Ordinary Differential Equations (13): Introduction to Partial Differential Equations

When physical quantities change not only with time but also with space, we enter the realm of partial differential equations (PDEs). Heat conducting through a metal rod, waves spreading across water, electromagnetic fields propagating through space — these phenomena all require PDEs to describe. This chapter introduces the basic concepts and classification of PDEs, along with solution methods for the three classical equations.

From ODEs to PDEs

Why Do We Need Partial Differential Equations?

In previous chapters, the unknown functions we studied depended on only one independent variable, such as or. But in the real world, many physical quantities depend simultaneously on multiple variables:

• Temperature field: depends on spatial position and time
• Wave function: depends on position and time
• Fluid velocity field: three-dimensional space plus time

When the unknown function depends on multiple independent variables, partial derivatives appear in the differential equation — this is a partial differential equation (PDE).

Basic Concepts

Definition: A partial differential equation is an equation containing an unknown function and its partial derivatives.

Order: The order of the highest partial derivative appearing in the equation.

Linear vs Nonlinear: - Linear: The unknown function and its derivatives appear linearly - Nonlinear: Contains products, powers, etc., of the unknown function or derivatives

Examples:

1. One-dimensional heat equation (linear, second-order):
2. One-dimensional wave equation (linear, second-order):
3. Laplace equation (linear, second-order):
4. Burgers equation (nonlinear, second-order):

Classification of PDEs

Standard Form of Second-Order Linear PDEs

The general form of a second-order linear PDE (two independent variables):

Classification Criteria

By analogy with the quadratic curve, PDEs are classified according to the discriminant:

Discriminant	Type	Physical Prototype	Characteristics
	Hyperbolic	Wave equation	Information propagation, finite speed
	Parabolic	Heat equation	Diffusion process, irreversible
	Elliptic	Laplace equation	Steady state, equilibrium

Physical Intuition

Hyperbolic (wave): Information propagates at finite speed. If you throw a stone into a lake, ripples spread gradually, but distant points are temporarily unaffected.

Parabolic (diffusion): Information propagates instantly to infinity (though the effect is small). Heat immediately begins to spread to the surroundings.

Elliptic (equilibrium): No time evolution, describes steady states. Examples include electrostatic fields and steady-state temperature distributions.

Initial-Boundary Value Problems

Why Do We Need Boundary Conditions?

ODEs need initial conditions to determine solutions. PDEs need not only initial conditions (for evolution equations) but also boundary conditions (because there is a spatial domain).

Types of Boundary Conditions

Let the domain boundary be:

First kind (Dirichlet): Given function values

Second kind (Neumann): Given normal derivatives

Third kind (Robin/Mixed): Linear combination of function values and derivatives

Well-Posedness

A PDE problem is well-posed if: 1. Existence: A solution exists 2. Uniqueness: The solution is unique 3. Stability: The solution depends continuously on the data

The Heat Equation

Physical Derivation

Consider heat conduction in a thin rod. Letbe the temperature at pointat time.

Fourier's law of heat conduction: Heat flux is proportional to temperature gradient

Energy conservation:Combining gives the heat equation:whereis the thermal diffusivity.

Separation of Variables

Consider the problem:$

$

Assume solution form:Substituting into the equation:Separating variables:This yields two ODEs:$

$

Solving the spatial part (Sturm-Liouville problem):

Boundary conditionslead to:

Solving the temporal part:

General solution:

Finite Difference Methods for the Heat Equation

Explicit scheme (FTCS):

Using forward difference for time and central difference for space:Rearranging:where.

Stability condition:

Implicit Scheme (Crank-Nicolson)

The Crank-Nicolson scheme is unconditionally stable:This is a tridiagonal linear system.

The Wave Equation

Physical Derivation

Consider small vibrations of a taut string. Letbe the vertical displacement at pointat time.

By Newton's second law:This gives the wave equation:whereis the wave speed.

d'Alembert's Solution

For an unbounded domain initial value problem:The solution is:This shows the solution consists of superposition of left-traveling and right-traveling waves.

Finite Difference for the Wave Equation

Explicit scheme:Rearranging:where.

CFL condition:(Courant-Friedrichs-Lewy)

Laplace and Poisson Equations

Physical Background

Laplace equation:Describes steady-state, source-free fields: - Steady-state temperature distribution (no internal heat source) - Electrostatic potential (charge-free region) - Velocity potential of incompressible flow

Poisson equation:Used when there are sources (charges, heat sources, etc.).

Properties

Maximum principle: Harmonic functions (solutions of Laplace's equation) do not attain extreme values in the interior; extrema occur only on the boundary.

Mean value property: The value at the center of a circle equals the average value on the circumference.

Finite Difference Method

Using the five-point stencil:That is:This is a large sparse linear system.

Iterative Methods

For large-scale problems, direct solution is too slow. Common iterative methods:

Jacobi iteration:

Gauss-Seidel iteration:

SOR (Successive Over-Relaxation):The optimal relaxation factor is approximately.

Stability Analysis of Numerical Methods

Von Neumann Stability Analysis

Basic idea: Expand the error in Fourier modes and analyze the growth factor of each mode.

Let error, substitute into the difference scheme, and find conditions for.

CFL Condition

For the wave equation, the CFL condition states: The numerical information propagation speed must be greater than or equal to the physical information propagation speed.

Physical wave speed:Numerical information propagation speed:CFL condition:, i.e.,.

Advanced Topics

Spectral Methods

Idea: Expand the solution in global basis functions (such as Fourier series, Chebyshev polynomials).

Advantages: For smooth solutions, can achieve exponential convergence (spectral accuracy).

Disadvantages: Difficult to handle complex boundaries; not suitable for non-smooth solutions.

Introduction to Finite Element Methods

The Finite Element Method (FEM) is the most commonly used numerical method for PDEs in engineering.

Basic steps: 1. Weak form: Multiply PDE by test function and integrate 2. Discretization: Approximate with finite-dimensional space 3. Assembly: Construct stiffness matrix and load vector 4. Solve: Solve linear system

Multigrid Methods

Problem: Iterative methods (like Jacobi, Gauss-Seidel) converge slowly for low-frequency errors.

Multigrid idea: 1. Iterate a few steps on fine grid (eliminate high-frequency errors) 2. Restrict residual to coarse grid 3. Solve on coarse grid (low-frequency errors become high-frequency on coarse grid) 4. Prolongate coarse grid correction back to fine grid 5. Repeat

Complexity:(optimal)

Summary

In this chapter, we entered the world of partial differential equations:

Equation Type	Physical Model	Typical Numerical Methods	Stability Condition
Parabolic (heat equation)	Diffusion, heat conduction	FTCS, CN	(FTCS)
Hyperbolic (wave equation)	Wave propagation, vibration	Central difference	CFL:
Elliptic (Laplace)	Steady state, equilibrium	Iterative, direct	Always stable

Key concepts:

1. Classification: Discriminantdetermines equation type
2. Well-posedness: Existence, uniqueness, stability
3. Stability analysis: Von Neumann analysis, CFL condition
4. Numerical methods: Finite differences, finite elements, spectral methods

This chapter marks our transition from ODEs to PDEs. PDEs are a vast field, and we have only scratched the surface. I hope this introduction lays the foundation for further study in computational mathematics, fluid mechanics, electromagnetism, and other fields!