Exercises of Integrals and Integro-Differentials Equations E-Book

Simone Malacrida

• Herausgeber: Simone Malacrida
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2022

Beschreibung

In this book, exercises are carried out regarding the following mathematical topics:
solving integral equations
solving integral-differential equations
calculus of variations
Initial theoretical hints are also presented to make the conduct of the exercises understandable.

Leseprobe

Table of Contents

"Exercises of Integrals and Integro-Differentials Equations"

INTRODUCTION

THEORETICAL OUTLINE

EXERCISES

"Exercises of Integrals and Integro-Differentials Equations"

SIMONE MALACRIDA

In this book, exercises are carried out regarding the following mathematical topics:

solving integral equations

solving integral-differential equations

calculus of variations

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

Integral equations of Fredholm and Volterra

Calculation of variations

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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

INTRODUCTION

In this exercise book some examples of calculations related to integral and integro-differential equations are carried out.

These equations, often snubbed compared to their "cousins" differential equations, however are crucial in the calculus of variations and in the physical and technological applications of this calculus, from astrophysics to mechanics.

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

Introduction and definitions

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An integral equation is an equation that presents the unknown under the sign of integral.

Actually, whenever you solve a differential equation, the solution formula is an integral equation, so we have already said a lot about such equations in previous chapters. A linear integral equation has a form like this:

Where K(x,z) is the kernel of the equation (which can be real or complex, symmetric or antisymmetric) and f(x) is the known term.

If f(x) is different from zero we speak of equations of the second kind, if it is equal to zero we speak of equations of the first kind.

Integral equations of Fredholm and Volterra

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In integral equations, the integral is defined so we have integration extremes.

If these extremes are fixed we speak of integral equation of Fredholm, if instead one of the extremes is variable in x the equation is called of Volterra.

The Fredholm operator is defined as a bounded linear operator between Banach spaces having a finite-dimensional core and con-core.

Moreover, saying T a Fredholm operator (from a space X to a Y) and S a linear and bounded operator (from the space Y to that X) we have that

are compact operators on X and Y.

The index of a Fredholm operator is defined as follows:

The set of Fredholm operators forms an open set in Banach space of bounded and continuous linear operators.

The index of the composition of two Fredholm operators is equal to the sum of the indices of the single operators, furthermore the added Fredholm operator has the opposite index with respect to the starting one.

Finally, given a Fredholm operator and a compact one, their convolution returns again a Fredholm operator having the same index as the starting one.

The tensor product between a Banach space and its dual is a complete space endowed with the following norm: