Table of Contents
Table of Contents
“Exercises of Partial Differential Equations”
INTRODUCTION
THEORETICAL OUTLINE
EXERCISES
“Exercises of Partial Differential Equations”
“Exercises of Partial Differential Equations”
SIMONE MALACRIDA
In this book, exercises are carried out regarding the following mathematical topics:
solving first-order partial-derivative differential equations
solving second-order partial derivative differential equations: elliptic, parabolic and hyperbolic
weak problem formulation
Initial theoretical hints are also presented to make the conduct of the exercises understandable.
Simone Malacrida (1977)
Engineer and writer, has worked on research, finance, energy policy and industrial plants.
ANALYTICAL INDEX
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INTRODUCTION
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I – THEORETICAL OUTLINE
Introduction and definitions
Resolution methods
Second order equations
Weak wording
Remarkable differential equations
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II – EXERCISES
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 11
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
Exercise 23
Exercise 24
Exercise 25
Exercise 26
Exercise 27
Exercise 28
Exercise 29
Exercise 30
Exercise 31
Exercise 32
Exercise 33
INTRODUCTION
INTRODUCTION
In this workbook, some examples of calculations relating to partial differential equations are carried out.
Furthermore, the main theorems used in differential analysis are presented.
Partial differential equations represent one of the high points in the study of mathematical analysis.
The majority of physical events are governed precisely by differential equations of this type and their resolution is, in general, not an easy task.
The weak formulation, which we are going to expose here, is the connection point between analytical resolution (which is possible in a few cases) and numerical resolution.
In order to understand in more detail what is presented in the resolution of the exercises, the theoretical reference context is recalled in the first chapter.
What is presented in this workbook is generally addressed in advanced mathematical analysis courses (analysis 3 and beyond).
I
THEORETICAL OUTLINE
THEORETICAL OUTLINE
Introduction and definitions
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A partial differential equation is a differential equation where the partial derivatives of a function of several variables appear:
Where k is an integer called order of the equation and is the maximum degree of the derivative present in the equation.
A partial differential equation is said to be linear if:
If f(x)=0 then the equation is called homogeneous.
If the equation is in this form, it is said to be semi-linear :
While it is called quasi-linear if it can be expressed as follows:
It goes without saying that it is possible to construct systems of partial differential equations.
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A problem related to these equations is said to be well posed if the solution exists, is unique and depends continuously on the data provided.
Let's say right away that, even more than ordinary differential equations, partial differential equations depend on the initial conditions and boundary conditions and that, at the same time, the analytical solutions of these equations are difficult to extrapolate and not of absolute validity.
In this context, all those numerical resolution methods assume a primary role.
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Resolution methods
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For equations in two variables, the first order equation is given by:
Having used this notation to indicate the partial derivative operation:
A general solution is given by the full integral :
If it is not possible to derive this integral, a system of ordinary differential equations is solved by means of the method of characteristics .
This method constitutes, together with the method of separation of the variables, one of the few analytical methods for solving partial differential equations.
The method makes it possible to find curves, called characteristics, along which the partial differential equation is similar to an ordinary differential equation.
Given a first order quasi-linear partial differential equation:
The characteristic curve equations are given by:
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Second order equations
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A partial differential equation in two variables of the second order is given by:
By Schwarz's theorem, mixed second derivatives are equal.
It is possible to distinguish the partial differential equations of the second order into three types as this quantity, called delta, varies:
If this quantity is negative, the equation is called elliptic, if it is zero it is called parabolic, if it is positive it is called hyperbolic.
If the variables are n, instead of two, the equation is elliptic if the eigenvalues are all positive or all negative, it is parabolic if they are all positive or all negative, except one which is zero, it is hyperbolic if there is only one negative eigenvalue (positive), while all others are positive (negative).
We point out that, for hyperbolic equations, the method of characteristics is valid.
An existence and uniqueness result for partial differential equations having analytic coefficients and associated Cauchy problems is given by the Cauchy-Kovalevskaya theorem.
The limit of this theorem is given by the fact that existence is local and does not ensure a global solution over the whole domain of definition.
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Weak wording
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One point that distinguishes partial differential equations from ordinary ones is the weak formulation of the problem.
With this diction we intend to find solutions to a problem, called weak, which are understood as distributions and not as classical functions.
Therefore the spaces of solutions of partial differential equations are, generally, the Sobolev and Hilbert spaces.
Given a Hilbert space with norm and scalar product and a bilinear form b(u,v) in it such that, if F is a generic functional, we have:
if this form is continuous and coercive , i.e. if:
Impressum
Verlag: BookRix GmbH & Co. KG
Tag der Veröffentlichung: 23.04.2023
ISBN: 978-3-7554-4006-2
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