Triharmonic Equation Research Articles

Some Asymptotic Properties of Solutions to Triharmonic Equations

Some Asymptotic Properties of Solutions to Triharmonic Equations

Infinitely many solutions for a p(x)-triharmonic equation with Navier boundary conditions

Abstract In this work, we will study the existence of an infinity of solutions of a Navier problem governed by the p ⁢ ( x ) {p(x)} -triharmonic operator using the theory of Ljusternick–Shrilemann and the theory of the variable exponent Sobolev spaces.

ДЕЯКІ ПРЕДСТАВЛЕННЯ ТРИГАРМОНІЙНИХ ФУНКЦІЙ

In this paper, a range of results have been obtained that enable one to consider the theory of game dynamics problems as an environment for constructing important mathematical objects. Namely, the triharmonic equation is integrated in the Cartesian coordinates with specially selected boundary conditions. The triharmonic Poisson integral for the upper half-plane, which belongs to the class of positive operators, is constructed. The functional dependence of the triharmonic operator on periodic functions is considered, and an integral with a delta-shaped kernel is obtained, which can be decomposed into three constant-sign fractions. The analysis of the asymptotic behavior of the triharmonic kernel shows the consistency of the obtained results with the previously known results. Keywords: triharmonic equation, upper half-plane, Fourier transform, Fourier series, positive operator.

ДЕЯКІ АСИМПТОТИЧНІ ВЛАСТИВОСТІ РОЗВ’ЯЗКІВ ТРИГАРМОНІЙНИХ РІВНЯНЬ

The author considers the optimization problem for the triharmonic equation in the presence of specific boundary conditions. As a result, the triharmonic Poisson integral was constructed in Cartesian coordinates for the upper half-plane. The asymptotic properties of this operator on Lipschitz classes in a uniform metric were studied. An exact equality was found for the upper bound of the deviation of the Lipschitz class functions from the triharmonic Poisson integral defined in Cartesian coordinates for the upper half-plane in the metric space. The results obtained in the article demonstrate the connection between the methods of approximation theory and the principles of optimal decision theory. Keywords: optimization problem, class of Lipschitz functions, uniform metric, triharmonic Poisson integral.

Nine-point compact sixth-order approximation for two-dimensional nonlinear elliptic partial differential equations: Application to bi- and tri-harmonic boundary value problems

Nine point sixth order compact numerical approximations are suggested to solve 2D nonlinear elliptic partial differential equations (NLEPDEs) and for the estimation of normal derivatives on a uniform rectangular grid subject to Dirichlet boundary conditions. We deliberate error analysis and reveal that, under specific conditions, our method converges to the sixth order. In addition, we extend our technique to vector form in order to solve the system of NLEPDEs. In application, we discuss nine-point compact sixth order approximations for bi- and tri-harmonic elliptic boundary value problems. Numerical experiments are carried out on several benchmark problems including bi- and tri-harmonic equations, and verified the sixth order convergence of the proposed methods.

A decoupled finite element method for the triharmonic equation

A decoupled finite element method for the triharmonic equation in two dimensions is developed in this paper. The triharmonic equation can be decoupled into two biharmonic equations and one symmetric tensor-valued Stokes equation. Two biharmonic equations are discretized by the Morley element. An H1(S)-conforming finite element in arbitrary dimension is constructed and applied for solving the tensor-valued Stokes equation. The convergence analysis is presented for the resulting discrete method. Numerical results are provided to verify the theoretical convergence rates.

The triharmonic equation on the Heisenberg group

Abstract Consider the equation { - Δ H 3 ⁢ u = f ⁢ ( ξ , u , ∇ H ⁡ u , ∇ H 2 ⁡ u , ∇ H 3 ⁡ u , ∇ H 4 ⁡ u , ∇ H 5 ⁡ u ) in ⁢ Ω , u > 0 in ⁢ Ω , u = ∂ ∂ ⁡ ν ⁢ ( ∇ H 2 ⁡ u ) = ∂ ∂ ⁡ ν ⁢ ( ∇ H 3 ⁡ u ) = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\Delta_{H}^{3}u&\displaystyle=f(\xi,u% ,\nabla_{H}u,\nabla_{H}^{2}u,\nabla_{H}^{3}u,\nabla_{H}^{4}u,\nabla_{H}^{5}u)&% &\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=\dfrac{\partial}{\partial\nu}(\nabla_{H}^{2}u)=% \dfrac{\partial}{\partial\nu}(\nabla_{H}^{3}u)=0&&\displaystyle\phantom{}\text% {on }\partial\Omega,\end{aligned}\right. where Ω is a domain of the finite-dimensional space ℍ n {\mathbb{H}^{n}} and f is a positive and bounded function. We prove the existence of a solution for the above equation. In addition, we prove the uniqueness and the cylindrical symmetry of the solution.

РЕШЕНИЕ ЗАДАЧИ РИКЬЕ–НЕЙМАНА ДЛЯ ПОЛИГАРМОНИЧЕСКОГО УРАВНЕНИЯ В ШАРЕ

In this paper, an elementary solution for polyharmonic equations is determined and its properties are given. This elementary solution coincides with previously known elementary solutions of biharmonic and triharmonic equations. Using the elementary solution, an integral representation of the solutions of a non-homogeneous polyharmonic equation in a bounded domain with a smooth boundary is found. Based on the integral representation, the solvability of the Riquier–Neumann problem is investigated. First, the concept of the Green's function of the Riquier–Neumann problem is defined, and then the Green's function is proved. Using the integral representation of the solutions of the polyharmonic equation and the Green's function of the Riquier–Neumann problem, the integral representation of the solution of the Riquier–Neumann problem in a unit ball is found. An example of the solution of the Neumann problem for the Poisson equation with the simplest right-hand side is given, which is necessary in what follows. On the basis of the Green's function of the Riquier–Neumann problem, a theorem on the integral representation of the solution of the Riquier–Neumann boundary value problem with boundary data, the integral of which over the unit sphere vanishes, is proved. In conclusion, on the basis of the theorem, an example of calculating the solution of the Riquier–Neumann problem with boundary functions coinciding with the traces of homogeneous harmonic polynomials on a unit sphere is given.

On properties of positive solutions to nonlinear tri-harmonic and bi-harmonic equations with negative exponents

In this paper, we investigate various properties (e.g. nonexistence, asymptotic behavior, uniqueness and integral representation formula) of positive solutions to nonlinear tri-harmonic equations in [Formula: see text] ([Formula: see text]) and bi-harmonic equations in [Formula: see text] with negative exponents. Such kind of equations arise from conformal geometry.

NUMERICAL METHOD FOR SOLVING THE DIRICHLET BOUNDARY VALUE PROBLEM FOR NONLINEAR TRIHARMONIC EQUATION

In this work, we consider the Dirichlet boundary value problem for nonlinear triharmonic equation. Due to the reduction of the problem to operator equation for the pair of the right hand side function and the unknown second normal derivative of the function to be sought, we design an iterative method at both continuous and discrete levels for numerical solution of the problem. Some examples demonstrate that the numerical method is of fourth order convergence. When the right hand side function does not depend on the unknown function and its derivatives, the numerical method gives more accurate results in comparison with the results obtained by the interior method of Gudi and Neilan.

High-resolution compact numerical method for the system of 2D quasi-linear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications

In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach.

Construction of triharmonic Bézier surfaces from boundary conditions

The surface of partial differential equation (PDE surface) is a surface that satisfies the PDE with boundary conditions, which can be applied in surface modeling and construction. In this paper, the construction of tensor product Bézier surfaces of triharmonic equation from different boundary conditions is presented. The internal control points of the resulting triharmonic Bézier surface can be obtained uniquely by the given boundary condition. Some representative examples show the effectiveness of the presented method.

The Green Function of the Dirichlet Problem for the Triharmonic Equation in the Ball

An explicit representation of the Green function of the Dirichlet problem for the triharmonic equation in the unit ball of space of dimension greater than 2 is given.

Finite Morse index solutions of the H\xe9non Lane\u2013Emden equation

In this paper, we are concerned with Liouville-type theorems of the Hénon Lane–Emden triharmonic equations in whole space. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence.

Lipschitz Continuity for the Solutions of Triharmonic Equation

Let D be the unit disk in the complex plane C and denote T=∂D. Write Hom+T,∂Ω for the class of all sense-preserving homeomorphism of T onto the boundary of a C2 convex Jordan domain Ω. In this paper, five equivalent conditions for the solutions of triharmonic equations ∂z∂z¯3ω=ff∈CD¯ with Dirichlet boundary value conditions ωzz¯zz¯T=γ2∈CT,ωzz¯T=γ1∈CT and ωT=γ0∈Hom+T,∂Ω to be Lipschitz continuous are presented.

Solving the triharmonic equation over multi-patch planar domains using isogeometric analysis

We present a framework for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. Our approach is based on the construction of a globally C 2 -smooth isogeometric spline space which is used as discretization space. The generated C 2 -smooth space consists of three different types of isogeometric functions called patch, edge and vertex functions. All functions are entirely local with a small support, and numerical examples indicate that they are well-conditioned. The construction of the functions is simple and works uniformly for all multi-patch configurations. While the patch and edge functions are given by a closed form representation, the vertex functions are obtained by computing the null space of a small system of linear equations. Several examples demonstrate the potential of our approach for solving the triharmonic equation.

A Cubic H3-Nonconforming Finite Element

The lowest degree of polynomial for a finite element to solve a 2kth-order elliptic equation is k. The Morley element is such a finite element, of polynomial degree 2, for solving a fourth-order biharmonic equation. We design a cubic $$H^3$$ -nonconforming macro-element on two-dimensional triangular grids, solving a sixth-order tri-harmonic equation. We also write down explicitly the 12 basis functions on each macro-element. A convergence theory is established and verified by numerical tests.

Space of [formula omitted]-smooth geometrically continuous isogeometric functions on planar multi-patch geometries: Dimension and numerical experiments

We study the space of C2-smooth isogeometric functions on bilinearly parameterized multi-patch domains Ω⊂R2, where the graph of each isogeometric function is a multi-patch spline surface of bidegree (d,d), d∈{5,6}. The space is fully characterized by the equivalence of the C2-smoothness of an isogeometric function and the G2-smoothness of its graph surface (cf. Groisser and Peters (2015), Kapl et al. (2015)). This is the reason to call its functions C2-smooth geometrically continuous isogeometric functions.In particular, the dimension of this C2-smooth isogeometric space is investigated. The study is based on the decomposition of the space into three subspaces and is an extension of the work Kapl and Vitrih (2017) to the multi-patch case. In addition, we present an algorithm for the construction of a basis, and use the resulting globally C2-smooth functions for numerical experiments, such as performing L2 approximation and solving triharmonic equation, on bilinear multi-patch domains. The numerical results indicate optimal approximation order.

Space of [formula omitted]-smooth geometrically continuous isogeometric functions on two-patch geometries

The space of C2-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is considered. The investigation of the dimension of the spaces of biquintic and bisixtic C2-smooth geometrically continuous isogeometric functions on such domains is presented. In addition, C2-smooth isogeometric functions are constructed to be used for performing L2-approximation and for solving triharmonic equation on different two-patch geometries. The numerical results indicate optimal approximation order.

Riemann boundary value problem for triharmonic equation in higher space.

We mainly deal with the boundary value problem for triharmonic function with value in a universal Clifford algebra: Δ3[u](x) = 0, x ∈ R n∖∂Ω, u +(x) = u −(x)G(x) + g(x), x ∈ ∂Ω, (D j u)+(x) = (D j u)−(x)A j + f j(x), x ∈ ∂Ω, u(∞) = 0, where (j = 1,…, 5) ∂Ω is a Lyapunov surface in R n, D = ∑k=1 n e k(∂/∂x k) is the Dirac operator, and u(x) = ∑A e A u A(x) are unknown functions with values in a universal Clifford algebra Cl(V n,n). Under some hypotheses, it is proved that the boundary value problem has a unique solution.