Biharmonic Functions Research Articles

The dynamical problem for the infinite elastic layer with a cylindrical cavity

The wave field of an infinite elastic layer weakened by a cylindrical cavity is constructed. The ideal contact conditions are given on the upper layer’s face and bottom face is rigidly fixed. The normal dynamic tensile load is applied to a cylindrical cavity’s surface at the initial moment of time. The Laplace and finite sin- and cos- Fourier integral transforms are applied successively directly to axisymmetric equations of motion and to the boundary conditions, on the contrary to the traditional approaches, when integral transforms are applied to solutions’ representation through harmonic and biharmonic functions. This operation leads to a one-dimensional vector inhomogeneous boundary value problem with respect to unknown displacements’ transformations. The problem is solved using a matrix differential calculus, which leads to an integral equation solved with a method of orthogonal polynomials. The field of initial displacements is derived after application of inverse integral transforms. The case of the steady-state oscillations was investigated. The normal stress on the rigidly fixed face of the elastic layer is constructed and investigated depending on the mechanical and dynamic parameters. Formulas for determining the normal stress for large values ​​of natural vibration frequencies were constructed.

Explicit solution of the Dirichlet boundary value problem of elasticity for porous infinite strip

The object of the present paper is to consider the Dirichlet boundary value problem of the coupled linear quasi-static theory of elasticity for porous isotropic elastic infinite strip. The general representations of a regular solutions of a system of considered equations for a homogeneous isotropic medium are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. Using the Fourier method, the Dirichlet BVP is solved effectively (in quadratures) for the infinite strip.

A Numerical Method for Boundary Value Problems for a Homogeneous Equation with the Squared Laplace Operator with the Use of Interpolating Wavelets

We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces $$h_{\infty}(K)$$ of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.

Research in the Field of Geometric Analysis at Volgograd State University

This article discusses the main directions of research in geometric analysis, which were conducted and are being carried out by the scientific mathematical school of Volgograd State University. The results of the founder of the scientific school, Doctor of Physics and Mathematics, Professor Vladimir Mikhailovich Miklyukov and his students are summarized. These results concern the solution of a number of problems in the field of quasiconformal flat mappings and mappings with bounded distortion of surfaces and Riemannian manifolds, the theory of minimal surfaces and surfaces of prescribed mean curvature, surfaces of zero mean curvature in Lorentz spaces, as well as problems associated with the study of the stability of such surfaces. In addition, the results of the study of various classes of triangulations — an object that appears at the junction of research in the field of geometric analysis and computational mathematics — are noted. Besides, this review discusses papers that use the Fourier decomposition method for solutions of the Laplace — Beltrami equations and the stationary Schr¨odinger equation with respect to the eigenfunctions of the corresponding boundary value problems. In particular, the authors give the results on finding capacitive characteristics that allowed for the first time to formulate and prove the criteria for the fulfillment of various theorems of Liouville type and the solvability of boundary value problems on model and quasimodel Riemannian manifolds. The role of the equivalent function method is also indicated in the study of such problems on manifolds of a fairly general form. In addition to this, the article gives an overview of the results concerning estimates of calculating error integral functionals and convergence of piecewise polynomial solutions of nonlinear variational type equations: minimal surface equations, equilibrium equations capillary surface and equations of biharmonic functions.

A Minimization Approach for Constructing Generalized Barycentric Coordinates and Its Computation

We are interested in constructing more generalized barycentric coordinates (GBC) over arbitrary polygon in the 2D setting. We propose a constrained minimization over the class of infinitely differentiable functions subject to the GBC constraints of preserving linear functions and the non-negativity condition. It includes the harmonic GBC, biharmonic GBC, maximum entropy GBC, local barycentric coordinates as special cases. We mainly show that the constrained minimization has a unique solution when the minimizing functional is strictly convex. Next we use a $$C^r$$ smoothness spline function space $$S^r_d(\triangle )$$ with $$r\ge 2$$ over a triangulation $$\triangle $$ of a polygon of interest in $$\mathbb {R}^2$$ to approximate the minimizer. One advantage of using smooth splines is that derivaties, e.g. the mean curvature and/or Gaussian curvature of spline GBC functions can be calculated. As the minimization restricted to the spline space $$S^r_d$$ certainly has a unique minimizer, we use the standard projected gradient descent (PGD) method to approximate the spline minimizer. To find the projection of each iteration, we shall explain an alternating projection algorithm (APA). A convergence of the APA and the convergence of the PGD with the APA will be presented. As an example of this approach, a new kind of biharmonic GBC functions which preserve the nonnegativity is constructed. Finally, we have implemented the PGD method based on bivariate splines of arbitrary degree d and arbitrary smoothness r over arbitrary triangulation as long as $$d>>r$$ . The surfaces of many new GBC’s will be shown. Some standard GBC applications will be demonstrated.

Generalized Clifford Algebras Associated to Certain Partial Differential Equations

In the classical Clifford analysis the Laplace operator is factorized by the Cauchy–Riemann operator $$\Delta =\overline{D}D$$ . The consequence is all components of a monogenic function are harmonic functions. In more general situation, suppose that we are given a linear partial differential equation. We wish to find a generalized Clifford algebra such that all components of a generalized monogenic function taking values in that algebra satisfy the given partial differential equation. For instance, in order to represent biharmonic functions in the theory of plane elasticity, in 1934 L. Sobrero introduced a hypercomplex algebra which is generated by the imaginary e with the rule $$(e^2+1)^2=0$$ . In this paper we introduce an extension of the idea of L. Sobrero to construct some generalized Clifford algebras in order to cover more partial differential equations in higher dimensions.

First Principles Derivation of Displacement and Stress Function for Three-Dimensional Elastostatic Problems, and Application to the Flexural Analysis of Thick Circular Plates

In this study, stress and displacement functions of the three-dimensional theory of elasticity for homogeneous isotropic bodies are derived from first principles from the differential equations of equilibrium, the generalized stress – strain laws and the geometric relations of strain and displacement. It is found that the stress and displacement functions must be biharmonic functions. The derived functions are used to solve the elasticity problem of finding stresses and displacement fields in a thick circular plate with clamped edges for the case of uniformly distributed transverse load over the plate domain. Superposition of second to sixth order Legendre polynomials which are biharmonic functions are used in the thick circular plate problem as the stress function with the unknown constants as the parameters to be determined. Use of the stresses and displacement fields derived in terms of the stress and displacement function yielded the stress fields and displacement fields in terms of the unknown constants of the biharmonic stress function. Enforcement of the boundary conditions yielded the unknown constants, leading to a complete determination of the stress and displacement function for the stress fields and the displacement fields. The solutions obtained are comparable to solutions in the technical literature.

Fractional thermoelasticity problem for an infinite solid with a penny-shaped crack under prescribed heat flux across its surfaces.

The time-nonlocal generalization of the Fourier law with the 'long-tail' power kernel can be interpreted in terms of fractional calculus and leads to the time-fractional heat conduction equation with the Caputo derivative. The theory of thermal stresses based on this equation was proposed by the first author (J. Therm. Stresses 28, 83-102, 2005 (doi:10.1080/014957390523741)). In the present paper, the fractional heat conduction equation is solved for an infinite solid with a penny-shaped crack in the case of axial symmetry under the prescribed heat flux loading at its surfaces. The Laplace, Hankel and cos-Fourier integral transforms are used. The solution for temperature is obtained in the form of integral with integrands being the generalized Mittag-Leffler function in two parameters. The associated thermoelasticity problem is solved using the displacement potential and Love's biharmonic function. To calculate the additional stress field which allows satisfying the boundary conditions at the crack surfaces, the dual integral equation is solved. The thermal stress field is calculated, and the stress intensity factor is presented for different values of the order of the Caputo time-fractional derivative. A graphical representation of numerical results is given. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Thermal Stressed State of a Half Space with Free or Rigidly, Smoothly, or Flexibly Fastened Boundary Under the Conditions of Heat Insulation in a Domain Located in the Plane Parallel to the Boundary

We construct the Green functions of the stationary problems of heat conduction and thermoelasticity for half spaces with free or rigidly, smoothly, or flexibly fastened boundaries subjected to the action of heat dipoles in the case where the temperature of the boundaries is kept equal to zero or the conditions of heat insulation are imposed on these boundaries. In this case, harmonic double-layer potentials are used for the solution of the problem of heat conduction, as well as thermoelastic potentials of displacements in the infinite space with two heat dipoles located symmetrically about the boundary. To satisfy the boundary conditions on the boundary, we construct the Love biharmonic functions. We present explicit expressions for temperature, displacements, and stresses, which can be used for the determination of the thermoelastic state of the half space caused by the perturbation of a given heat flux by a thin heatinsulating inclusion parallel to the boundary.

On plane Λ-fractional linear elasticity theory

ABSTRACT Non-local plane elasticity problems are discussed in the context of Λ-fractional linear elasticity theory. Adapting the Λ-fractional derivative along with the Λ-fractional space, where geometry and mechanics are valid in the conventional way, non-local plane elasticity problems are solved with the help of biharmonic functions. Then, the results are transferred into the initial plane. Applications are presented to homogeneous and the fractional beam bending problem.

Замкнутость бигармонической системы базисных потенциалов

The shift systems of the fundamental solution of biharmonic equation are considered. The shift sequence belongs to the complement of the bounded single-bond domain $Q$, so that all system functions satisfy the biharmonic equation in $Q$. The space of biharmonic functions $G_{2}(Q)$ is introduced as a closure in the $L_{2}(Q)$ norm of the linear shell of all kinds of shifts of the fundamental solution of biharmonic equation. The following problem is considered: what conditions the countable sequence of shifts must have so that the linear shell closure matches the entire $G _ {2} (Q) $ space, or, in other words, when is the considered system complete in $G _ {2} (Q)$? The problem for the harmonic case was considered by V.G. Lezhnev, he introduced a sufficient basis condition and proved the completeness and linear independence of shift systems of the fundamental solution of Laplace's equation.The work generalizes the basis condition to the biharmonic case and proves the closedness and linear independence of biharmonic system. The basis condition is related to the singularity condition of biharmonic functions. The proof relies on the property of continuity of potentials with a biharmonic nucleus and on the Rikier's boundary value problem solution.

New explicit solutions to the p-Laplace equation based on isoparametric foliations

In contrast to an infinite family of explicit examples of two-dimensional p-harmonic functions obtained by G. Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of p-harmonic and biharmonic functions. Remarkably, for some distinguished values of p and the ambient dimension n this yields first examples of rational and algebraic p-harmonic functions. Moreover, we show that there are no p-harmonic polynomials of the isoparametric type. This supports a negative answer to a question proposed in 1980 by J. Lewis.

An exact solution of the dynamical problem for the infinite elastic layer with a cylindrical cavity

The wave field of an infinite elastic layer weakened by a cylindrical cavity is constructed in this paper. The ideal contact conditions are given on the upper and bottom faces of the layer. The normal dynamic tensile load is applied to a cylindrical cavity's surface at the initial moment of time. The Laplace and finite $sin-$ and $cos-$ Fourier integral transforms are applied successively directly to axisymmetric equations of motion and to the boundary conditions, on the contrary to the traditional approaches, when integral transforms are applied to solutions' representation through harmonic and biharmonic functions. This operation leads to a one-dimensional vector homogeneous boundary value problem with respect to unknown transformations of displacements. The problem is solved using matrix differential calculus. The field of initial displacements is derived after application of inverse integral transforms. The case of the steady-state oscillations was investigated. The normal stress on the faces of the elastic layer are constructed and investigated depending on the mechanical and dynamic parameters.

New Biharmonic Functions on the Compact Lie Groups mathbf{SO}(n), mathbf{SU}(n), mathbf{Sp}(n)

We develop a new scheme for the construction of explicit complex-valued proper biharmonic functions on Riemannian Lie groups. We exploit this and manufacture many infinite series of uncountable families of new solutions on the special unitary group mathbf{SU}(n). We then show that the special orthogonal group mathbf{SO}(n) and the quaternionic unitary group mathbf{Sp}(n) fall into the scheme. As a by-product we obtain new harmonic morphisms on these groups. All the constructed maps are defined on open and dense subsets of the corresponding spaces.

Bi-Laplacians on graphs and networks

We study the differential operator $A=\frac{d^4}{dx^4}$ acting on a connected network $\mathcal{G}$ along with $\mathcal L^2$, the square of the discrete Laplacian acting on a connected discrete graph $\mathsf{G}$. For both operators we discuss well-posedness of the associated {linear} parabolic problems \[ \frac{\partial u}{\partial t}=-Au,\qquad\frac{df}{dt}=-\mathcal L^2 f, \] on $L^p(\mathcal{G})$ or $\ell^p(\mathsf{V})$, respectively, for $1\leq p\leq\infty$. In view of the well-known lack of parabolic maximum principle for all elliptic differential operators of order $2N$ for $N>1$, our most surprising finding is that, after some transient time, the parabolic equations driven by $-A$ may display Markovian features, depending on the imposed transmission conditions in the vertices. Analogous results seem to be unknown in the case of general domains and even bounded intervals. Our analysis is based on a detailed study of bi-harmonic functions complemented by simple combinatorial arguments. We elaborate on analogous issues for the discrete bi-Laplacian; a characterization of complete graphs in terms of the Markovian property of the semigroup generated by $-\mathcal L^2$ is also presented.

The Approximation of the Fourth-Order Partial Differential Equations in the Class of the Piecewise Polynomial Functions on the Triangular Grid

The present work determines the deviation of the piecewise cubic almost-solution of the biharmonic equation and derives the general formula (3) for its calculation. Based on this concept, we obtained an approximation of the equation. A number of numerical calculations were carried out in order to confirm the obtained formula (see pictures 1 and 2) experimentally. In general, for all selected biharmonic functions, the deviation value "ΔΔ(𝑢) turned out to be, as expected, rather small even with a relatively small number of triangulation nodes. On average, with 25 ≤ 𝑁 ≤ 35, the absolute error was about 0, 0001, which gives an approximately asymptotic estimate of 𝑂(ℎ2) when the partitioning step is ℎ → 0.

On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

The ∞‐Bilaplacian is a third‐order fully nonlinear PDE given by urn:x-wiley:0749159X:media:num22295:num22295-math-0001In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p‐Bilaplacian urn:x-wiley:0749159X:media:num22295:num22295-math-0002We prove convergence of the numerical solution to the weak solution of and show that we are able to pass to the limit p → ∞. We perform various tests aimed at understanding the nature of solutions of and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of ‐solutions.

Avkhadiev–Becker Type Univalence Conditions for Biharmonic Mappings

In this paper we consider complex-valued biharmonic functions that are locally univalent. We construct families of biharmonic univalent mappings of the unite disc similar to Avkhadiev- Becker type conditions for analytic functions. Also, we investigate the case where biharmonic functions are defined on the exterior of the unit disc. In this case we obtain three analogs of Avkhadiev–Becker type conditions of univalence.

Biharmonic functions on spheres and hyperbolic spaces

We construct new explicit proper r-harmonic functions on the standard n-dimensional hyperbolic spaces Hn and spheres Sn for any r≥1 and n≥2.

A note on biharmonic functions on the Thurston geometries

We construct new explicit proper biharmonic functions on the $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $H^2\times\rn$ and $S^2\times\rn$.