Chapter 6

Introduction to Partial Differential Equations

Heat Equation • Wave Equation • Separation of Variables • Fourier Series

Learning Objectives

Definition and Basic Concepts

Definition: Partial Differential Equation (PDE) A partial differential equation is an equation involving partial derivatives of an unknown function $u$ that depends on several independent variables — for example $u(x,t)$ or $u(x,y)$.
Key Terminology

Order of a PDE: The highest order of derivatives appearing in the equation.

Linear PDE: The unknown $u$ and its derivatives appear only in linear form (no products, no nonlinear functions of $u$).

Homogeneous: Every term contains $u$ or one of its derivatives.

Non-Homogeneous: Contains a "forcing" term $p(x,t)$ that does not involve $u$.

Examples of Classification

Example

The equation $u_t - 3 u_{xx} = 0$ is a second-order, linear, homogeneous PDE (the heat equation with $K=3$).

The equation $u_t - 3 u_{xx} = \sin x$ is second-order, linear, non-homogeneous (forcing term $\sin x$).

The equation $u_t + u\, u_x = 0$ is first-order, non-linear (the product $u \, u_x$ violates linearity).

The Three Classical Second-Order PDEs

Most PDEs encountered in engineering reduce to one of three canonical forms. Each models a fundamentally different physical phenomenon, and the method of solution is essentially the same for all three.

1. Heat (Diffusion) Equation $$\frac{\partial u}{\partial t} = K\,\frac{\partial^2 u}{\partial x^2}, \qquad K > 0$$

First-order in time, second-order in space. Models heat conduction, mass diffusion, and any process where a quantity spreads out to equilibrate. Solutions decay over time.

2. Wave Equation $$\frac{\partial^2 u}{\partial t^2} = a^2\,\frac{\partial^2 u}{\partial x^2}, \qquad a > 0$$

Second-order in both time and space. Models vibrating strings, sound waves, electromagnetic waves. Solutions oscillate — energy is conserved.

3. Laplace's Equation $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$

Second-order in two spatial variables, no time. Models steady-state temperature, electrostatic potential, and incompressible fluid flow. Solutions are harmonic.

Why Are These "Classical"?

Every second-order linear PDE in two variables can be classified as parabolic (like heat), hyperbolic (like wave), or elliptic (like Laplace). The qualitative behaviour — decay, oscillation, or equilibrium — is determined entirely by this classification.

Chapter Roadmap

This chapter is organized into five focused pages. Work through them in order, or jump to the topic you need.

6.2

Heat Equation & Separation of Variables

Derive the separation method. Work through all three cases $\lambda < 0$, $\lambda = 0$, $\lambda > 0$ and discover why only one produces non-trivial solutions.

 6.3

Three Types of Boundary Conditions

Homogeneous Dirichlet, non-homogeneous Dirichlet (with and without source), and Neumann. For Neumann, $\lambda = 0$ becomes non-trivial!

 6.4

The Wave Equation

Same spatial problem as the heat equation, but the temporal ODE is second-order — producing oscillations instead of decay. Two initial conditions are needed.

 6.5

Visualizations & Practice

Interactive heat & wave simulators with adjustable parameters, a 6-question self-assessment quiz, a summary table, and a quick-reference card.
وَفَوْقَ كُلِّ ذِي عِلْمٍ عَلِيمٌ
"And above every knower is one more knowing."
— Quran 12:76

Partial differential equations open a door to phenomena across an entire universe — heat flow, wave propagation, electromagnetism, quantum mechanics. Every solution you compute is a small window onto how nature actually works.

← Back to Home