Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2019 | Book

Quasilinear Equations Read chapter Read first chapter

Essentials of Partial Differential Equations

With Applications

Authors: Prof. Dr. Marin Marin, Prof. Dr. Andreas Öchsner

Publisher: Springer International Publishing

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

This book offers engineering students an introduction to the theory of partial differential equations and then guiding them through the modern problems in this subject.

Divided into two parts, in the first part readers already well-acquainted with problems from the theory of differential and integral equations gain insights into the classical notions and problems, including differential operators, characteristic surfaces, Levi functions, Green’s function, and Green’s formulas. Readers are also instructed in the extended potential theory in its three forms: the volume potential, the surface single-layer potential and the surface double-layer potential. Furthermore, the book presents the main initial boundary value problems associated with elliptic, parabolic and hyperbolic equations. The second part of the book, which is addressed first and foremost to those who are already acquainted with the notions and the results from the first part, introduces readers to modern aspects of the theory of partial differential equations.

Table of Contents

Frontmatter

Classical Solutions

Frontmatter
Chapter 1. Quasilinear Equations
Abstract
In this book, we study mainly second-order partial differential equations. Let \(\Omega \subset \mathrm{I}\!\mathrm{R}^n\) be bounded. Denote \(x=(x_{1}, x_{2},\ldots , x_{n})\in R^n\). Let \(u:\ \Omega \rightarrow \mathrm{I}\!\mathrm{R}\) be a sufficiently smooth function and denote
$$ u_{x_i}=\frac{\partial u}{\partial x_i}, i=1,2,\dots , n \quad \text {and} \quad u_{x_ix_j}=\frac{\partial ^2 u}{\partial x_i\partial x_j}, i, j=1,2,\dots , n. $$
Marin Marin, Andreas Öchsner
Chapter 2. Differential Operators of Second Order
Abstract
Let \(\Omega \) be a domain (therefore an open and convex set) in the space \(\mathrm{I}\!\mathrm{R}^n\). The most general form of a differential linear operator of second order is
$$ L u=\sum _{i=1}^n\sum _{j=1}^na_{ij}(x)\frac{\partial ^2 u}{\partial x_{i} \partial x_{j}} +\sum _{i=1}^nb_{i}(x)\frac{\partial u}{\partial x_i}+c(x)u,$$
where \(a_{ij}=a_{ji}(x)\in C^2(\Omega ),\;b_{i}=b_{i}(x)\in C^1(\Omega ),\; c=c(x)\in C^0(\Omega )\) are given functions, and \(u=u(x)\in C^2(\Omega )\) is the unknown function. As usual, we denote by x the vector with n components \(x=(x_1,x_2,\ldots , x_n)\).
Marin Marin, Andreas Öchsner
Chapter 3. The Theory of Potential
Abstract
The Newtonian potential, or the potential of volume, associated to the Laplace equation \(\Delta u=0\) is, by definition, the following improper integral:
$$\begin{aligned} U(\xi )=-\frac{1}{(n-2)\omega _n}\int _{\Omega }\frac{f(x)}{r^{n-2}}\mathrm {d}x, \end{aligned}$$
where \(r=r_{\xi x}=\left| \overline{\xi x}\right| =\sqrt{ \sum \limits _{i=1}^n\left( x_i-\xi _i\right) ^2}\), and \(\omega _n\) is the area of the unit sphere from the n-dimensional space \(\mathrm{I}\!\mathrm{R}^n\).
Marin Marin, Andreas Öchsner
Chapter 4. Boundary Value Problems for Elliptic Operators
Abstract
Consider the bounded domain \(\Omega \subset \mathrm{I}\!\mathrm{R}^n,\;n\ge 3\) with boundary \(\partial \Omega \) and the linear operator of second order
$$\begin{aligned} Lu=\sum _{i=1}^n\sum _{j=1}^na_{ij}(x)\frac{\partial ^2 u}{\partial x_i\partial x_j}+\sum _{i=1}^nb_{i}(x)\frac{\partial u}{\partial x_i}+c(x)u. \end{aligned}$$
Marin Marin, Andreas Öchsner
Chapter 5. Operational Calculus
Abstract
A useful tool in approaching ordinary differential equations and also partial differential equations is the Laplace transform, which will be studied in this paragraph.
Marin Marin, Andreas Öchsner
Chapter 6. Parabolic Equations
Abstract
The prototype of a parabolic equation is given by the equation of propagation of heat in a body. Let \(\Omega \) be a bounded domain from \({\mathrm{I}}\!{\mathrm{R}}^n\) having boundary \(\partial \Omega \) and \(\overline{\Omega }= \Omega \cup \partial \Omega \). For a constant of time \(T>0\), arbitrarily fixed, consider the interval of time \(\mathcal{T}_T\) given by
$$ \mathcal{T}_T=\{t\;:0<t\le T\},\overline{\mathcal{T}_T}=\{t\;:0\le t\le T\}. $$
Marin Marin, Andreas Öchsner
Chapter 7. Hyperbolic Equations
Abstract
The prototype of hyperbolic equations is considered to be the equation of the oscillating chord, also known as the wave equation.
Marin Marin, Andreas Öchsner

Solutions in Distributions

Frontmatter
Chapter 8. Elements of Distributions
Abstract
One of the essential reasons for which the concept of distribution was introduced is the necessity to solve a differential equation in weaker conditions of regularity.
Marin Marin, Andreas Öchsner
Chapter 9. Integral Formulas
Abstract
A partial differential equation of order k is any relation of the form
$$\begin{aligned} F\left( x, u,\frac{\partial u}{\partial x_{1}},\ldots ,\frac{\partial u}{\partial x_{n}},\ldots ,\frac{\partial ^{k}u}{\partial x_{1}^{k}},\ldots , \frac{\partial ^{k}u}{\partial x_{n}^{k}}\right) =0, \end{aligned}$$
that is, a relationship between the n-dimensional variable \(x=(x_{1}, x_{2},\ldots , x_{n})\), \(x\in \mathrm{I}\!\mathrm{R}^{n}\), the function \(u=u(x)\) and the partial derivatives of the function u of order less than or equal to k.
Marin Marin, Andreas Öchsner
Chapter 10. Partial Differential Equations of the First Order
Abstract
In this chapter, we consider partial differential equations of the form.
Marin Marin, Andreas Öchsner
Chapter 11. Linear Partial Differential Equations of Second Order
Abstract
Let us consider the differential operator
$$L(u)(x)\!=\!\sum _{i, j=1}^{n}a_{ij}(x)\frac{\partial ^{2} u}{\partial x_{i}\partial x_{j}}(x)\!+\! \sum _{j=1}^{n}a_{j}(x)\frac{\partial u}{\partial x_{j}}(x)\!+\!a_{0}(x)u(x),$$
in which the coefficients \(a_{ij}\;(i, j=\overline{1,n}),\; a_{j}\,(j=\overline{1,n})\) and \(a_{0}\) are regular functions which depend only on one variable x and which are defined on an open set \(\Omega \) from \(R^{n}\), where \(\Omega \) is not necessarily bounded.
Marin Marin, Andreas Öchsner
Chapter 12. Harmonic Functions
Abstract
We call a harmonic function on the open set \(\Omega \subset \mathrm{I}\!\mathrm{R}^{n}\), any function u which is twice continuously differentiable on \(\Omega \) and which verifies the equation \(\Delta u(x)=0,\forall x\in \Omega \), where \(\Delta \) is the operator of Laplace
$$\Delta u=\sum \limits _{k=1}^n\frac{\partial ^2 u}{\partial x_k^2}.$$
Marin Marin, Andreas Öchsner
Chapter 13. Weak Solutions of Classical Problems
Abstract
The Sobolev spaces, which will be defined in the following, are spaces on which weak solutions can be defined (in a sense to be defined later) for classical boundary value problems.
Marin Marin, Andreas Öchsner
Chapter 14. Regularity of the Solutions
Abstract
We recall for beginners some very helpful inequalities in the following.
Marin Marin, Andreas Öchsner
Chapter 15. Weak Solutions for Parabolic Equations
Abstract
In this chapter, we will deal with the study of the following problem:
$$\begin{aligned}&\frac{\partial u}{\partial t}(t,x)-\Delta u(t,x)=f(t,x), \forall (t, x)\in Q_{T}=(0,T)\times \Omega ,\nonumber \\&u(0,x)=u_{0}(x),\;\forall x\in \Omega ,\\&u(t, x)=0,\;\forall (t, x)\in (0,T)\times \partial \Omega ,\nonumber \end{aligned}$$
where \(\Omega \) is an open set from \(R^{n}\) whose boundary \(\partial \Omega \), if it exists, is assumed to be a regular surface. Here, we denoted by T a strictly positive real number. The problem (15.1.1) is the problem of the heat propagation and is a prototype for parabolic differential equations of second order.
Marin Marin, Andreas Öchsner
Chapter 16. Weak Solutions for Hyperbolic Equations
Abstract
In principle, in this chapter, we will study the wave equation, which constitutes the prototype of the hyperbolic equations. Let \(\Omega \) be an open set from \(\mathrm{I}\!\mathrm{R}^n\) and T a real number \(T>0\). Then, the Cauchy problem, associated with the wave equation, consists of
$$\begin{aligned}&\frac{\partial ^2u}{\partial t^2}(t,x)- \Delta u(t, x)=0, \; \forall (t, x)\in Q_T,\\&u(0,x)=u_0(x), \;\forall x\in \Omega , \\&\frac{\partial u}{\partial t}(0,x)=u_1(x),\;\forall x\in \Omega , \end{aligned}$$
where \(Q_T\) is the notation for the cylinder \(Q_T=(0,T)\times \Omega \).
Marin Marin, Andreas Öchsner
Backmatter
Metadata
Title
Essentials of Partial Differential Equations
Authors
Prof. Dr. Marin Marin
Prof. Dr. Andreas Öchsner
Copyright Year
2019
Electronic ISBN
978-3-319-90647-8
Print ISBN
978-3-319-90646-1
DOI
https://doi.org/10.1007/978-3-319-90647-8

Premium Partners

    Image Credits