Inhomogeneous Boundary Value Problem Research Articles

A Localized Collocation Method Based on Nonsingular Fundamental Solutions for Steady-State Heat Conduction Analysis in Heterogeneous Materials

This paper further develops a localized nonsingular method of fundamental solutions (LNMFS) for effectively solving heat conduction problems in functionally graded materials (FGMs). In this numerical framework, the LNMFS approach is proposed as a solution for the inhomogeneous boundary value problems, where a pseudo-spectral Chebyshev collocation scheme (CCS) is employed for the approximation of the corresponding particular solutions. In the proposed LNMFS, the artificial boundary that is present in the existing method of fundamental solutions (MFS) is eliminated. Accordingly, the singularities are replaced by the normalized area integrals of the fundamental solution over small circular disks, which encompass the source nodes that intersect with the collocation nodes. The proposed method inherits the high accuracy of the nonsingular method of fundamental solutions (NMFS). Additionally, in contrast to the NMFS, the LNMFS not only possesses the capability to achieve solutions with a notable level of accuracy in nonharmonic boundary condition but also overcomes the constraint of exclusively simulating heat conduction issues in FGMs that exhibit specific spatial variations. The feasibility and accuracy of the recently developed method are demonstrated by means of comparing it with analytical solutions for several numerical examples.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

Solvability analysis for the Boussinesq model of heat transfer under the nonlinear Robin boundary condition for the temperature.

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

The solvability of inhomogeneous boundary-value problems in Sobolev spaces

The aim of the paper is to develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be overdetermined or underdetermined. They may contain derivatives, of the unknown vector-valued function, whose integer or fractional orders exceed the order of the differential equation. Similar problems arise naturally in various applications. The theory introduces the notion of a rectangular number characteristic matrix of the problem. The index and Fredholm numbers of this matrix coincide, respectively, with the index and Fredholm numbers of the inhomogeneous boundary-value problem. Unlike the index, the Fredholm numbers (i.e., the dimensions of the problem kernel and co-kernel) are unstable even with respect to small (in the norm) finite-dimensional perturbations. We give examples in which the characteristic matrix can be explicitly found. We also prove a limit theorem for a sequence of characteristic matrices. Specifically, it follows from this theorem that the Fredholm numbers of the problems under investigation are semicontinuous in the strong operator topology. Such a property ceases to be valid in the general case.

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance’s concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop a mathematical apparatus for studying the inhomogeneous boundary value problems under consideration. It is based on using a weak solution of the boundary value problem and on the construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce initial problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to the boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance’s concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions.

DIRECT SOLUTION OF THE DYNAMICAL ELASTICITY PROBLEM FOR A QUARTER SPACE

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load is concentrated at the point on another face. The problem was solved by the direct application of the integral Laplace and Fourier transforms to the motion equations and the boundary conditions. This operation leads to the one-dimensional vector inhomogeneous boundary value problem with respect to unknown displacement’s transformants. The problem was solved using the matrix differential calculus. A fundamental matrix and a decreasing solution to the corresponding homogenous matrix equation were constructed with a basic residue theorem. A singular integral equation was obtained in the process by satisfying unrealized boundary condition. Weakly convergent part of the equation was summed up. The behavior of the unknown function had been analyzed based on its mechanical sense. The form of unknown function was expressed as a series on Laguerre polynomials. The original displacements’ field was found after an application the inverse integral transforms.

The Riemann problem in a half-plane for generalized analytic functions with a supersingular point on the contour of the boundary condition

In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on the real axis for a generalized equation Cauchy–Riemann with supersingular coefficients. To solve the problem, it was necessary to derive a structural formula for the general solution of this equation and to investigate the solvability of the Riemann boundary value problem of the theory of analytic functions with an infinite index due to the power-order vorticity point.

Analysis of Inhomogeneous Boundary Value Problems for Generalized Boussinesq Model of Mass Transfer

Analysis of Inhomogeneous Boundary Value Problems for Generalized Boussinesq Model of Mass Transfer

A large strain anisotropic ductile damage model — Effective driving forces and gradient-enhancement of damage vs. plasticity

A novel anisotropic ductile damage model at finite strains is proposed in this work. The model is formulated based on the fictitious configuration concept and includes deformation induced evolution of anisotropic damage properties. As a key contribution, different effective driving forces for both, plasticity and damage are elaborated in the context of a so-called two-surface formulation, i.e. the consideration of two admissible (but not independent) domains — one related to plastic flow and one related to damage evolution. The investigations include the introduction of effective stresses driving plastic flow on the one hand, and effective driving forces related to the evolution of anisotropic damage on the other. In addition and as an alternative to the introduction of effective driving forces, contributions influencing the respective threshold values related to plastic flow and damage evolution are proposed. Specification of the underlying constitutive equations together with the introduction of a suitable algorithmic implementation allows the model to predict experimentally observed material behaviour. Moreover, fundamental properties of the model are discussed by means of non-proportional loading paths generated by combinations of loading under uniaxial tension and simple shear. In view of the simulation of inhomogeneous boundary value problems, the implementation of the model into a finite element formulation is discussed. Since softening, respectively material degradation and necking phenomena, may results in mesh-dependent simulation results, an appropriate regularisation must be included in the model. In this work, gradient-enhancements in the form of a so-called micromorphic approach are included. In particular, gradients of scalar-valued hardening variables related to plasticity and damage are accounted for. A detailed analysis of the necessity of these gradient-enhancements of both, damage and plasticity is carried out based on the proposed multi-surface anisotropic ductile damage model. It turns out that gradient-enhanced regularisation of the damage contribution alone is not sufficient to regularise the problem (and vice versa, i.e. gradient-enhanced regularisation of the plasticity contribution alone is not sufficient as well) but contributions related to both phenomena, i.e. damage and plasticity, need to be regularised.

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.

On the Effect of Irregularity of the Domain Boundary on the Solution of a Boundary Value Problem for the Laplace Equation

We consider an inhomogeneous boundary value problem with mixed boundary conditions for the Laplace equation in a domain representing a perturbationof a rectangle where one of its sides is replaced by some curve of minimal smoothness. An estimate is obtained for the difference between the solutions of the perturbed and unperturbed problems in the norm of the Sobolev space on their common domain.

Linear parabolic equation with Dirichlet white noise boundary conditions

We study inhomogeneous Dirichlet boundary value problems associated to a linear parabolic equation dudt=Au with strongly elliptic operator A on bounded and unbounded domains with white noise boundary data. Our main assumption is that the heat kernel of the corresponding homogeneous problem enjoys the Gaussian type estimates taking into account the distance to the boundary. Under mild assumptions about the domain, we show that A generates a C0-semigroup in weighted Lp-spaces where the weight is an appropriate power of the distance to the boundary. We also prove some smoothing properties and exponential stability of the semigroup. Finally, we reformulate the Cauchy-Dirichlet problem with white noise boundary data as an evolution equation in the weighted space and prove the existence of Markovian solutions and invariant measures.

Asymptotic estimations of the solution for a singularly perturbed equation with unlimited boundary conditions

The paper studies a two-point boundary value problem with unlimited boundary conditions for a linear singularly perturbed differential equation. Asymptotic estimates are given for a linearly independent system of solutions of a homogeneous perturbed equation. Auxiliary, so-called boundary functions, the Cauchy function are defined. For sufficiently small values of the parameter, estimates for the Cauchy function and boundary functions are found. An algorithm for constructing the desired solution of the boundary value problem has been developed. A theorem on the solvability of a solution to a boundary value problem is proved. For sufficiently small values of the parameter, an asymptotic estimate for the solution of the inhomogeneous boundary value problem is established. The initial conditions for the degenerate equation are determined. The formula is determined; the phenomena of the initial jump are studied.

Краевая задача Гильберта для одного класса обобщенных аналитических функций с сингулярной линией

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on the real axis for one generalized equation Cauchy — Riemann with singular coefficient. A structural formula for the general solution of this equation is obtained under constraints leading to an infinite index of the accompanying Hilbert boundary value problem for analytic functions. The study of the solvability of the latter is the basis for solving the boundary value problem for generalized analytical functions. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in terms of the characteristics of the coefficient of the boundary condition. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in the class of bounded analytic functions. This study made it possible to obtain a formula for solving the Hilbert problem already for generalized analytic functions. We also formulated the conditions under which the problem has no solution, has a single solution or an infinite number of solutions.

On an inhomogeneous slip-inflow boundary value problem for a steady viscous compressible channel flow

We prove the existence and uniqueness of a strong solution to the steady isentropic compressible Navier–Stokes equations with inflow boundary condition for density and mixed boundary conditions for the velocity around a shear flow. In particular, the Dirichlet boundary conditions on the inflow and outflow part of the boundary and the full Navier boundary conditions on the wall for the velocity field are considered. For our result, there are no restrictions on the amplitude of friction coefficients α, and only the assumption that the viscosity coefficient μ is appropriately large is required. One of the substantial ingredients of our proof is an elegant transformation induced by the flow field. With the help of this transformation, we can overcome the difficulties caused by the hyperbolicity of the continuity equation, establish the a priori estimates for a linearized system and apply the fixed point argument.

Unified Theory of Unsteady Planar Laminar Flow in the Presence of Arbitrary Pressure Gradients and Boundary Movement

A general unified solution of the plane Couette–Poiseuille–Stokes–Womersley incompressible linear fluid flow in a slit in the presence of oscillatory pressure gradients with periodic synchronous vibrating boundaries is presented. Oscillatory flow remains stable and laminar with no-slip boundary conditions applied. Eigenfunction expansion method is used to obtain the exact analytical solution of the general linear inhomogeneous boundary value problem. Fourier expansion of arbitrary harmonic pressure gradient and non-harmonic wall oscillations was used to calculate arbitrary driving of the fluid. In-house developed optimized computational fluid dynamics marching-in-time finite-volume method was used to test and verify all analytical results. A number of particular transients, steady-state and combined flows were obtained from the general analytical result. Generalized Stokes and Womersley flows were solved using the analytical computations and numerical experiments. The combined effects of periodic non-harmonic wall movements with oscillatory pressure gradients offers rich and interesting flow patterns even for a linear Newtonian fluid and may be particularly interesting for pumping-assist microfluidic devices. The main motivation for developing a unified solution of the unsteady laminar planar Couette–Stokes–Poiseuille–Womersley flow originates in a need for, but is not limited to, in-depth exploration of flow patterns in hemodynamic and microfluidic pumping applications.

A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method.

Localized singular boundary method for the simulation of large-scale problems of elliptic operators in complex geometries

In this paper, the novel localized singular boundary method (LSBM) in conjunction with the Chebyshev collocation scheme (CCS) is developed to simulate large-scale inhomogeneous problems governed by elliptic operators. By means of the Gauss–Lobatto collocation points, the CCS is employed to obtain the particular solution. With the obtained particular solution, the inhomogeneous elliptic boundary value problem can be transformed into a classical homogeneous problem which can be solved by the LSBM. In the LSBM, the whole computational domain with distributed nodes is divided into a series of overlapping circle-shaped sub-domains. For each sub-domain, the moving least square method and the conventional singular boundary method (SBM) are utilized. Considering that the matrix generated by the CCS-LSBM is sparse, this scheme is granted the tremendous potential to address large-scale problems. Numerical experiments are presented to verify the effectiveness and accuracy of the proposed scheme for problems with mixed boundary conditions, sophisticated analytical solutions, and complex computational domains, respectively.

On one approach to the solution of miscellaneous problems of the theory of elasticity

Herein, a miscellaneous contact problem of the theory of elasticity in the upper half-plane is considered. The boundary is a real semi-axis separated into four parts, on each of which the boundary conditions are set for the real or imaginary part of two desired analytical functions. Using new unknown functions, the problem is reduced to an inhomogeneous Riemann boundary value problem with a piecewise constant 2 × 2 matrix and four singular points. A differential equation of the Fuchs class with four singular points is constructed, the residue matrices of which are found by the logarithm method of the product of matrices. The single solution of the problem is represented in terms of Cauchy-type integrals when the solvability condition is met.