Analysis And Differential Equations

✍ Scribed by Odile Pons

• Publisher: WSPC
• Year: 2022
• Tongue: English
• Leaves: 305
• Edition: 2
• Category: Library

✦ Synopsis

The book presents advanced methods of integral calculus and optimization, the classical theory of ordinary and partial differential equations and systems of dynamical equations. It provides explicit solutions of linear and nonlinear differential equations, and implicit solutions with discrete approximations. The main changes of this second edition are: the addition of theoretical sections proving the existence and the unicity of the solutions for linear differential equations on real and complex spaces and for nonlinear differential equations defined by locally Lipschitz functions of the derivatives, as well as the approximations of nonlinear parabolic, elliptic, and hyperbolic equations with locally differentiable operators which allow to prove the existence of their solutions; furthermore, the behavior of the solutions of differential equations under small perturbations of the initial condition or of the differential operators is studied.

✦ Table of Contents

Contents
Preface
Preface of the second edition
1. Introduction
1.1 Differential equations
1.2 Second order differential equations
1.3 Differential equations for functions on Rp
1.4 Multidimensional differential equations
1.5 Overview
1.6 Exercises
2. Expansions with orthogonal polynomials
2.1 Introduction
2.2 Laguerre’s polynomials
2.3 Hermite’s polynomials
2.4 Legendre’s polynomials
2.5 Generalizations
2.6 Bilinear functions
2.7 Hermite’s polynomials in R2
2.8 Legendre’s polynomials in R2
2.9 Exercises
3. Differential and integral calculus
3.1 Differentiability of functions
3.2 Maximum and minimum of functions
3.3 Euler-Lagrange conditions
3.4 Integral calculus
3.5 Partial derivatives of elliptic functions
3.6 Applications
3.7 Exercises
4. Linear differential equations
4.1 First order differential equations in R+
4.2 Existence and unicity of solutions
4.3 Behavior of the solutions under small perturbations
4.4 Second order linear differential equations in R+
4.5 Sturm-Liouville second order differential equations
4.6 Applications
4.7 Nonlinear differential equations
4.8 Differential equations of higher orders
4.9 Differential equations in C
4.10 Exercises
5. Linear differential equations in Rp
5.1 Introduction
5.2 Laplace’s differential equation
5.3 Potential equations
5.4 Heat conduction equations
5.5 Wave differential equations
5.6 Parabolic, elliptic and hyperbolic equations
5.7 Elasticity equations
5.8 Exercises
6. Partial differential equations
6.1 Partial differential equations in R2
6.2 First order linear partial differential equations
6.3 Second order linear partial differential equations
6.4 Multidimensional differential equations
6.5 Epidemics
6.6 Lotka-Volterra equations
6.7 Birth-and-death differential equations
6.8 Differential equations for multi-states dynamic models
6.9 Poincaré-Lorenz differential system
6.10 Exercises
7. Special functions
7.1 Eulerian functions
7.2 Airy function
7.3 Bessel’s function
7.4 Boyd’s function
7.5 Hermite’s function
7.6 Laguerre’s functions
7.7 Hydrogen atom equation and others
7.8 Exercises
8. Solutions
8.1 Integral and differential calculus
8.2 Orthogonal polynomials
8.3 Calculus and optimization
8.4 Linear and nonlinear differential equations
8.5 Linear differential equations in Rp
8.6 Partial differential equations
8.7 Special functions
8.8 Programs
Bibliography
Index