Spherical Harmonics Expansion of Fundamental Solutions and Their Derivatives for Homogeneous Elliptic Operators

2017 ◽  
Vol 08 (03n04) ◽  
pp. 1740006
Author(s):  
Vincenzo Gulizzi ◽  
Ivano Benedetti ◽  
Alberto Milazzo
Keyword(s):  
Differential Operator ◽  
Fundamental Solution ◽  
Spherical Harmonics ◽  
Homogeneous Function ◽  
Three Dimensional ◽  
Partial Differential Operator ◽  
Fundamental Solutions ◽  
Numerical Implementation ◽  
Isotropic Elasticity ◽  
Spherical Harmonics Expansion

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

scholarly journals Directional Method of Fundamental Solutions for Three-dimensional Laplace Equation

2017 ◽  
Vol 9 (6) ◽  
pp. 112 ◽  
Author(s):  
Chein-Shan Liu ◽  
Zhuojia Fu ◽  
Chung-Lun Kuo
Keyword(s):  
Fundamental Solution ◽  
Laplace Equation ◽  
Logarithmic Singularity ◽  
Three Dimensional ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Simple Extension ◽  
Two Dimensional ◽  
Problem Domain ◽  
Interior Domain

We propose a simple extension of the two-dimensional method of fundamental solutions (MFS) to a two-dimensional like MFS for the numerical solution of the three-dimensional Laplace equation in an arbitrary interior domain. In the directional MFS (DMFS) the directors are planar orientations, which can take the geometric anisotropy of the problem domain into account, and more importantly the order of the logarithmic singularity with $\ln R$ of the new fundamental solution is reduced than that of the conventional three-dimensional fundamental solution with singularity $1/r$. Some numerical examples are used to validate the performance of the DMFS.

REGULARIZATION TECHNIQUES FOR THE METHOD OF FUNDAMENTAL SOLUTIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341004 ◽  
Author(s):  
CSABA GÁSPÁR
Keyword(s):  
Fundamental Solution ◽  
Solution Process ◽  
Partial Differential Operator ◽  
Fundamental Solutions ◽  
Higher Order ◽  
Method Of Fundamental Solutions ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Approximation Theorems ◽  
Regularization Techniques

A special regularization method based on higher-order partial differential equations is presented. Instead of using the fundamental solution of the original partial differential operator with source points located outside of the domain, the original second-order partial differential equation is approximated by a higher-order one, the fundamental solution of which is continuous at the origin. This allows the use of the method of fundamental solutions (MFS) for the approximate problem. Due to the continuity of the modified operator, the source points and the boundary collocation points are allowed to coincide, which makes the solution process simpler. This regularization technique is generalized to various problems and combined with the extremely efficient quadtree-based multigrid methods. Approximation theorems and numerical experiences are also presented.

A Fundamental Solution for a Nonelliptic Partial Differential Operator

1979 ◽  
Vol 31 (5) ◽  
pp. 1107-1120 ◽  
Author(s):  
Peter C. Greiner
Keyword(s):  
Vector Field ◽  
Differential Operator ◽  
Fundamental Solution ◽  
Half Plane ◽  
Partial Differential Operator ◽  
Holomorphic Vector Field ◽  
Partial Differential ◽  
Upper Half Plane ◽  
Cauchy Riemann ◽  
Image Position

Let(1)and set(2)Here . Z is the “unique” (modulo multiplication by nonzero functions) holomorphic vector-field which is tangent to the boundary of the “degenerate generalized upper half-plane”(3)In our terminology t = Re z1. We note that ℒ is nowhere elliptic. To put it into context, ℒ is of the type □b, i.e. operators like ℒ occur in the study of the boundary Cauchy-Riemann complex. For more information concerning this connection the reader should consult [1] and [2].

scholarly journals A Fundamental Solution of N. Zeilon's Operator

2000 ◽  
Vol 86 (2) ◽  
pp. 273 ◽  
Author(s):  
Peter Wagner
Keyword(s):  
Differential Operator ◽  
Fundamental Solution ◽  
Partial Differential Operator ◽  
Elliptic Integrals ◽  
Partial Differential

In this paper, we resume earlier work of N. Zeilon and of J. Fehrman and derive an explicit re- presentation by elliptic integrals of a fundamental solution of the partial differential operator $\partial_1^3+\partial_2^3+\partial_3^3$.

scholarly journals Codomains for the Cauchy-Riemann and Laplace operators inℝ2

2008 ◽  
Vol 6 (1) ◽  
pp. 71-87
Author(s):  
Lloyd Edgar S. Moyo
Keyword(s):  
Differential Operator ◽  
Fundamental Solution ◽  
Laplace Operator ◽  
Constant Coefficient ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
Nonzero Constant ◽  
Space Of Distributions ◽  
Cauchy Riemann

A codomain for a nonzero constant-coefficient linear partial differential operatorP(∂)with fundamental solutionEis a space of distributionsTfor which it is possible to define the convolutionE*Tand thus solving the equationP(∂)S=T. We identify codomains for the Cauchy-Riemann operator inℝ2and Laplace operator inℝ2. The convolution is understood in the sense of theS′-convolution.

A fundamental solution for a nonelliptic partial differential operator, II

2018 ◽  
Vol 16 (03) ◽  
pp. 407-433
Author(s):  
Peter Greiner ◽  
Yutian Li
Keyword(s):  
Vector Field ◽  
Differential Operator ◽  
Fundamental Solution ◽  
Half Plane ◽  
Special Functions ◽  
Partial Differential Operator ◽  
Principal Result ◽  
Tangential Vector ◽  
Tangential Vector Field ◽  
Upper Half Plane

Let [Formula: see text] denote the holomorphic tangential vector field to the generalized upper-half plane [Formula: see text]. In our terminology, [Formula: see text]. Consider the [Formula: see text] operator on the boundary of [Formula: see text], [Formula: see text]; note that [Formula: see text] is nowhere elliptic, but it is subelliptic with step three. The principal result of this paper is the derivation of an explicit fundamental solution [Formula: see text] to [Formula: see text]. Our approach is based on special functions and their properties.

scholarly journals Three-dimensional fundamental solution in transversely isotropic thermoelastic diffusion material

2012 ◽  
Vol 39 (2) ◽  
pp. 165-184 ◽  
Author(s):  
Rajneesh Kumar ◽  
Vijay Chawla
Keyword(s):  
Fundamental Solution ◽  
Transverse Isotropy ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Fundamental Solutions ◽  
Thermoelastic Diffusion ◽  
Diffusion Media ◽  
Stress Components ◽  
Steady State Problem ◽  
Special Case

The aim of the present investigation is to study the fundamental solution for three dimensional problem in transversely isotropic thermoelastic diffusion medium. After applying the dimensionless quantities, two displacement functions are introduced to simplify the basic threedimensional equations of thermoelastic diffusion with transverse isotropy for the steady state problem. Using the operator theory, we have derived the general expression for components of displacement, mass concentration, temperature distribution and stress components. On the basis of general solution, three dimensional fundamental solutions for a point heat source in an infinite thermoelastic diffusion media is obtained by introducing four new harmonic functions. From the present investigation, a special case of interest is also deduced to depict the effect of diffusion.

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