Fundamental Solutions of Linear Partial Differential Operators: Theory and Practice

✍ Scribed by Norbert Ortner, Peter Wagner (auth.)

• Publisher: Springer International Publishing
• Year: 2015
• Tongue: English
• Leaves: 407
• Edition: 1
• Category: Library

✦ Synopsis

This monograph provides the theoretical foundations needed for the construction of fundamental solutions and fundamental matrices of (systems of) linear partial differential equations. Many illustrative examples also show techniques for finding such solutions in terms of integrals. Particular attention is given to developing the fundamentals of distribution theory, accompanied by calculations of fundamental solutions.

The main part of the book deals with existence theorems and uniqueness criteria, the method of parameter integration, the investigation of quasihyperbolic systems by means of Fourier and Laplace transforms, and the representation of fundamental solutions of homogeneous elliptic operators with the help of Abelian integrals.

In addition to rigorous distributional derivations and verifications of fundamental solutions, the book also shows how to construct fundamental solutions (matrices) of many physically relevant operators (systems), in elasticity, thermoelasticity, hexagonal/cubic elastodynamics, for Maxwell’s system and others.

The book mainly addresses researchers and lecturers who work with partial differential equations. However, it also offers a valuable resource for students with a solid background in vector calculus, complex analysis and functional analysis.

✦ Table of Contents

Front Matter....Pages i-xii
Distributions and Fundamental Solutions....Pages 1-117
General Principles for Fundamental Solutions....Pages 119-179
Parameter Integration....Pages 181-250
Quasihyperbolic Systems....Pages 251-331
Fundamental Matrices of Homogeneous Systems....Pages 333-369
Back Matter....Pages 371-398

✦ Subjects

Partial Differential Equations; Integral Transforms, Operational Calculus; Functional Analysis