A speculative extension of the differential operator definition to fractal via the fundamental solution

2018 ◽  
Vol 28 (11) ◽  
pp. 113105
Author(s):  
Wen Chen ◽  
Fajie Wang
Keyword(s):  
Differential Operator ◽  
Fundamental Solution ◽  
Operator Definition

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 On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations

2015 ◽  
Vol 7 (1) ◽  
pp. 101-107 ◽  
Author(s):  
M.M. Osypchuk
Keyword(s):  
Cauchy Problem ◽  
Differential Operator ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Stable Process ◽  
Pseudo Differential Operator ◽  
The Cauchy Problem ◽  
Vector Valued Function ◽  
Vector Valued

A fundamental solution for some class of pseudo-differential equations is constructed by the method based on the theory of perturbations. We consider a symmetric $\alpha$-stable process in multidimensional Euclidean space. Its generator $\mathbf{A}$ is a pseudo-differential operator whose symbol is given by $-c|\lambda|^\alpha$, were the constants $\alpha\in(1,2)$ and $c>0$ are fixed. The vector-valued operator $\mathbf{B}$ has the symbol $2ic|\lambda|^{\alpha-2}\lambda$. We construct a fundamental solution of the equation $u_t=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$ with a continuous bounded vector-valued function $a$.

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 $\partial_1^3+\partial_2^3+\partial_3^3$.

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.

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