2500 Solved Problems in Differential Equations

Это введение в теорию распределений с использованием решенных задач

The aim of this App (2500 Solved Problems in Differential Equations) is to provide a comprehensive introduction to the theory of distributions, by the use of solved problems. Although written for mathematicians, it can also be used by a wider audience, including engineers and physicists.

The first six chapters deal with the classical theory, with special emphasis on the concrete aspects. The reader will find many examples of distributions and learn how to work with them. At the beginning of each chapter the relevant theoretical material is briefly recalled.

The last chapter is a short introduction to a very wide and important field in analysis which can be considered as the most natural application of distributions, namely the theory of partial differential equations. It includes exercises on the classical differential operators and on fundamental solutions, hypoellipticity, analytic hypoellipticity, Sobolev spaces, local solvability, the Cauchy problem, etc.

Table of Contents

1 Basic Concepts

2 An Introduction to Modeling and Qualitative Methods

3 Classifications of First-Order Differential Equations

4 Separable First-Order Differential Equations

5 Exact First-Order Differential Equations

6 Linear First-Order Differential Equations

7 Applications of First-Order Differential Equations

8 Linear Differential Equations: Theory of Solutions

9 Second-Order Linear Homogeneous Differential Equations with Constant Coefficients

10 nth-Order Linear Homogeneous Differential Equations with Constant Coefficients

11 The Method of Undetermined Coefficients

12 Variation of Parameters

13 Initial-Value Problems for Linear Differential Equations

14 Applications of Second-Order Linear Differential Equations

15 Matrices

17 Reduction of Linear Differential Equations to a System of First-Order Equations

18 Graphical and Numerical Methods for Solving First-Order Differential Equations

19 Further Numerical Methods for Solving First-Order Differential Equations

20 Numerical Methods for Solving Second-Order Differential Equations Via Systems

22 Inverse Laplace Transforms

23 Convolutions and the Unit Step Function

24 Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transforms

25 Solutions of Linear Systems by Laplace Transforms

26 Solutions of Linear Differential Equations with Constant Coefficients by Matrix Methods

27 Power Series Solutions of Linear Differential Equations with Variable Coefficients

View sub-sections28 Series Solutions Near a Regular Singular Point

29 Some Classical Differential Equations

30 Gamma and Bessel Functions

31 An Introduction to Partial Differential Equations

32 Second-Order Boundary-Value Problems

33 Eigenfunction Expansions

34 An Introduction to Difference Equations

B Some Comments about Technology

ANSWERS: Answers to Supplementary Problems

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