Introduction

From the oscillation of strings to the diffusion of heat, from gravitational waves rippling through spacetime to quantum particles governed by Schrödinger’s equation — partial differential equations (PDEs) serve as the language of change in multiple dimensions.

In this post, we will journey through the foundational ideas of PDEs, their classification, and some iconic examples that shape both theoretical mathematics and modern physics.

What is a Partial Differential Equation?

A partial differential equation (PDE) is an equation that involves partial derivatives of an unknown function with respect to more than one independent variable.

Mathematically, if $u=u(x_1,x_2,\dots,x_n)$ is a function of several variables, then a PDE relates $u$ , its partial derivatives (like $\large$\frac{\partial u}{\partial x_i},\frac{\partial^2u}{\partial x_i\partial x_j}$$ , etc.), and possibly the independent variables themselves.

Contrast this with an ordinary differential equation (ODE), where the unknown function depends on only one independent variable. For example:

ODE: $\large$\frac{dy}{dx}+y=e^x,y=y(x)$$
PDE: $\large$\frac{\partial u}{\partial t}=\alpha\frac{\partial^2u}{\partial x^2},u=u(x,t)$$

In the PDE above, $\large$u=u(x,t)$$ depends on both space and time — a hallmark of PDEs: they often describe how physical systems evolve in space and time.

Standard Examples of PDEs

The Heat Equation (Diffusion of Temperature): $\large$\frac{\partial u}{\partial t}=\alpha\frac{\partial^2u}{\partial x^2},u=u(x,t)$$
The Wave Equation (Vibrations, Sound, Light): $\large$\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}$$
Laplace’s Equation (Steady-State Heat, Electrostatics): $\large$\nabla^2u=0$$
Poisson’s Equation (Gravitational / Electrostatic Potentials): $\large$\nabla^2u=f(x)$$

Each of these models different physical phenomena but shares a deep mathematical kinship rooted in the structure of the equations.

Classification of Second Order PDEs

Let’s focus on second-order PDEs in two variables, often written as: $\boxed{A(x,y)\frac{\partial^2u}{\partial x^2}+B(x,y)\frac{\partial^2u}{\partial x\partial y}+C(x,y)\frac{\partial^2u}{\partial y^2}+D(x,y)\frac{\partial u}{\partial x}+E(x,y)\frac{\partial u}{\partial y}+F(x,y)u+G(x,y)=0}$

This form can be linear or non-linear, depending on whether the equation is linear in $u$ and its derivatives.

Linear vs. Non-linear PDEs

Linear PDEs: The unknown function and all its derivatives appear linearly. Example: $\large$\frac{\partial u}{\partial t}-\frac{\partial^2u}{\partial x^2}=0$$
Non-linear PDEs: Involve non-linear terms like $\large$u^2,(\frac{\partial u}{\partial x})^2$$ , etc. Example: $\large$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0$$ (Burgers’ equation)

Classification of Linear Second Order PDEs

When a second-order PDE is linear, we can classify it using the discriminant of the second-order part: $\large$D=B^2-4AC$$

Based on the value of $\large$D$$ , we classify the equation as:

Type	Condition on $\large$D$$	Example
Elliptic	Negative	Laplace’s or Poisson’s Equation
Parabolic	Vanishes	Heat Equation
Hyperbolic	Positive	Wave Equation

This classification echoes the conic sections — ellipse, parabola, and hyperbola — and dictates the nature of solutions and boundary conditions.

Parabolic PDE: The Heat Equation

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

Physical meaning: Describes how heat diffuses over time.
Type: Parabolic, since $\large$D=0$$ .
Features: Irreversible time evolution, smoothing of initial data.

Elliptic PDE: Poisson’s Equation

$\large$\nabla^2u\equiv\frac{\partial^2u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=f(x,y)$$

Physical meaning: Describes steady-state distributions like electrostatic potential given a charge density $\large$f$$ .
Type: Elliptic, since discriminant $\large$D$$ is negative.
Features: Solutions are smooth (under reasonable conditions), no time dependence.

Hyperbolic PDE: The Wave Equation

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

Physical meaning: Describes vibrations, waves in strings, sound waves, etc.
Type: Hyperbolic, since $\large$D>0$$ .
Features: Supports finite-speed propagation and wavefronts.

Why Does This Classification Matter?

Each class of PDE behaves fundamentally differently:

Elliptic: Boundary value problems, smooth solutions.
Parabolic: Initial-boundary value problems, diffusion and decay.
Hyperbolic: Initial value problems, wave propagation and causality.

Understanding the classification gives insight into:

The nature of the physical phenomenon being modeled.
The type of initial and boundary data needed.
Appropriate analytical and numerical solution techniques.

Toward Deeper Waters

This post has just scratched the surface of the immense world of PDEs. In future entries, we will explore methods of solution (like separation of variables, Fourier transforms, Green’s functions), and delve into nonlinear PDEs that arise in general relativity, fluid dynamics, and quantum field theory.

Stay tuned to Manifolds Unfolded as we unfold the tapestry of equations that shape the fabric of the physical and geometric world.

Manifolds Unfolded

Partial Differential Equations: Classifying the Dynamics of the Continuum

Introduction

What is a Partial Differential Equation?

Standard Examples of PDEs