Nonhomogeneous Biharmonic Equation Research Articles

On Schwarz–Pick-Type Inequality and Lipschitz Continuity for Solutions to Nonhomogeneous Biharmonic Equations

The purpose of this paper is to study the Schwarz–Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: $$\Delta (\Delta f)=g$$ , where g : $$\overline{{\mathbb D}}\rightarrow {\mathbb {C}}$$ is a continuous function and $$\overline{{\mathbb D}}$$ denotes the closure of the unit disk $${\mathbb D}$$ in the complex plane $${\mathbb {C}}$$ . In fact, we establish the following properties for these solutions: First, we show that the solutions f do not always satisfy the Schwarz–Pick-type inequality $$\begin{aligned} \frac{1-|z|^2}{1-|f(z)|^2} \le C, \end{aligned}$$ where C is a constant. Second, we establish a general Schwarz–Pick-type inequality of f under certain conditions. Third, we discuss the Lipschitz continuity of f, and as applications, we get the Lipschitz continuity with respect to the distance ratio metric and the Lipschitz continuity with respect to the hyperbolic metric.

Dean Vortex for Laminar Flows in Curved Pipes with Various Cross-Sections

Curved pipes flows are encountered in different areas such as heat transfer, chaotic mixing, separation of mixtures in pipes, or blood circulation among others and exhibit a variety of characteristics depending on the ranges of Dean numbers and pipe curvatures. Studies on curved pipes flows usually consider the cases of circular, elliptical, and rectangular shapes for the cross sections of the pipe. The present work extends the availability of asymptotic analytical solutions to new ranges of cross-sectional shapes while considering fully developed steady state flows at low Dean numbers. The new shapes are given by a polar equation R* (q) satisfying the relation 1-R^(*2) (q)+dR^(*y) (q)sin(yq)=0 where d and y are parameters. The zero-order azimuthal velocity profiles for various cross-sections are given by exact analytical solutions. Solutions for the nonhomogeneous biharmonic equation for the secondary flows are given by using exact expressions for the particular solutions. Furthermore, the Fourier series decomposition of the solution is adopted to determine the integration constants that allow satisfying the non-slip boundary conditions. Solutions are presented for semi triangular (y=3) , square (y=4), and pentagonal (y=5) cross sections shapes. It is found that the velocity distribution and the Dean’s vortexes intensities are modified in function of the cross-section shapes.

Least squares weighted residual method for finding the elastic stress fields in rectangular plates under uniaxial parabolically distributed edge loads

In this work, the least squares weighted residual method is used to solve the two-dimensional (2D) elasticity problem of a rectangular plate of in-plane dimensions 2a 2b subjected to parabolic edge tensile loads applied at the two edges x = a. The problem is expressed using Beltrami–Michell stress formulation. Airy’s stress function method is applied to the stress compatibility equation, and the problem is expressed as a boundary value problem (BVP) represented by a non-homogeneous biharmonic equation. Airy’s stress functions are chosen in terms of one and three unknown parameters and coordinate functions that satisfy both the domain equations and the boundary conditions on the loaded edges. Least squares weighted residual integral formulations of the problems are solved to determine the unknown parameters and thus the Airy stress function. The normal and shear stress fields are determined for the one-parameter and the three-parameter coordinate functions. The solutions for the stress fields are found to satisfy the stress boundary conditions as well as the domain equation. The presented solutions for the Airy stress function and the normal stresses and shear stress fields are identical with solutions obtained by using variational Ritz methods, Bubnov–Galerkin methods and agree with results obtained by Timoshenko and Goodier.

BOUNDARY SCHWARZ LEMMA FOR SOLUTIONS TO NONHOMOGENEOUS BIHARMONIC EQUATIONS

We establish a boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations.

On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.

On Solvability of the Neumann Boundary Value Problem for Non-homogeneous Biharmonic Equation

In this work we investigate the Neumann boundary value problem in the unit ball for a nonhomogeneous biharmonic equation. It is well known, that even for the Poisson equation this problem does not have a solution for an arbitrary smooth right hand side and boundary functions; it follows from the Green formula, that these given functions should satisfy a condition called the solvability condition. In the present paper these solvability conditions are found in an explicit form for the natural generalization of the Neumann problem for the non-homogeneous biharmonic equation. The method used is new for these type of problems. We first reduce this problem to the Dirichlet problem, then use the Green function of the Dirichlet problem recently found by T. Sh. Kal’menov and D. Suragan

A meshless method for nonhomogeneous polyharmonic problems using method of fundamental solution coupled with quasi-interpolation technique

This paper proposes a meshless method based on coupling the method of fundamental solutions (MFS) with quasi-interpolation for the solution of nonhomogeneous polyharmonic problems. The original problems are transformed to homogeneous problems by subtracting a particular solution of the governing differential equation. The particular solution is approximated by quasi-interpolation and the corresponding homogeneous problem is solved using the MFS. By applying quasi-interpolation, problems connected with interpolation can be avoided. The error analysis and convergence study of this meshless method are given for solving the boundary value problems of nonhomogeneous harmonic and biharmonic equations. Numerical examples are also presented to show the efficiency of the method.

Two [formula omitted]-methods to generate Bézier surfaces from the boundary

Two methods to generate tensor-product Bézier surface patches from their boundary curves and with tangent conditions along them are presented. The first one is based on the tetraharmonic equation: we show the existence and uniqueness of the solution of Δ 4 x → = 0 with prescribed boundary and adjacent to the boundary control points of a n × n Bézier surface. The second one is based on the nonhomogeneous biharmonic equation Δ 2 x → = p , where p could be understood as a vectorial load adapted to the C 1 -boundary conditions.

The Dbar formalism for certain linear non-homogeneous elliptic PDEs in two dimensions

We discuss two different approaches for the analysis of the Poisson and of the non-homogeneous biharmonic equations in two dimensions. The first approach yields the solution as an integral in the complex $z$-plane (the physical plane), involving explicitly the given boundary conditions. The second approach yields an integral in the complex $k$-plane (the Fourier plane), involving the Fourier transforms of the given boundary conditions. For simple boundary value problems, such as certain problems formulated in the half complex plane, the first approach is easier. However, for more complicated problems, such as those formulated in the interior of an equilateral triangle, it appears that only the second approach can be used. Furthermore, the second approach also seems more efficient for numerical computations.

The non-homogeneous biharmonic plate equation: fundamental solutions

Real materials and structural components are often non-homogeneous, either by design or because of the physical composition and imperfections in the underlying material. Thus, analytical solutions for non-homogeneous materials under mechanical loads are of considerable interest to engineers and have widespread applications, given the prevalence of these materials in fields as diverse as aerospace, construction, electronics, etc. More precisely, those are essentially composites with carefully manufactured properties that yield desirable mechanical characteristics and properties, such as optimal arrangement of the material, minimum weight, etc. To this end, the displacement fundamental solution (or Green’s function) corresponding to a point force for the non-homogenous biharmonic equation in two dimensions are derived in this work by employing a conformal mapping technique in conjunction with the Radon transformation. These functions, besides being useful in their own right, can also be used within the context of integral equation formulations for the solution of boundary-value problems. Finally, a series of numerical examples that deal with the non-homogeneous plate on elastic foundation problem serve to illustrate the present method.

Deformation of a Superconducting Cylindrical Tube in a Perpendicular Magnetic Field

Stresses and displacements are obtained for an elastic superconductor in the form of an infinitely long, circular cylindrical tube subjected to an initially uniform magnetic field perpendicular to the tube's axis. The effect of the magnetic field penetration on the stressed state is assessed by solving the problem, firstly taking into account this penetration and, secondly, neglecting it. The problem involves the solution of a non-homogeneous biharmonic equation in two dimensions for the stress function. The special case of a circular cylinder is also given. Numerical calculations estimate the variation between the two cases for some quantities of practical interest. It is shown, in particular, that neglecting the penetration of the magnetic field in the elastic superconductor may induce a non negligible error in the appreciation of the stressed state.

Deformation of an infinite elliptic cylindrical conductor carrying a uniform axial current

The formulation and solution of a two-dimensional magnetothermoelastostatic boundary value problem is presented for an infinitely long, elliptic cylindrical conductor carrying a steady, uniformly distributed electric current. A linear dependence of the magnetic permeability tensor on strain is taken into account. The analysis involves the solution of a non-homogeneous biharmonic equation for the stress function. The purpose of this work is to evaluate expressions for the stresses inside the conductor and for the displacement on its surface. Experimentalists could use this to determine two physical constants that describe the dependence of the magnetic permeability on strain.

Quasi-static analysis of viscoplastic plates by the boundary element method

A direct boundary element method (BEM) is developed for the determination of the time-dependent inelastic deflection of plates of arbitrary planform and under arbitrary boundary conditions to general lateral loading history. The governing differential equation is the nonhomogeneous biharmonic equation for the rate of small transverse deflection. The boundary integral formulation is derived by using a combination of the BEM and finite element methodology. The plate material is modelled as elastic-viscoplastic. Numerical examples for sample problems are presented to illustrate the method and to demonstrate its merits.

A boundary element formulation for planar time-dependent inelastic deformation of plates with cutouts

A boundary element formulation for planar, time-dependent, inelastic deformation problems for bodies with cutouts is presented in this paper. A stress function description for these nonlinear problems leads to a non-homogeneous biharmonic equation for the stress function rate. An integral representation of the solution uses modified kernels which guarantee that the cutout boundary is traction free for all time. This incorporation of the effect of the cutout on the stress field into the kernels leads to an accurate determination of stresses in the near field of the cutout. Illustrative analytical examples for circular plates with circular cutouts are presented in this paper. In a companion paper [14], numerical solutions are presented for problems of finite plates with very narrow elliptic cutouts. These problems are of considerable importance in inelastic fracture.

Application of boundary integral method to elastoplastic analysis of V-notched beams

The boundary integral equation method was applied in the solution of the plane elastoplastic problems. The use of this method was illustrated by obtaining stress and strain distributions for a number of specimens with a single edge notch and subjected to pure bending. The boundary integral equation method reduced the non-homogeneous biharmonic equation to two coupled Fredholm-type integral equations. These integral equations were replaced by a system of simultaneous algebraic equations and solved numerically in conjunction with the method of successive elastic solutions.

Magneto-thermo-elastic stresses in an infinitely long cylindrical conductor carrying a uniformly distributed axial current

A formulation of two-dimensional magneto-thermo-elastostatic boundary value problems is presented in this paper. It is found that since the Lorentz pondermotive body force is conservative in this case, a nonhomogeneous biharmonic equation for the stress function can be derived. The effects of the Lorentz body force and Joule heating on the elastic deformation can thus be easily examined. As an illustrative example, stresses induced in an infinitely long cylindrical conductor carrying a uniformly distributed axial current is analysed.

Hermitian Methods for the Approximate Solution of Partial Differential Equations

General procedures, which are based on the use of Hermitian functions and employ a scheme of Hermitian collocation, are given for the approximate solution of partial differential equations. This procedure is extended, by utilizing local approximations of the function set, to a form of localized Hermitian collocation. For a particular class of Hermitian functions and subsequent local approximations, the increases in numerical accuracy over conventional Lagrangian difference techniques are illustrated with reference to the non-homogeneous biharmonic equation. Where iterative solution procedures are employed this increased numerical accuracy is achieved, at least for the example given, with a decrease in total computational effort.

A method of elastic-plastic plane stress and strain analysis

Plane boundary value problems are formulated in terms of an elastic-plastic stress function and the plastic strains. The governing relation is a non-homogeneous biharmonic equation. It is applicable to the total or deformation theories and to the incremental theories. Sample problems illustrate the practicality of the method.