation in isotropic solid bodies are discussed. Relations that completely define the entire set of eigenfunctions (normal waves) are derived by solving a homogeneous boundary-value problem of dynamic elasticity. The basic solutions of the boundary-value problem of dynamic elasticity are considered assuming that the source of spatially developed Lamb waves is not axially symmetric. The complex roots of the dispersion equation for symmetric and antisymmetric Lamb waves are quantitatively estimated.