Exercises of Partial Differential Equations E-Book

Simone Malacrida

• Herausgeber: BookRix
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2023

Beschreibung

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.

Leseprobe

Simone Malacrida

Exercises of Partial Differential Equations

BookRix GmbH & Co. KG81371 Munich

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.