Formulation Of Boundary Value Problem Research Articles

Variational formulation for fractional inhomogeneous boundary value problems

The steady state fractional convection diffusion equation with inhomogeneous Dirichlet boundary is considered. By utilizing standard boundary shifting trick, a homogeneous boundary problem is derived with a singular source term which does not belong to $$L^2$$ anymore. The variational formulation of such problem is established, based on which the finite element approximation scheme is developed. Inf-sup conditions for both continuous case and discrete case are demonstrated thus the corresponding well-posedness is verified. Furthermore, rigorous regularity analysis for the solutions of both original equation and dual problem is carried out, based on which the error estimates for the finite element approximation are derived. Numerical results are presented to illustrate the theoretical analysis.

Об упрощении сложных уравнений применением блочных элементов

There are several approaches aimed at simplifying complex partial differential equations or their systems involved in the formulation of boundary value problems by introducing simpler, but in a larger number of differential equations. Their solutions allow us to describe solutions to complex boundary value problems. However, to implement this approach, it is necessary to construct solutions of simplified boundary value problems for arbitrary boundary conditions in solvability spaces boundary value problem. In some cases, this can be done using the block element method. The block element method, which has a topological basis, reveals both global and local properties of solutions to boundary value problems for partial differential equations. At the same time, it can be used to study and solve more complex boundary value problems by applying relations that describe certain equations of the continuum by means of relatively simple equations, for example, Helmholtz. To do this, we need to construct solutions of the Helmholtz equations that satisfy boundary conditions that contain completely arbitrary values, rather than partial values, set at the boundary of functions. In relation to the Helmholtz equations, this is achieved using the block element method. Examples of constructing solutions to boundary value problems for Helmholtz equation for Dirichlet and Neumann problems and a comparative analysis of solutions are given in this article.

Encoding electromagnetic transformation laws for dimensional reduction

AbstractElectromagnetic phenomena are mathematically described by solutions of boundary value problems. For exploiting symmetries of these boundary value problems in a way that is offered by techniques of dimensional reduction, it needs to be justified that the derivative in symmetry direction is constant or even vanishing. A generalized notion of symmetry can be defined with different directions at every point in space, as long as it is possible to exhibit unidirectional symmetry in some coordinate representation. This can be achieved, for example, when the symmetry direction is given by the direct construction out of a unidirectional symmetry via a coordinate transformation which poses a demand on the boundary value problem. Coordinate independent formulations of boundary value problems do exist but turning that theory into practice demands a pedantic process of backtranslation to the computational notions. This becomes even more challenging when multiple chained transformations are necessary for propagating a symmetry. We try to fill this gap and present the more general, isolated problems of that translation. Within this contribution, the partial derivative and the corresponding chain rule for multivariate calculus are investigated with respect to their encodability in computational terms. We target the layer above univariate calculus, but below tensor calculus.

Some Boundary-Value Problems of Fractional-Differential Mobile–Immobile Migration Dynamics in a Profile Filtration Flow

Within the framework of fractional-differential mathematical model, the formulation of boundary-value problems of convective diffusion of soluble substances with regard to immobilization under the conditions of stationary profile filtration of groundwater from reservoir to drainage is performed. In case of averaging the filtration rate over the complex potential region, closed solutions of boundary-value problems, corresponding to classical and nonlocal boundary conditions are obtained. In the general case of the variable filtration velocity, a technique is developed for the numerical solution to the boundary-value problem of convective diffusion in a fractional-differential formulation, the problems of parallelizing computations are covered, and the results of the computer experiments are presented.

Homogenization of time-harmonic Maxwell’s equations in nonhomogeneous plasmonic structures

We carry out the homogenization of time-harmonic Maxwell’s equations in a periodic, layered structure made of two-dimensional (2D) metallic sheets immersed in a heterogeneous and in principle anisotropic dielectric medium. In this setting, the tangential magnetic field exhibits a jump across each sheet. Our goal is the rigorous derivation of the effective dielectric permittivity of the system from the solution of a local cell problem via suitable averages. Each sheet has a fine-scale, inhomogeneous and possibly anisotropic surface conductivity that scales linearly with the microstructure scale, d. Starting with the weak formulation of the requisite boundary value problem, we prove the convergence of its solution to a homogenization limit as d approaches zero. The effective permittivity and cell problem express a bulk average from the host dielectric and a surface average germane to the 2D material (metallic layer). We discuss implications of this analysis in the modeling of plasmonic crystals.

CONDITIONS OF EXISTENCE THE TRACE OF FUNCTIONS FROM SPACE WITH MULTIWEIGHTED DERIVATIVES IN A SPECIAL POINT

When solving differential equations with variable coefficients, especially when the coefficients degenerate at the boundary of a given domain, problems arise in the formulation of boundary value problems. Usually, differential equations with variable coefficients are investigated in a suitable weight functional space. Often in the role of such spaces the weight Sobolev space or various generalizations are considered, which are currently sufficiently studied. However, in some cases, when the coefficients of the considered differential equation are strongly degenerate, the formulation of boundary value problems becomes problematic. In this work, we consider the so-called space with multiweighted derivatives, where after each derivative, the function is multiplied by the weight function and then the next derivative is taken. By controlling the behavior of the weight functions on the boundary, strongly degenerate equations can be investigated. Here we investigate the existence of traces on the boundary of a function from such spaces.

Construction of the Scheme of Substitution of Magnetic Systems of Electrical Devices Using Connected Field Potentials

The problem of developing an algorithm for the formalized construction of equivalent circuits for electro-mechanical devices containing several magnetic systems, the relative position of which in space can change, is considered and solved. It is assumed that the magnetic systems are made of anisotropic electrical steel with non-linear characteristics. Generalized formulations of boundary value problems describing the distribution of the magnetic field in the considered electrical systems and their variational formulations are given. A combined method of finite elements and complex boundary elements is described, which makes it possible to calculate the field in magnetic systems and in an area not filled with steel. A formalized procedure for constructing magnetic equivalent circuits for electrical devices using conjugate potentials that have the meaning of the scalar poten-tial and the magnetic field flux function is described. It is shown that the use of conjugate potentials makes it possible to directly find the magnetic fluxes and magnetic voltages used for constructing equivalent circuits without numerical differentiation and integration. A step-by-step algorithm for finding the equivalent circuit topology and component branch equations is described. An example of building a chain model of an electro-magnet is given. The estimation of the error in calculating its characteristics using a magnetic equivalent cir-cuit is carried out.

Natural vibrations of truncated conical shells of variable thickness

The paper presents the results of studying the natural frequencies of circular truncated conical shells, the thickness of which varies along the length according to different laws. The behavior of elastic structure is described in the framework of the classical theory of shells based on the Kirchhoff-Love hypotheses. The corresponding geometric and physical relations together with the equations of motion are reduced to a system of ordinary differential equations for new unknowns. The solution to the formulated boundary value problem is found using Godunov's orthogonal sweep method involving the numerical integration of differential equations by the fourth order Runge-Kutta method. The natural frequencies of vibrations are evaluated using a combination of a step-wise procedure and subsequent refinement by the interval bisection method. The reliability of the obtained results is verified by making a comparison with the known numerical-analytical solutions. The dependences of the minimum vibration frequencies obtained at shell thicknesses subject to a power-law variation (linear and quadratic, having symmetric and asymmetric shapes) and harmonic variation (with positive and negative curvature) are investigated for shells with different combinations of boundary conditions (free support, rigid and cantilever fastening), cone angles and linear dimensions. The results of the study confirm the existence of configurations, which provide a significant increase in the frequency spectrum in comparison with shells of constant thickness under the same limitations on the weight of the structure.

Optimal control of spacecraft motion in the vicinity of Eros 433 asteroid

The article describes an algorithm for optimizing controlled motion of a spacecraft equipped with low-thrust electric propulsion engines maneuvering in the vicinity of an object with an irregular gravitational field (asteroid Eros 433). A mathematical model of the objects gravitational potential and a model of spacecraft motion are presented. The Pontryagin maximum principle is used to get the time-optimal control program. The formulated boundary value problem is solved numerically by the modified Newton method. The described algorithm can be used to solve similar problems of low-thrust flight dynamics in the vicinity of objects with irregular gravitational fields.

Basic theorems and wave propagation in a piezothermoelastic medium with dual phase lag

The basic governing equations for an anisotropic piezothermoelastic solid with dual phase lag are presented and used to study the problem. Some basic theorems like variational principle, uniqueness theorem and theorem of reciprocity are established for the assumed model. Also, we characterize an alternative formulation of the mixed initial boundary value problem. These theorems are also summarized for a special case of orthotropic piezothermoelastic solid with dual phase lag. We formulated the plane wave propagation in an orthotropic piezothermoelastic solid with dual-phase-lag model. The non-trivial solution of the system is insured by a quartic equation whose roots represent the complex velocities of four attenuating waves in the medium. Various characteristics of the waves like phase velocity, attenuation quality factor, specific heat loss and penetration depth are computed numerically and shown graphically for three different models with the direction of propagation in three-dimensional space. Some special cases are also deduced from the present investigation.

An alternative formulation of the magnetostatic boundary value problem

We present an alternative formulation of the magnetostatic boundary value problem that is useful for calculating the magnetic field around a magnetic material placed in the vicinity of steady currents. The formulation differs from the standard approach in that a single-valued scalar potential plus a vector field that depends on the given currents but not on the magnetic material are used to obtain the total magnetic field instead of a magnetic vector potential. We illustrate the method with examples.

GENERIC-based formulation and discretization of initial boundary value problems for finite strain thermoelasticity

A new variational formulation for finite strain thermoelastodynamics is proposed. The variational formulation emanates from the GENERIC formalism and makes possible the free choice of the thermodynamic state variable. In particular, one may choose the absolute temperature, the internal energy density or the entropy density as state variable. To solve initial boundary value problems, the GENERIC formalism is extended to open systems. The discretization in time makes use of the standard mid-point rule. Depending on the choice of the thermodynamic state variable structure-preserving schemes are obtained. For example, choosing the internal energy as state variable yields a new energy–momentum consistent scheme. Thus the newly developed GENERIC-based weak form is particularly well suited for the design of structure-preserving methods. Numerical investigations are presented which confirm the theoretical findings and shed light on the numerical stability of the newly developed schemes.

To the Modeling of multilayer Thin Prismatic Bodies

Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies were also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the fifth approximation of the micropolar theory of five-layer thin prismatic bodies are considered in more detail.

О СМЕШАННОЙ ЗАДАЧЕ E ДЛЯ ВЫРОЖДАЮЩИХСЯ НА ГРАНИЦЕ ОБЛАСТИ ПАРАБОЛИЧЕСКИХ УРАВНЕНИЙ 2ГО ПОРЯДКА

Исследуется вопрос об однозначной разрешимости первой смешанной задачи для вырождающегося параболического уравнения 2го порядка в случае, когда граничная и начальная функции принадлежат пространствам типа L2. Данная тематика берет начало с классических работ Ф. Рисса 1 и Литтлвуда и Пэли 2, посвященных граничным значениям аналитических функций. Дальнейшее развитие этой тематики для равномерно эллиптических уравнений получило в работах В. П. Михайлова, А. К. Гущина 3 9. Условие гладкости границы (Q C2) можно ослабить (см. 7). При наиболее слабых ограничениях на гладкость границы (и на коэффициенты уравнения) критерий существования граничного значения установлен в 7 9. При этом, как показано в работе 9, все направления принятия граничных значений для равномерно эллиптических уравнений оказываются равноправными, решение обладает свойством, аналогичным свойству непрерывности по совокупности переменных. В случае вырождения уравнения на границе области, когда направления не являются равноправными, ситуация более сложная. При этом постановка первой краевой задачи определяется типом вырождения. В случае, когда значения соответствующей квадратичной формы вырождающегося эллиптического уравнения на векторе нормали отличны от нуля (вырождение типа Трикоми), корректна задача Дирихле, и свойства такого вырождающегося уравнения весьма близки к свойствам равномерно эллиптического уравнения. В частности, в этой ситуации справедливы аналоги теорем Рисса 1 и Литтлвуда Пэли 2, 3. В случае вырождения типа Келдыша ситуация более сложная. Постановка первой краевой задачи и поведение решения вблизи границы определяются порядком вырождения уравнения, а в случае сильного вырождения коэффициентами при младших членах. Вопросам разрешимости первой краевой задачи для вырождающихся эллиптических и параболических уравнений посвящено большое число работ. Достаточно отметить работы Ф. Трикоми 10, М. В. Келдыша11, А. В. Бицадзе 12, С. А. Терсенова 13, И. М. Петрушко 14, 15, О. А. Олейник, Е. В. Радкевича 16, Г. Фикеры 17 и др. Из недавних работ можно отметить 18. В настоящей работе рассматривается случай сильного вырождения параболического уравнения 2го порядка, когда соответствующая квадратичная форма убывает как r(x) и постановка первой смешанной задачи определяется коэффициентом при первой производной по нормали. The paper investigates the question of the unique solvability of the first mixed problem for a degenerate secondorder parabolic equation in the case when the boundary and initial functions belong to spaces of type L2. This topic originates from the classical works of F. Riesz 1 and Littlewood and Paley 2 devoted to the boundary values of analytic functions. Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criterion for the existence of a boundary value was established in 7 9. The boundary smoothness condition (Q C2) can be weakened (see 10). Moreover, as shown in 9, all directions of the adoption of boundary values for uniformly elliptic equations turn out to be equal the solution has the property similar to the property of continuity in the set of variables. In the case of degeneration of the equation at the boundary of the region, when the directions are not equal, the situation is more complicated. Moreover, the formulation of the first boundary value problem is determined by the type of degeneracy. In the case when the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are nonzero (Tricomi type degeneracy), the Dirichlet problem is correct, and the properties of such degenerate equation are very close to the properties of a uniformly elliptic equation. In particular, in this situation, analogs of the Riesz theorems 1 and Littlewood Paley 2, 3 are valid. In the case of degeneracy of the Keldysh type, the situation is more complicated. The statement of the first boundaryvalue problem and the behavior of the solution near the boundary are determined by the order of degeneracy of the equation, and in the case of strong degeneracy, by the coefficients of the lower terms. A large number of papers have been devoted to the solvability of the first boundaryvalue problem for degenerate elliptic and parabolic equations. Note the works of F. Tricomi 11, M. V. Keldysh 12, A. V. Bitsadze 13, S. A. Tersenov 14, I. M. Petrushko 15, O. A. Oleinik and E. V. Radkevich 16, G. Fichera 17, etc. From recent works it is worth noting 18. In this paper, we consider the case of strong degeneracy of a secondorder parabolic equation when the corresponding quadratic form decreases as r(x) and the formulation of the first mixed problem is determined by the coefficient of the first normal derivative.

Modeling the Thermal Response of a Cast-Iron–Concrete System under Active Thermal Non-Destructive Control

In this paper, we perform the mathematical modeling of the thermophysical experiment of diagnosing a space under a cast-iron tubing of a vertical tubing support using optical lock-in thermography. We present the formulation and numerical solution of a two-dimensional boundary value problem of non-stationary heat exchange, including heat transfer due to thermal conductivity, natural convection, and radiation. The defect in the under-tubing space is detected based on an analysis of the distribution of the phase characteristics of the temperature fluctuations at the tubing boundary (which is available for observation) calculated by the digital lock-in correlation. The effect of the heating frequency, heating time, and noise on the distribution of phase characteristics is studied to optimize the process of active thermal non-destructive control. Based on the use of the reference temperature distribution at the tubing boundary with a defect-free tubing space, an algorithm for mathematically processing noisy data is proposed. The algorithm includes the Kalman filter, the Rauch-Tung-Striebel smoothing procedure, and a smoothing spline method with the criterial selection of the smoothing parameter. The results of the computational experiments illustrate the effectiveness of the presented approach.

Solving biharmonic equation as an optimal control problem using localized radial basis functions collocation method

Solving fourth or higher order differential equations using localized numerical methods as the finite element, the finite difference and the localized radial basis functions (LRBFs), show difficulties to get accurate results. So many authors adopted iterative methods to deal with biharmonic equation by decoupling it into two Poisson’s problems. In this paper we investigate the formulation of a mixed fourth order boundary value problem as an optimal control one. Then, we establish a new iterative method by coupling an optimization iterative scheme and localized radial basis functions meshless collocation method to deal with the numerical solution of such problem. To transform the problem into an optimal control one, we firstly construct the constraints functions by splitting the biharmonic equation into two coupled Laplace equations. The Neumann boundary condition is used as energy-like error functional to be minimized. Theoretical analysis of the existence and uniqueness of the solution of such formulated optimization problem and its equivalence to the initial biharmonic problem are also demonstrated. Finally we show the effectiveness of the proposed method by solving problems in both convex and non-convex regular and irregular domain.

On the Modeling of Five-Layer Thin Prismatic Bodies

Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity, similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies are also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the micropolar theory of K-layer thin prismatic bodies are considered. From here, we can easily get the statements of the initial-boundary value problems for the five-layer thin prismatic bodies.

Boundary element analysis of 3D shell‐like rigid electrically conducting inclusions in anisotropic thermomagnetoelectroelastic solids

AbstractThe paper presents general boundary element approach for analysis of thermomagnetoelectroelastic solids containing shell‐like electrically conducting inclusions with high magnetic permittivity. The latter are modeled with opened surfaces with certain boundary conditions on their faces. Rigid displacement and rotation, along with constant electric and magnetic potentials of inclusions are accounted for in these boundary conditions. Formulated boundary value problem is reduced to a system of singular boundary integral equations, which is solved numerically by the boundary element method. Special attention is paid to the field singularity at the front line of a shell‐like inclusion. Special shape functions are introduced, which account for this square‐root singularity and allow accurate determination of field intensity factors. Approaches for fast and accurate numerical evaluation of anisotropic thermomagnetoelectroelastic kernels are discussed, which are crucial in derivation of fast and precise boundary element approach. Numerical examples are presented.

Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as a linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the neglected effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

Approximate Solution and its Convergence Analysis for Hypersingular Integral Equations

This paper propose a residual based Galerkin method with Legendre polynomial as a basis functions to find the approximate solution of hypersingular integral equations. These equations occur quite naturally in the field of aeronautics such as problem of aerodynamics of flight vehicles and during mathematical modeling of vortex wakes behind aircraft. The analytic solution of these kind of equations is known only for a particular case ( m(x,t) =0 in Eqn (1)). Also, in these singular integral equations which occur during the formulation of many boundary value problems, the known function m(x,t) in (Eqn (1)) is not always zero. Our proposed method find the approximate solution by converting the integral equations into a linear system of algebraic equations which is easy to solve. The convergence of sequence of approximate solutions is proved and error bound is obtained theoretically. The validation of derived theoretical results and implementation of method is also shown with the aid of numerical illustrations.