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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