General Boundary Value Problems Research Articles

Numerical solution of a generalized boundary value problem for the modified Helmholtz equation in two dimensions

We propose numerical schemes for solving the boundary value problem for the modified Helmholtz equation and generalized impedance boundary condition. The approaches are based on the reduction of the problem to the boundary integral equation with a hyper-singular kernel. In the first scheme the hyper-singular integral operator is treated by splitting off the singularity technique whereas in the second scheme the idea of numerical differentiation is employed. The solvability of the boundary integral equation and convergence of the first method are established. Exponential convergence for analytic data is exhibited by numerical examples.

ФУНКЦИИ ГРИНА НЕКОТОРЫХ КРАЕВЫХ ЗАДАЧ ДЛЯ БИГАРМОНИЧЕСКИХ ОПЕРАТОРОВ И ИХ КОРРЕКТНЫЕ СУЖЕНИЯ

In this paper, a constructive method is given for constructing the Green function of the Dirichlet problem for a biharmonic equation in a multidimensional ball. The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise. There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the construction of the Green function is an open problem. For the ball, the Green function of the internal and external Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases. Finding general correct boundary value problems for differential equations is always an urgent problem. The abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of symmetric operators with finite defect indices was developed in detail. Many problems for partial differential equations lead to operators with infinite defect indices. In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain minimum operator, independently of each other, in terms of the inverse operator. In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's function.

GREEN'S FUNCTIONS AND CORRECT RESTRICTIONS OF THE POLYHARMONIC OPERATOR

In this paper, for completeness of presentation, we give explicitly the Green's functions for the classical problems - Dirichlet, Neumann, and Robin for the Poisson equation in a multidimensional unit ball. There are various ways of constructing the Green's function of the Dirichlet problem for the Poisson equation. For many types of areas, it is built explicitly. Recently, there has been renewed interest in the explicit construction of Green's functions for classical problems. The Green's function of the Dirichlet problem for a polyharmonic equation in a multidimensional ball is constructed in an explicit form, and for the Neumann problem the construction of the Green's function remains an open problem. The paper gives a constructive way of constructing the Green's function of Dirichlet problems for a polyharmonic equation in a multidimensional ball. Finding general well-posed boundary value problems for differential equations is always an urgent problem. In this paper, we briefly outline the theory of restriction and extension of operators and describe well-posed boundary value problems for a polyharmonic operator.

A Unified Variational Approach to Discontinuous Differential Equations

A direct variational technique involving Clarke generalized gradient is used to treat general boundary value problems with discontinuous nonlinearities. Based on the theory of positive definite symmetric operators it is established the nonsmooth variational form of the regularized inclusions which give the Filippov solutions of the discontinuous problems. These solutions reduce to classical solutions in case that a transversality condition on the set of discontinuities is satisfied. The results apply to a wide class of concrete boundary value problems of different orders. Two illustrative examples are given.

A fractional spectral collocation method for general Caputo two-point boundary value problems

In this paper we consider a fractional spectral collocation method for solving linear multi-term fractional differential equations involving Caputo-type fractional derivative. By using integral equation method, the boundary value problem of the fractional differential equation is reformulated as an equivalent Volterra-Fredholm integral equation with weakly singular kernels. Then we apply the fractional spectral collocation method to solve the resulting equation. We ensure that the approximate equation has a unique solution and the approximate solution arrives at the optimal convergence order in the infinite norm. At last, numerical examples are given to confirm the theoretical results.

Existence-uniqueness and fixed-point iterative method for general nonlinear fourth-order boundary value problems

Existence and uniqueness results are established for the general nonlinear fourth-order two-point boundary value problem with general linear homogeneous boundary conditions. The proofs are based on the Banach fixed-point theorem under the conditions that the nonlinear term is bounded and Lipschitz in a specific region. A fixed-point iterative method for finding the solution of the problem is also proposed with error estimates. The resulting iterative solutions automatically satisfy the boundary conditions provided that the initial iterate does so. Two examples are given to illustrate the applicability of the proposed approach and the efficiency of the iterative method.

Flexoelectric effect in dielectrics under a dynamic load

Health monitoring of structures can be improved if a more universal two-way electromechanical coupling, the flexoelectric effect, is utilized. It is a coupling between the electric polarization and the strain gradients. In the direct flexoelectricity, the electric polarization is induced by strain gradients, which yield also a finite higher-order stress tensor in the considered phenomenological theory. Because of the size-effect in higher-grade theories of continua, the polarization in piezoelectric solids under a non-uniform strains in nano-sized structures is significantly influenced by flexoelectricity. This effect is substantially enhanced near the crack defects, in regions with large strain gradients. The mixed finite element method (FEM) is developed from the variational formulation of flexoelectric boundary value problems. The C0 continuous approximation is applied independently for both the displacements and displacement gradients. The kinematic constraints between them are satisfied by collocation at some internal points of elements. The developed computational method is applied to general 2D boundary value problems with cracks under a dynamic load. The influence of flexoelectricity on the induced electric potential and the crack opening displacement is investigated.

Обобщенная краевая задача для дифференциального уравнения второго порядка с дробной производной

A generalized boundary value problem for a second-order differential equation with a fractional derivative is solved. An explicit representation of the solution of the problem under study is constructed, a condition for unique solvability is found, and a uniqueness theorem for the solution is proved.

МАТЕМАТИЧНА МОДЕЛЬ І МЕТОД РОЗВ’ЯЗАННЯ УЗАГАЛЬНЕНОЇ ЗАДАЧІ НЕЙМАНА ТЕПЛООБМІНУ ПАРАБОЛОЇДА ОБЕРТАННЯ

В даний час недостатньо вивчені питання про розподіл температурних полів в заготовках при новому способу нагрівання, що здійснюється шляхом обертання заготовок в магнітному полі постійного струму, який створюється у збудниках з надпровідними обмотками, без знання яких неможливо здійснити його технічну реалізацію з високими техніко-економічними показниками. При цьому невелике число наукових робіт присвячено дослідженням електромагнітних і теплових явищ всередині заготовки при її обертанні в магнітному полі постійного струму. Тому вельми актуальною є задача розробки математичних методів моделювання температурних полів в заготовках при індукційному нагріву металу з використанням інноваційної технології, розв’язанню якої присвячена ця робота. В статті побудована нова узагальнена просторова математична модель розрахунку температурних полів в заготовках у вигляді параболоїда обертання, що обертається з постійною кутовою швидкістю, з урахуванням кінцевої швидкості поширення тепла у вигляді крайової задачі математичної фізики, а також знаходження розв’язків отриманої крайової задачі. Вперше побудована математична модель розрахунку полів температури в параболоїді обертання, який обертається, з урахуванням кінцевої швидкості поширення тепла, у вигляді крайової задачі математичної фізики для гіперболічного рівняння теплопровідності з граничними умовами Неймана. Побудоване інтегральне перетворення для двовимірного кінцевого простору, із застосуванням якого знайдено температурне поле у вигляді збіжних рядів по функціям Фур’є. Знайдений розв’язок узагальненої крайової задачі теплообміну параболоїда обертання, який обертається, з урахуванням скінченності величини швидкості поширення тепла, може знайти застосування при моделюванні температурних полів, які виникають при індукційному нагріві, що здійснюється шляхом обертання заготовок в магнітному полі постійного струму, який створюється у збудниках з надпровідними обмотками.

RETRACTED: Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps

AbstractIt is known that the study of boundary value and mixed problems for integrable linear equations encounters significant difficulties of a fundamental nature. Exceptions are problems with boundary conditions of a special type, which are often called integrable or linearizable. The purpose of this article is to study the asymptotic behaviors of solutions of singularly perturbed general boundary value problems with boundary jumps for higher‐order equations. Using the Schlesinger–Birghof theorem, we constructed a fundamental system of solutions of a homogeneous perturbed equation of conditionally stable type in the critical case. Initial boundary functions are constructed based on the fundamental system of solutions. An analytical representation is found, the existence and uniqueness of a solution to this boundary value problem are proved. Asymptotic estimates of the solution and its derivatives are derived from the analytical representation of the solution of the given boundary value problem. The limit passage of solution of the perturbed problem to the solution of the unperturbed problem is proved. The conditions of the existence of jumps are found. The values of boundary jumps are determined. As a result, a class of boundary value problems is highlighted that has possessing of phenomenon of boundary jumps.

Mathematical analysis of static and dynamic three-dimensional models of thermo-electro-magneto-elastic solids

This paper is devoted to the investigation of three-dimensional models of thermo-electro-magneto-elastic solids made of a multidomain inhomogeneous anisotropic material. General boundary and initial boundary value problems corresponding to the static and dynamic models are studied where, on certain parts of the boundary, mechanical displacement, electric and magnetic potentials and temperature vanish and, on the corresponding remaining parts of the boundary, the mechanical stress vector and components of the electric displacement, magnetic induction and heat flux along the outward normal vector of the boundary are given. Variational formulations of the boundary and initial boundary value problems are obtained and, applying them, existence and uniqueness results and the continuous dependence of solutions on given data, in suitable factor spaces of Sobolev spaces or spaces of vector-valued distributions, are proved.

Adjoint state method for time-harmonic scattering problems with boundary perturbations

Knowing how the solution to time-harmonic wave scattering problems depends on medium properties and boundary conditions is pivotal in wave-based inverse problems, e.g. for imaging. This paper is devoted to the exposition of a computationally efficient method, called the adjoint state method, that allows to quantify the influence of media properties, directly and through boundary conditions, in the study of acoustic, electromagnetic and elastic time-harmonic waves. Firstly, the adjoint state method is derived for general boundary value problems. A continuous (rather than discrete) formalism is adopted in order to highlight the role of the boundary terms. Then, the method is applied systematically to acoustic, electromagnetic and elastic scattering problems with impedance boundary conditions, making use of the similitude between the three problems. Finally, numerical examples solved using the finite element method are presented to demonstrate the validity of the proposed method.

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Solving Advection-Diffusion Equations via Sobolev Space Notions

In this paper, the time-dependent advection-diffusion equation is studied. After introducing these equations in various engineering fields such as gas adsorption, solid dissolution, heat and mass transfer in falling film or pipe and other equations similar to transport phenomena, a new method has been proposed to find their solutions. Among the various works on solving these PDEs by numerical and somewhat analytical methods, a general analytical framework for solving these equations is presented. Using advanced components of Sobolev spaces, weak solutions and some important integral inequalities, an analytical method for the existence and uniqueness of the weak solution of these PDEs is presented, which is the best solution in the proposed structure. Then, with a reduced system of ODE, one can solve the problem of the general parabolic boundary value problem, which includes PDE transport phenomena. Besides, the new approach supports the infinite propagation speed of disturbances of (time-dependent) diffusion-time equations in semi-infinite media.

On a class of linear continuous-discrete systems with discrete memory

A class of linear functional differential systems with continuous and discrete times and discrete memory is considered. An explicit representation of the principal components to the general solution representation such as the fundamental matrix and the Cauchy operator is derived. The obtained representation is given in terms of the system parameters and opens a way towards efficient studying general linear boundary value problems and control problems with respect to a fixed collection of linear on-target functionals. In the study of the problems mentioned above outside the class under consideration, the systems with discrete memory can be employed as model or approximating ones. This can be useful as applied to systems with aftereffect under studying rough properties that hold under small perturbations of the parameters.

Analytical Solutions for a General Mixed Boundary Value Problem Associated with Motions of Fluids with Linear Dependence of Viscosity on the Pressure

An unsteady flow of incompressible Newtonian fluids with linear dependence of viscosity on the pressure between two infinite horizontal parallel plates is analytically studied. The fluid motion is induced by the upper plate that applies an arbitrary time-dependent shear stress to the fluid. General expressions for the dimensionless velocity and shear stress fields are established using a suitable change of independent variable and the finite Hankel transform. These expressions, that satisfy all imposed initial and boundary conditions, can generate exact solutions for any motion of this type of the respective fluids. For illustration, three special cases with technical relevance are considered and some important observations and graphical representations are provided. An interesting relationship is found between the solutions corresponding to motions induced by constant or ramptype shear stresses on the boundary. Furthermore, for validation of the results, the steady-state solutions corresponding to oscillatory motions are presented in different forms whose equivalence is graphically proved.

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

Abstract The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 {\mathcal{C}} in the non-classical setting when solutions are sought in the Bessel potential spaces ℍ p s ⁢ ( 𝒞 ) {\mathbb{H}^{s}_{p}(\mathcal{C})} , 1 p < s < 1 + 1 p {\frac{1}{p}<s<1+\frac{1}{p}} , 1 < p < ∞ {1<p<\infty} . Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain Ω α ⊂ ℝ 2 {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J. 27 2020, 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2 × 2 {2\times 2} systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes ℝ + {\mathbb{R}^{+}} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys. 60 2013, 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731].

A Ritz-based finite element method for a fractional-order boundary value problem of nonlocal elasticity

We present the analytical formulation and the finite element solution of a fractional-order nonlocal continuum model of a Euler-Bernoulli beam. Employing consistent definitions for the fractional-order kinematic relations, the governing equations and the associated boundary conditions are derived based on variational principles. Remarkably, the fractional-order nonlocal model gives rise to a self-adjoint and positive-definite system accepting a unique solution. Further, owing to the difficulty in obtaining analytical solutions to this boundary value problem, a finite element model for the fractional-order governing equations is presented. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the nonlocal response of a Euler-Bernoulli beam subjected to various loading and boundary conditions. The fractional-order positive definite system will be used here to address some paradoxical results obtained for nonlocal beams through classical integral approaches to nonlocal elasticity. Although presented in the context of a 1D Euler-Bernoulli beam, the f-FEM formulation is very general and could be extended to the solution of any general fractional-order boundary value problem.

Crack analysis of nano-sized thermoelectric material structures

General 2D boundary value problems with cracks in nano-sized structures of thermoelectric materials are analyzed by the finite element method (FEM). The size-effect phenomenon observed in nano-sized structures is considered by strain-gradients in the constitutive equation for the high-order stress tensor. The variational principle is applied for derivation of coupled thermo-electro-mechanical governing equations. The thermo-electrical fields are fully coupled and the mechanical fields are coupled with temperature only. The mixed FEM with the C0 continuous interpolation for displacements and strains is developed. A collocation approach for kinematic constraints between strains and displacements is applied. The influence of the size effect on variation of crack opening displacements is demonstrated by some numerical examples.

Eigenvalues with respect to a weight for general boundary value problems on networks

In this work we analyze self-adjoint boundary value problems on networks for Schrödinger operators, in which a part of the boundary with a Neumann condition is always considered. We first characterize when the energy is positive semi-definite on the space of functions satisfying the null boundary conditions. To do this, the fundamental tools are the Doob transform and the discrete version of the trace function. Then, we raise eigenvalue problems with respect to a weight for general boundary value problems and we prove the discrete version of the Mercer Theorem. Finally, we apply the obtained results to a Dirichlet-Robin boundary value problem on a star-like network.