Inhomogeneous Biharmonic Equation Research Articles

Spherical Spline Solutions of an Inhomogeneous Biharmonic Equation

Spherical Spline Solutions of an Inhomogeneous Biharmonic Equation

A Volterra edge dislocation problem in second-order elasticity theory

A displacement function technique has been developed to study plane strain problems in second-order elasticity theory for incompressible elastic materials. The formulation culminates in the development of a single inhomogeneous biharmonic equation of the form [Formula: see text] for the second-order displacement function [Formula: see text]. In the second-order, the isotropic stress associated with constitutive behaviour is governed by an inhomogeneous harmonic equation of the form [Formula: see text]. Both functions [Formula: see text] and [Formula: see text] depend only on the solution to the analogous problem in classical elasticity. While there are similarities to the Airy stress function, the displacement function approach enables the evaluation of the first- and second-order displacement fields purely through its derivatives, thus eliminating the introduction of arbitrary rigid body terms normally associated with formulations where the strains derived from a stress function approach need to be integrated. The displacement function approach is applied to develop a second-order elasticity solution to the problem of a Volterra edge dislocation.

Mathematical Modeling of Elastically Deformed States of Thin Isotropic Plates Using Chebyshev Polynomials

Abstract. In this paper a method for solving an inhomogeneous biharmonic equation while modeling elastically deformed states of thin isotropic rectangular plates using a system of orthogonal Chebyshev polynomials of the first kind is proposed. The method is based on representation of a solution to the initial biharmonic equation as a finite sum of Chebyshev series by each independent variable in combination with matrix transformations and properties of Chebyshev polynomials. The problem is examined for the case when a transverse load acts on the plate, and the hinge fastening along the edges of the plate is taken as boundary conditions. Using the extremes and zeros of Chebyshev polynomials of the first kind as collocation points, the boundary value problem is reduced to a system of linear algebraic equations. Decomposition coefficients of desired function with respect to Chebyshev polynomials act as unknowns in this system. As the comparison showed, the results obtained by this method with a high degree of accuracy coincide with similar results derived using analytical approach that are given in the article. The paper also presents the results of calculations using the proposed method in the case when two opposite edges of the plate are pinched and two others are pivotally fixed. The comparison with similar results of modeling the stress-strain states of rectangular plates which are presented in the open sources is carried out.

Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

Mathematical modeling of bending of a thin orthotropic plate clamped along the contour

Within the framework of Kirchhoff’s theory, a new approach to constructing a solution to the problem of modeling the bending of a thin rectangular orthotropic plate clamped along the contour, which is under the influence of a load normally distributed over its surface, is proposed. the solution to the inhomogeneous biharmonic equation for an orthotropic plate is obtained in the form of a partial sum of a double series in Chebyshev polynomials of the first kind. To find the coefficients in this expansion, the boundary value problem is reduced by the collocation method to a system of linear algebraic equations in matrix form using the properties of these polynomials. Based on matrix and differential transformations, expressions for bending moments and shearing forces are obtained. the results of calculations of the bending of the middle surface of the plate under different loads on the plate are presented, which demonstrate the effectiveness of the proposed approach.

Some remarks on the inhomogeneous biharmonic NLS equation

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation iut+Δ2u+λ|x|−b|u|αu=0,where λ=±1 and α, b>0. In the subctritical case, we improve the global well-posedness result obtained in Guzmán and Pastor (2020) for dimensions N=5,6,7 in the Sobolev space H2(RN). The fundamental tools to establish our results are the standard Strichartz estimates related to the linear problem and the Hardy-Littlewood inequality. Results concerning the energy-critical case, that is, α=8−2bN−4 are also reported. More precisely, we show well-posedness and a stability result with initial data in the critical space Ḣ2.

Poroelastic solution of a wellbore in a swelling rock with non-hydrostatic stress field

The stress distribution around a circular borehole has been studied extensively. The existing analytical poroelastic solutions, however, often neglect the complex interactions between the solid and fluid that can significantly affect the stress solution. This is important in unconventional gas reservoirs such as coalbeds and shale formations. In order to address this limitation, this paper presents the development of a poroelastic solution that takes into account the effect of gas sorption-induced strain. The solution considers drainage of the reservoir fluid through a vertical wellbore in an isotropic, homogenous, poroelastic rock with non-hydrostatic in situ stress field. The sorption-induced shrinkage of coal is modelled using a Langmuir-type curve which relates the volumetric change to the gas pressure. The redistributed stress field around the wellbore after depletion is found by applying Biot's definition of effective stress and Airey's stress functions, which leads to a solution of the inhomogeneous Biharmonic equation. Two sets of boundary conditions were considered in order to simulate the unsupported cavity (open-hole) and supported cavity (lined-hole) cases. The implementation was verified against a numerical solution for both open-hole and lined-hole cases. A comparative study was then conducted to show the significance of the sorption-induced shrinkage in the stress distribution. Finally, a parametric study analysed the sensitivity of the solution to different poroelastic parameters. The results demonstrate that the developed solution is a useful tool that can be employed alongside complex flow models in order to conduct efficient, field-scale coupled hydro-mechanical simulations, especially when the stress-dependent permeability of the reservoir is of concern.

On the solution of the inhomogeneous biharmonic equation

On the solution of the inhomogeneous biharmonic equation

Boundary Element Method for Solving an Inhomogeneous Biharmonic Equation with a Right-Hand Side Containing the Unknown Function and Its Derivatives

Boundary Element Method for Solving an Inhomogeneous Biharmonic Equation with a Right-Hand Side Containing the Unknown Function and Its Derivatives

Green's functions of some boundary value problems for the biharmonic equation

In this paper the Green's functions for three boundary value problems for the biharmonic equation are investigated. First, an integral representation of solutions to the inhomogeneous biharmonic equation is given. Then the Green's function of the Dirichlet problem is found and an integral representation of the solution to the Dirichlet problem in terms of the Green's function is given. After that, the Green's function of the Navier problem and the integral representation of the solution to the Navier problem are presented. To study the Neumann-2 problem, the Green's function of the Neumann problem for the Poisson equation is discussed and on its basis the Green's function of the Neumann-2 problem is constructed. To illustrate the results obtained, solutions of the three considered homogeneous problems for the polynomial right-hand side of the equation are found.

Non-Euclidean model for description of residual stresses in planar deformations

We introduced the Airy stress function in the non-Euclidean model of a continuous medium, for which the Saint-Venant compatibility condition for deformations is not satisfied. For this function, an inhomogeneous biharmonic equation is obtained whose right side is determined by the incompatibility of deformations. The internal stresses was shown to be composed of elastic stresses and stresses determined by the incompatibility of deformations. The theoretical results obtained are used to analyse experimental data on the study of residual stresses.

Semi-analytical solution by Cartesian harmonic polynomials for a problem of deformation of a long, current-carrying elastic cylindrical conductor

Harmonic Cartesian polynomial and rational functions are shown to provide a simple way of obtaining a semi-analytical solution to the uncoupled, two-dimensional problem of thermomagnetoelasticity for a long, transversely isotropic, elastic cylinder carrying an axial, steady electric current. The proposed method involves the solution of a difficult inhomogeneous biharmonic equation for the stress function, and may be invariably used for general geometries of the normal cross-section of the cylinder, for various thermal and mechanical boundary conditions, and also to solve the problem of a long non-conducting cylinder in a transverse magnetic field. At all stages of the solution, there is no need to smoothen the cross-sectional contour in the presence of singular points, as for other methods. Results are provided for the elliptic and rectangular normal cross-sections under the Dirichlet thermal condition and prescribed uniform normal boundary displacement. Quantities of practical interest are represented graphically and discussed. The results may be of interest in determining the deformations occurring in current-carrying metallic parts of structures and machines, and in the central regions of busbars in electric power plants.

Inequality Estimates of an Inhomogeneous Semilinear Biharmonic Equation in Entire Space

In this paper, we consider the inequality estimates of the positive solutions for the inhomogeneous biharmonic equation (*)$$-\Delta^{2} u+u^{p}+f(x)=0 \; \rm{in} \; \mathbb{R}^{n},$$ where Δ2 is the biharmonic operator, p > 1, n ≥ 5 and 0 ≢ f ∈ C(ℝn) is a given nonnegative function. We obtain different inequality estimates of Eq.(*), with which the necessary conditions of existence on f and p are also established.

Regularization of an initial inverse problem for a biharmonic equation

In this paper, we consider the problem of finding the initial distribution for the linear inhomogeneous biharmonic equation. The problem is severely ill-posed in the sense of Hadamard. In order to obtain a stable numerical solution, we propose two regularization methods to solve the problem. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in mathcal{L}^{2} uniformly with respect to the space coordinate under some a priori assumptions on the solution. Finally, in order to increase the significance of the study, numerical results are presented and discussed illustrating the theoretical findings in terms of accuracy and stability.

The radial basis function-differential quadrature method for elliptic problems in annular domains

Abstract We employ a radial basis function (RBF) - differential quadrature (DQ) method for the numerical solution of elliptic boundary value problems in annular domains. With an appropriate selection of collocation points, for any choice of RBF, both the coefficient and right hand side matrices in the systems appearing in this discretization possess block circulant structures. These linear systems can thus be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy–Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples.

Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Let be the class of all sense‐preserving homeomorphic self‐mappings of . The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk with associated boundary value conditions and . Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when and are small enough.

Optimization of the boundary conditions of an inhomogeneous biharmonic equation

Mathematical model construction of complicate physical phenomenon often leads to the setting and solving problems of parameters optimal control in differential equations in partial derivatives. Chosen equation with boundary and initial conditions is usually mathematical model basis of the object, which is under analysis. Optimal control of right-hand side function in non-linear problem for inhomogeneous biharmonic has been investigated. With the help of various gradient methods the problems of parameters control in such equations are solved successfully. Herewith linear problem is solved with the potential method on every step. In this work biharmonic potentials with logarithmic singularity are under consideration. That is why parameters optimization in these problems are conducted together with elimination of their incorrectness. It should be noted that system optimal control problems, described with biharmonic equations in irregular shape region, are studied poorly. Therefore the object of investigation is one of boundary conditions for inhomogeneous biharmonic equation. The circumstance which complicated the problem was irregularity of function domain. Clearly that this problem can be solved with computational mathematics methods. Without precise solution of linear problem, it is impossible to use gradient method, build converging iterative process, and obtain precise solution of optimization problem. Algorithm for linear boundary value problem solution with boundary integral equations overcomes this problem successfully. Physical examples of numerical implementation have been presented, analysis of obtained solutions have been conducted. Their accuracy, algorithm simplicity and time spent evidence about this approach promising for practical results obtaining in plate theory and mathematical physics problems successful numerical solving

Solving a Problem of Robin Type for Biharmonic Equation

We investigate existence and uniqueness conditions for solution to one Robin type problem for inhomogeneous biharmonic equation in the unit ball. We construct polynomial solution to the problem when the boundary functions of the problems are polynomials.

On solvability of some boundary value problems for a biharmonic equation with periodic conditions

In the paper we study questions about solvability of some boundary value problems with periodic conditions for an inhomogeneous biharmonic equation. The exact conditions for solvability of the problems are found.

Kansa-RBF algorithms for elliptic problems in regular polygonal domains

We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.