Introduction to Differential Equations E-Book

Simone Malacrida

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

Veröffentlichungsjahr: 2022

Beschreibung

The following mathematical topics are exposed in this book:
ordinary differential equations with methods of solving them
partial differential equations with methods of solving them
integral and integro-differential equations

Leseprobe

Table of Contents

“Introduction to Differential Equations”

INTRODUCTION

ORDINARY DIFFERENTIAL EQUATIONS

PARTIAL DIFFERENTIAL EQUATIONS

INTEGRAL AND INTEGRAL-DIFFERENTIAL EQUATIONS

“Introduction to Differential Equations”

SIMONE MALACRIDA

The following mathematical topics are exposed in this book:

ordinary differential equations with methods of solving them

partial differential equations with methods of solving them

integral and integro-differential equations

Simone Malacrida (1977)

Engineer and writer, has worked on research, finance, energy policy and industrial plants.

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ANALYTICAL INDEX

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INTRODUCTION

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I – ORDINARY DIFFERENTIAL EQUATIONS

Introduction and definitions

Resolution methods

Solutions

Remarkable differential equations

Autonomous systems

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II – PARTIAL DIFFERENTIAL EQUATIONS

Introduction and definitions

Resolution methods

Second order equations

Weak wording

Remarkable differential equations

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III - INTEGRAL AND INTEGRAL-DIFFERENTIAL EQUATIONS

Introduction and definitions

Integral equations of Fredholm and Volterra

Calculation of variations

INTRODUCTION

Differential equations represent the culmination of mathematical analysis.

After having introduced the fundamental operations of limit, derivative and integral and after having extended the concepts of analysis to functions of several variables or to complex ones, the study of analysis requires the resolution of equations with the present of derivatives and/or integrals.

The shocking aspect of differential equations is that they correspond to well-defined natural phenomena, i.e. Nature follows laws that can be expressed through differential equations.

Only with this formalism can all modern and contemporary physical theories be fully understood.

In this manual, ordinary differential, partial differential, integral and integro-differential equations will be presented and general mechanisms for solving them will be provided.

In addition, a detailed list of the main equations with their physical applications will be provided.

It seems superfluous to remember that in order to understand the above, it is necessary to have followed at least some courses in Mathematical Analysis and Geometry at university level.

I

ORDINARY DIFFERENTIAL EQUATIONS

Introduction and definitions

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A differential equation is a relationship between a function and some of its derivatives.

An equation defined in an interval of the set of real numbers in which the total derivatives of the function with respect to the unknown are present is called ordinary.

The order of the highest derivative in the equation is called the order of the equation.

One can generalize the defining set of an ordinary differential equation into an open and connected generic contained in the complex space of dimension greater than two.

A solution or integral of the ordinary differential equation is a function that satisfies the equation's relation.

An equation is said to be autonomous if the relation does not explicitly depend on the variable and is said to be written in normal form if it can be made explicit with respect to the derivative of maximum degree.

The equation is called linear if the solution is a linear combination of the derivatives according to this formula:

The term r(x) is calledsource and, if it is zero, the linear differential equation is called homogeneous .

In general, an ordinary differential equation of degree n has n linearly independent solutions, and any linear combination of them is itself a solution.

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Given a general solution of the homogeneous equation associated with an ordinary differential equation it is possible to find a particular solution of the equation.

This will be clarified shortly by the analytical methods of solving differential equations.

An ordinary differential equation of order n expressed in normal form can be reduced to a system of ordinary differential equations of order one in normal form, through the so-called first order reduction procedure.

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Resolution methods

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The ordinary differential equation of order n can be expressed as follows:

We can define coefficients such that: