Partial Differential Equations - Theory & Concepts

Partial Differential Equations

Learning Objectives

Define what constitutes a Partial Differential Equation (PDE).
Classify second-order linear PDEs as elliptic, parabolic, or hyperbolic.
Understand the role and structure of Fourier Series in solving PDEs.
Apply the method of Separation of Variables to solve linear PDEs.
Identify and understand the fundamental PDE models: Heat, Wave, and Laplace equations.

While ordinary differential equations (ODEs) involve unknown functions of a single variable, Partial Differential Equations (PDEs) involve unknown functions of multiple independent variables and their partial derivatives. They are essential for describing physical phenomena spread out over space and time, such as heat diffusion, wave propagation, and fluid dynamics.

Partial Differential Equation (PDE)

An equation containing an unknown function $u(x, y, \dots)$ of two or more independent variables and its partial derivatives ( $\frac{\partial u}{\partial x}, \frac{\partial^2 u}{\partial y^2}$ , etc.).

Classification of Second-Order Linear PDEs

A general second-order linear PDE in two variables $(x, y)$ takes the form:

General Second-Order Linear PDE

The standard form of a second-order linear PDE in two independent variables.

A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x} + E \frac{\partial u}{\partial y} + Fu = G

Variables

Symbol	Description	Unit
$u$	The unknown function dependent on x and y	-
$A, B, C$	Coefficients of the second-order partial derivatives	-
$D, E$	Coefficients of the first-order partial derivatives	-
$F, G$	Functions of x and y	-

The Discriminant Test

Similar to conic sections in geometry, PDEs are classified based on the discriminant $\Delta = B^2 - 4AC$ :

Elliptic ( $\Delta < 0$ ): Describes steady-state systems with no time dependence (e.g., 2D Laplace's Equation $u_{xx} + u_{yy} = 0$ ). Solutions are typically smooth and represent equilibrium states.
Parabolic ( $\Delta = 0$ ): Describes time-dependent dissipative processes (e.g., 1D Heat Equation $u_t = \alpha^2 u_{xx}$ ). Solutions diffuse and smooth out over time.
Hyperbolic ( $\Delta > 0$ ): Describes time-dependent wave propagation (e.g., 1D Wave Equation $u_{tt} = c^2 u_{xx}$ ). Solutions transport information at a finite speed (waves) and preserve discontinuities.

Fourier Series

To solve many linear PDEs on bounded domains, we must represent complex initial or boundary conditions as an infinite sum of simple sine and cosine waves. This decomposition is known as a Fourier Series.

Fourier Series Formula

Representation of a piecewise smooth periodic function as a sum of sines and cosines.

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right)

Variables

Symbol	Description	Unit
$f(x)$	The function to represent	-
$L$	Half of the period (period is 2L)	-
$a_0, a_n, b_n$	The Fourier coefficients computed via integrals	-

Computing Euler-Fourier Coefficients

STEP-BY-STEP

For $a_0$ : $a_0 = \frac{1}{L} \int_{-L}^{L} f(x) \, dx$
For $a_n$ : $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \, dx$
For $b_n$ : $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \, dx$

Visualizing Fourier Series

See how adding more terms (harmonics) to the series improves the approximation of square and sawtooth waves. Notice how higher frequencies are required to capture sharp corners (Gibbs Phenomenon).

Method of Separation of Variables

This is the primary analytical method for solving linear PDEs with homogeneous boundary conditions. It assumes the multivariable solution can be factored into a product of single-variable functions.

Method of Separation of Variables

STEP-BY-STEP

Assume Solution Form: Let $u(x,t) = X(x)T(t)$ .
Substitute into PDE: Plug the assumed form into the PDE. Rearrange algebraically to separate the variables onto opposite sides of the equation (e.g., all $x$ terms on the left, all $t$ terms on the right).
Set to Separation Constant: Since a function of $x$ equals a function of $t$ for all $x$ and $t$ , both sides must equal a constant, typically denoted $-\lambda$ . This converts the single PDE into two separate ordinary differential equations (ODEs).
Solve the Boundary Value Problem (BVP): Solve the spatial ODE ( $X(x)$ ) using the given homogeneous boundary conditions (like fixed ends of a string). This restricts $\lambda$ to specific discrete values called eigenvalues ( $\lambda_n$ ), which correspond to non-trivial solutions called eigenfunctions ( $X_n(x)$ ).
Solve the Temporal ODE: Solve the time-dependent ODE ( $T(t)$ ) for each found eigenvalue $\lambda_n$ . This usually yields exponential decay (Heat) or oscillatory sines/cosines (Wave).
Superposition: Construct the general solution by summing the infinite product solutions: $u(x,t) = \sum_{n=1}^\infty X_n(x)T_n(t)$ .
Apply Initial Conditions: Substitute $t=0$ into the series. Use the initial conditions and compute the Fourier coefficients to find the remaining arbitrary constants.

Fundamental PDE Models

Three classic linear PDEs form the foundation of this field. Each describes a fundamentally different physical process.

1. The 1D Heat Equation (Parabolic)

Models the diffusion of thermal energy in a rod over time.

Behavior: Applying Separation of Variables yields a spatial ODE $X'' + \lambda X = 0$ (sines/cosines) and a temporal ODE $T' + \alpha^2\lambda T = 0$ (exponential decay). The solution features terms like $e^{-\alpha^2\lambda_n t}\sin(\sqrt{\lambda_n}x)$ . High-frequency spatial variations (large $n$ ) decay very rapidly, meaning heat quickly smooths out sharp temperature differences.

1D Heat Equation

Models the parabolic diffusion of heat over space and time.

\frac{\partial u}{\partial t} = \alpha^2 \frac{\partial^2 u}{\partial x^2}

Variables

Symbol	Description	Unit
$u(x,t)$	The temperature at position x and time t	-
$\alpha^2$	Thermal diffusivity k/(c\rho)	-

2. The 1D Wave Equation (Hyperbolic)

Governs the small-amplitude, transverse vibrations of a stretched string, or acoustic waves in a pipe.

Behavior: Both the spatial ( $X$ ) and temporal ( $T$ ) ODEs are second-order and result in oscillatory sine/cosine solutions. The full series solution represents a superposition of standing waves (normal modes or harmonics) oscillating at discrete frequencies.

1D Wave Equation

Models hyperbolic wave propagation over space and time.

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Variables

Symbol	Description	Unit
$u(x,t)$	The wave displacement at position x and time t	-
$c$	The wave speed, \sqrt{T/\rho}	-

3. The 2D Laplace Equation (Elliptic)

Describes steady-state temperature distribution in a 2D plate (after a long time has passed), or electrostatic potential in a region devoid of charge.

Behavior: There is no time derivative. Separation of variables $u(x,y) = X(x)Y(y)$ leads to $X''/X = -Y''/Y = \lambda$ . If $\lambda > 0$ , $X$ is oscillatory (sines/cosines) and $Y$ is exponential (or hyperbolic sines/cosines $\sinh, \cosh$ ). The choice of the separation constant's sign depends entirely on the specific boundary conditions given on the edges of the rectangular domain.

2D Laplace Equation

Models elliptic steady-state systems.

\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0

Variables

Symbol	Description	Unit
$u(x,y)$	The steady-state quantity, such as temperature or potential	-

Interactive Simulation

Interact with the simulation below to visualize standing waves and normal modes in the 1D wave equation.

1D Wave Equation

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

Vibrating string fixed at both ends ( $x=0$ , $x=L$ ). The solution is a superposition of normal modes (standing waves).

Normal Mode (

n

Determines the spatial frequency $\sin(n\pi x/L)$ .

Wave Speed (

c

)1.0

u(x,t)

u_n(x,t) = \sin\left(\frac{1\pi x}{L}\right) \cos\left(\frac{1\pi (1.0) t}{L}\right)

Key Takeaways

PDE Classification: Second-order PDEs are classified as Elliptic ( $\Delta < 0$ , steady-state, e.g., Laplace's), Parabolic ( $\Delta = 0$ , diffusion, e.g., Heat), or Hyperbolic ( $\Delta > 0$ , propagation, e.g., Wave).
Separation of Variables: The core analytical technique to convert a single linear PDE into multiple ODEs by assuming the solution factors as $u(x,t) = X(x)T(t)$ .
Fourier Series: Used to express initial or boundary conditions as an infinite sum of sinusoidal waves, allowing us to perfectly match the eigenfunctions derived from the spatial ODE boundary value problem.
Heat Equation ( $u_t = \alpha^2 u_{xx}$ ): Results in exponential decay of temperature differences over time, smoothing the distribution.
Wave Equation ( $u_{tt} = c^2 u_{xx}$ ): Results in oscillatory motion in both space and time, supporting traveling and standing waves.
Laplace's Equation ( $u_{xx} + u_{yy} = 0$ ): Results in steady-state solutions built from combinations of trigonometric (oscillating) and hyperbolic (exponential) functions.

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