A partial differential equation (PDE) is an equation that relates an unknown function of multiple variables to its partial derivatives. If you’ve encountered ordinary differential equations, which involve functions of a single variable, PDEs are the natural extension: they describe how something changes with respect to two or more independent variables at once, like space and time. This makes them the mathematical language behind nearly every physical phenomenon that varies across both location and time, from heat spreading through metal to sound traveling through air.
PDEs vs. Ordinary Differential Equations
The distinction is straightforward. An ordinary differential equation (ODE) contains ordinary derivatives of a function that depends on one variable. A PDE contains partial derivatives of a function that depends on two or more variables. That’s the entire dividing line.
For example, an equation describing how a population grows over time involves one independent variable (time) and uses ordinary derivatives. But an equation describing how temperature changes across a metal rod over time involves two independent variables (position along the rod and time), so it requires partial derivatives and qualifies as a PDE. This jump from one variable to multiple variables is what makes PDEs dramatically more complex to solve, and also what makes them powerful enough to describe the real world, where things vary in space and time simultaneously.
Three Famous PDEs in Physics
Most introductions to PDEs start with three classical equations, each representing a fundamentally different type of physical behavior.
The Heat Equation
The heat equation describes how temperature spreads through a material over time. In one dimension, it says the rate of temperature change at a point is proportional to the curvature of the temperature profile at that point. The proportionality constant is called thermal diffusivity, which depends on how well the material conducts heat, its density, and its heat capacity. A high thermal diffusivity means heat spreads quickly. The characteristic time for heat to diffuse across a distance is proportional to that distance squared divided by the diffusivity, which is why a thin metal sheet heats through quickly but a thick one takes far longer.
The Wave Equation
The wave equation governs vibrating strings, sound, light, and any phenomenon where a disturbance propagates without energy loss. It connects the acceleration of a displacement (its second derivative in time) to the curvature of the displacement (its second derivative in space). As Richard Feynman described it in his physics lectures, functions of the form “position minus speed times time” naturally satisfy this equation, representing a shape that moves forward at a constant speed without changing form.
The Schrödinger Equation
In quantum mechanics, the Schrödinger equation plays the role that Newton’s second law plays in classical physics. As a Yale physics lecture put it, “just about anything you see in this room, or on this planet, anything you can see or use is really described by this equation.” It’s a PDE in which the unknown function (called the wave function) depends on position and time. Rather than telling you exactly where a particle is, the wave function gives you a probability density: its square at any point tells you the likelihood of finding the particle there. The equation governs how that probability distribution evolves over time.
How PDEs Are Classified
Second-order PDEs (those involving second derivatives) fall into three categories based on a quantity called the discriminant, which is calculated from the equation’s coefficients. This classification matters because it determines the equation’s fundamental behavior and the methods that work to solve it.
- Elliptic equations describe steady-state situations where nothing changes over time, like the temperature distribution in a room after it has fully equilibrated. There’s no “direction of flow” in the solution.
- Parabolic equations describe diffusion processes that evolve in one time direction. The heat equation is the classic example: temperature smooths out over time and never spontaneously sharpens.
- Hyperbolic equations describe wave propagation, where information travels at finite speed. The wave equation is the prototype: a disturbance at one point takes time to reach another.
These three types have fundamentally different characters. Elliptic equations produce smooth solutions influenced by conditions everywhere on the boundary. Parabolic equations smooth out initial irregularities as time progresses. Hyperbolic equations preserve and transport sharp features. Knowing which type you’re dealing with tells you what kind of physical behavior to expect and which solution strategies will work.
Boundary and Initial Conditions
A PDE on its own isn’t enough to pin down a unique solution. You also need to specify what happens at the edges of your domain (boundary conditions) and, for time-dependent problems, what the situation looks like at the start (initial conditions).
The two most common types of boundary conditions have distinct physical meanings. A Dirichlet condition fixes the value of the unknown function at the boundary. For the heat equation, this would mean holding the ends of a rod at specified temperatures. A Neumann condition fixes the rate of change (the gradient) of the function at the boundary. For the heat equation, this would mean specifying how much heat flows in or out at the ends, with zero gradient meaning the boundary is perfectly insulated.
Choosing the right boundary conditions isn’t just a mathematical formality. It’s how you encode the physical setup of your problem. The same PDE with different boundary conditions describes completely different physical situations and produces different solutions.
PDEs Beyond Physics
PDEs extend well beyond the physical sciences. One of the most consequential PDEs in finance is the Black-Scholes equation, which determines the fair price of stock options. The unknown function is the option price, which depends on two variables: the current stock price and time until expiration. The equation captures how the option’s value changes as the stock price fluctuates and as the expiration date approaches, incorporating the stock’s volatility and the prevailing interest rate. This single PDE, developed in the 1970s, became the foundation of modern options pricing and derivative markets.
In fluid dynamics, the Navier-Stokes equations describe the motion of fluids like water and air. They govern everything from weather patterns to airplane wing design. These equations are so fundamental, yet so difficult to analyze, that proving whether smooth solutions always exist remains one of seven Millennium Prize Problems identified by the Clay Mathematics Institute, carrying a million-dollar reward. As the Institute notes, “there is no proof for the most basic questions one can ask: do solutions exist, and are they unique?” We use these equations daily in engineering simulations, but their deepest mathematical properties remain unresolved.
Linear vs. Nonlinear PDEs
A PDE is linear if the unknown function and its derivatives appear only to the first power and aren’t multiplied together. The heat equation, wave equation, and Schrödinger equation are all linear. Linear PDEs have a powerful property called superposition: if you find two solutions, their sum is also a solution. This makes them far more tractable.
Nonlinear PDEs, where the unknown function or its derivatives appear in products or powers, are a different story entirely. The Navier-Stokes equations are nonlinear because fluid velocity appears multiplied by its own derivatives. This nonlinearity is what makes turbulence so difficult to predict and is closely related to why the Millennium Prize Problem remains open. Other nonlinear PDEs describe phenomena like shock waves and solitons (waves that maintain their shape over long distances). Nonlinear PDEs rarely have exact solutions, and even proving that solutions exist can be a major mathematical achievement.
How PDEs Are Solved Numerically
Most PDEs that arise in practice can’t be solved with pencil and paper. Instead, engineers and scientists use numerical methods that approximate the solution on a computer. The two most widely used approaches are fundamentally different in philosophy.
The finite difference method replaces derivatives with ratios of differences between nearby points on a grid. It’s conceptually simple: you lay down a mesh of points across your domain and approximate each derivative by looking at the difference in values between neighboring points. This works well for problems on simple, regular shapes like rectangles and boxes.
The finite element method takes a different approach. Instead of approximating derivatives directly, it works with a reformulation of the PDE that describes the solution’s overall behavior across the domain. The domain gets divided into small elements (often triangles in two dimensions), and the solution is approximated by simple functions within each element. This approach handles complex, irregular shapes much more naturally. It’s also better at accommodating Neumann-type boundary conditions, which can be tricky for finite differences near curved or irregular boundaries.
The finite element method requires less smoothness from the solution (only two derivatives, compared to four for finite differences in standard analyses), which matters for problems where the solution has sharp features. In practice, finite elements dominate in structural engineering, aerodynamics, and any field where complex geometry is the norm, while finite differences remain popular for problems on regular grids, like weather simulation and image processing.