Differential Equations

8.

Introduction to Partial Differential Equations

8.1.

Fundamental Concepts

8.1.1.

Definition of PDEs

8.1.2.

Dependent Variables

8.1.3.

Independent Variables

8.1.4.

Order Classification

8.1.5.

Linearity in PDEs

8.1.6.

Homogeneous vs Nonhomogeneous PDEs

8.2.

Classification of Second-Order PDEs

8.2.1.

General Second-Order Linear PDE Form

8.2.2.

Discriminant Analysis

8.2.3.

Elliptic Equations

8.2.4.

Parabolic Equations

8.2.5.

Hyperbolic Equations

8.2.6.

Physical Significance of Types

8.3.

Classical PDEs and Physical Applications

8.3.1.

8.3.1.1.

Physical Derivation

8.3.1.2.

Mathematical Formulation

8.3.1.3.

Boundary Conditions

8.3.1.4.

Initial Conditions

8.3.2.

8.3.2.1.

Physical Derivation

8.3.2.2.

Mathematical Formulation

8.3.2.3.

Boundary Conditions

8.3.2.4.

Initial Conditions

8.3.3.

Laplace's Equation

8.3.3.1.

Physical Significance

8.3.3.2.

Mathematical Formulation

8.3.3.3.

Boundary Value Problems

8.3.3.4.

Harmonic Functions

8.3.4.

Poisson's Equation

8.3.4.1.

Relationship to Laplace's Equation

8.3.4.2.

8.4.

Fourier Series Foundation

8.4.1.

Periodic Function Representation

8.4.2.

Fourier Coefficients

8.4.2.1.

Calculation Formulas

8.4.2.2.

Integration Techniques

8.4.3.

Convergence of Fourier Series

8.4.3.1.

Pointwise Convergence

8.4.3.2.

Uniform Convergence

8.4.3.3.

Gibbs Phenomenon

8.4.4.

Even and Odd Functions

8.4.4.1.

Fourier Cosine Series

8.4.4.2.

Fourier Sine Series

8.4.5.

Orthogonality Relations

8.4.5.1.

Sine and Cosine Orthogonality

8.4.5.2.

Inner Product Interpretation

8.4.6.

Extensions and Generalizations

8.4.6.1.

Half-Range Expansions

8.4.6.2.

Complex Fourier Series

8.5.

Method of Separation of Variables

8.5.1.

General Procedure

8.5.1.1.

Variable Separation Assumption

8.5.1.2.

Ordinary Differential Equation Formation

8.5.1.3.

Boundary Condition Application

8.5.2.

Application to Heat Equation

8.5.2.1.

Homogeneous Boundary Conditions

8.5.2.2.

Eigenvalue Problems

8.5.2.3.

Solution Construction

8.5.3.

Application to Wave Equation

8.5.3.1.

Standing Wave Solutions

8.5.3.2.

8.5.3.3.

General Solution Superposition

8.5.4.

Application to Laplace's Equation

8.5.4.1.

Rectangular Coordinates

8.5.4.2.

Cylindrical Coordinates

8.5.4.3.

Spherical Coordinates

8.5.5.

Sturm-Liouville Problems

8.5.5.1.

Eigenvalue Problems

8.5.5.2.

Orthogonal Eigenfunctions

8.6.

Boundary and Initial Conditions

8.6.1.

Types of Boundary Conditions

8.6.1.1.

Dirichlet Boundary Conditions

8.6.1.2.

Neumann Boundary Conditions

8.6.1.3.

Robin Boundary Conditions

8.6.1.4.

Periodic Boundary Conditions

8.6.2.

Initial Conditions for Evolution Equations

8.6.3.

Well-Posed Problems

8.6.3.1.

Existence of Solutions

8.6.3.2.

Uniqueness of Solutions

8.6.3.3.

Continuous Dependence on Data

8.6.4.

Physical Interpretation of Conditions

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7. Numerical Methods for Ordinary Differential Equations

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1. Introduction to Differential Equations