Introduction to PDEs

A New Kind of Equation

Until now, every differential equation we have studied involved functions of a single variable. Whether tracking population growth, analyzing oscillations, or solving systems of interacting quantities, the unknown function always depended on one independent variable, typically time.

But physical reality is richer than this. Temperature varies across space and changes over time. Waves propagate through a medium, their amplitude depending on both position and the moment of observation. Electric potential in a region depends on all three spatial coordinates.

To describe such phenomena, we need equations involving partial derivatives: derivatives with respect to multiple independent variables. These are partial differential equations, or PDEs.

The shift from ODEs to PDEs is not merely a matter of notation. It represents a fundamental change in the nature of solutions and the methods needed to find them.

From One Variable to Many

Consider the simple first-order ODE we know well:

The solution $y(t) = Ce^{-kt}$ is a function of one variable. At each instant $t$, there is exactly one value $y$. The solution is a curve.

Now imagine heat flowing through a rod. The temperature $u$ depends on where you measure along the rod (position $x$) and when you measure (time $t$). The temperature is a function of two variables: $u = u(x, t)$.

When we differentiate $u$ with respect to $t$ while holding $x$ fixed, we write $\partial u / \partial t$. When we differentiate with respect to $x$ while holding $t$ fixed, we write $\partial u / \partial x$. These are partial derivatives.

The ODE solution is a curve: at each time, one value. The PDE solution is a surface (or rather, a family of curves that evolve in time): at each position and each time, one value. This is why PDEs are fundamentally harder: we must track how an entire spatial profile changes, not just a single number.

The Heat Equation: Diffusion

Place a hot spot on a metal rod. What happens? The heat spreads out. Hot regions cool down, cold regions warm up, and eventually the temperature becomes uniform throughout. This smoothing process is diffusion.

The heat equation (or diffusion equation) captures this behavior:

Here $u(x, t)$ is temperature at position $x$ and time $t$. The constant $\alpha^2$ is the thermal diffusivity of the material.

The equation says: the rate at which temperature changes at a point equals a constant times the spatial curvature of the temperature profile. Where the profile curves downward (a local maximum), heat flows away and temperature decreases. Where it curves upward (a local minimum), heat flows in and temperature increases.

The heat equation erases sharp features. Discontinuities in initial conditions get smoothed out instantly. High-frequency oscillations decay rapidly while low-frequency modes persist longer. This is the mathematical essence of diffusion.

The Wave Equation: Propagation

Pluck a guitar string. The disturbance does not diffuse away; it travels back and forth, maintaining its shape. This is wave motion, fundamentally different from diffusion.

The wave equation describes oscillations and traveling waves:

Here $u(x, t)$ might be the displacement of a string, the pressure variation in a sound wave, or the electric field in electromagnetic radiation. The constant $c$ is the wave speed.

The key difference from the heat equation is the second time derivative. The heat equation is first-order in time; the wave equation is second-order. This second derivative allows for oscillation: not just approach to equilibrium, but genuine back-and-forth motion.

The wave equation preserves information. Initial disturbances split into left-traveling and right-traveling waves, each maintaining its shape as it propagates. The French mathematician d'Alembert showed that the general solution has the form $u = f(x - ct) + g(x + ct)$: any profile traveling right at speed $c$, plus any profile traveling left at speed $c$.

The Laplace Equation: Equilibrium

Sometimes we care not about how things evolve, but about the final steady state. What is the temperature distribution after a system has reached equilibrium? What is the electric potential in a region with given boundary conditions?

The Laplace equation describes these equilibrium situations:

There is no time derivative at all. The solution $u(x, y)$ does not evolve; it simply is. The equation says that the function equals its own average over any small circle around each point. Functions satisfying this are called harmonic functions.

The Laplace equation arises everywhere: electrostatics, gravitational potential, fluid flow, heat in steady state. Its solutions are completely determined by boundary values. Specify the temperature on the walls of a room, and the Laplace equation tells you the equilibrium temperature everywhere inside.

Each equation exhibits its characteristic behavior. Heat smooths. Waves propagate. Laplace equilibrates.

Why Boundary Conditions Matter More

For an ODE like $y'' + y = 0$, specifying $y(0)$ and $y'(0)$ uniquely determines the solution. Two numbers suffice because the solution space is two-dimensional.

For a PDE, the situation is fundamentally different. The solution is a function of multiple variables, and specifying its values along a boundary amounts to specifying an entire function, not just a few numbers.

Consider the heat equation on a rod of length $L$. To find $u(x, t)$, we need:

• An initial condition: $u(x, 0) = f(x)$ for $0 \le x \le L$
• Boundary conditions: what happens at $x = 0$ and $x = L$ for all $t > 0$

Different types of boundary conditions lead to qualitatively different solutions:

Dirichlet conditions specify the value of $u$ itself: "the ends of the rod are held at fixed temperatures."

Neumann conditions specify the derivative $\partial u / \partial x$: "the ends of the rod are insulated, so no heat flows through them."

Mixed conditions combine both types at different parts of the boundary.

The same equation with different boundary conditions gives different solutions. This is not a mathematical technicality: it reflects different physical setups. An insulated rod behaves differently from one whose ends are held in ice baths.

Classification: Parabolic, Hyperbolic, Elliptic

Just as quadratic equations classify into distinct types based on their discriminant, so do second-order linear PDEs. For an equation of the form:

the discriminant $B^2 - 4AC$ determines the type:

• Parabolic when $B^2 - 4AC = 0$: the heat equation
• Hyperbolic when $B^2 - 4AC > 0$: the wave equation
• Elliptic when $B^2 - 4AC < 0$: the Laplace equation

This classification is not just taxonomy. Each type has fundamentally different mathematical properties:

Parabolic equations describe diffusion processes. Information spreads in all directions simultaneously. Discontinuities are instantly smoothed. Solutions become smoother as time progresses.

Hyperbolic equations describe wave phenomena. Information travels along characteristics, curves in the $(x, t)$ plane along which disturbances propagate. Discontinuities persist but travel. The domain of dependence and influence is finite.

Elliptic equations describe equilibrium states. There is no preferred direction of information flow. The value at any point depends on values everywhere on the boundary. Boundary conditions must be specified on a closed curve surrounding the domain.

The same discriminant formula that classifies conic sections into ellipses, parabolas, and hyperbolas classifies PDEs. This is no coincidence: the characteristic curves of a PDE form a conic section in the tangent space.

Initial and Boundary Value Problems

The type of PDE determines what data we need to specify:

For the heat equation (parabolic):

• Initial condition: $u(x, 0) = f(x)$ (temperature profile at time zero)
• Boundary conditions: $u(0, t)$ and $u(L, t)$ for all $t > 0$

For the wave equation (hyperbolic):

• Initial conditions: $u(x, 0) = f(x)$ (initial displacement) and $\frac{\partial u}{\partial t}(x, 0) = g(x)$ (initial velocity)
• Boundary conditions: $u(0, t)$ and $u(L, t)$ for all $t > 0$

For the Laplace equation (elliptic):

• Boundary conditions only: $u$ specified on the entire boundary of the domain
• No initial conditions (there is no time variable)

Specifying too little data leaves the solution undetermined. Specifying too much may lead to no solution at all. The correct amount of data is dictated by the type of equation and the geometry of the domain.

Preview: Separation of Variables

How do we actually solve these equations? The most powerful technique for PDEs in simple geometries is separation of variables. The idea is elegant: assume the solution factors as a product of functions, each depending on only one variable.

For the heat equation, we try $u(x, t) = X(x)T(t)$. Substituting into the PDE and dividing by $XT$:

The left side depends only on $t$. The right side depends only on $x$. For these to be equal for all $x$ and $t$, both must equal the same constant, say $-\lambda$.

This gives us two ODEs:

We know how to solve ODEs. The boundary conditions determine which values of $\lambda$ are allowed (the eigenvalues), and the initial condition determines how to combine these solutions.

The full development of this method requires Fourier series, which we will study in an upcoming chapter. For now, the key insight is that separation of variables transforms one PDE into multiple ODEs, bringing all our previous techniques back into play.

What Lies Ahead

This chapter has introduced the landscape of partial differential equations:

• PDEs involve functions of multiple variables and partial derivatives
• The three canonical equations model diffusion (heat), propagation (wave), and equilibrium (Laplace)
• Boundary conditions are essential and must match the equation type
• Classification into parabolic, hyperbolic, and elliptic guides our intuition and methods

In the chapters that follow, we will develop the tools to solve these equations:

The Heat Equation will show separation of variables in action, leading us to Fourier sine series and the eigenvalue problem.

The Wave Equation will reveal d'Alembert's solution and the method of characteristics, showing how information propagates.

Fourier Series will provide the mathematical foundation for representing arbitrary functions as sums of sines and cosines.

The journey from ODEs to PDEs is a leap in sophistication, but the reward is the ability to model the continuous phenomena that fill the physical world: heat spreading through a solid, vibrations traveling along a string, electromagnetic waves carrying information across the void.

Key Takeaways

• Partial differential equations involve functions of multiple variables and describe phenomena where the unknown depends on both position and time (or multiple spatial dimensions)
• The heat equation $u_t = \alpha^2 u_{xx}$ models diffusion: sharp features smooth out and the system approaches equilibrium
• The wave equation $u_{tt} = c^2 u_{xx}$ models propagation: disturbances travel as waves without losing their shape
• The Laplace equation $u_{xx} + u_{yy} = 0$ models equilibrium states where the solution depends only on boundary values
• Boundary conditions (Dirichlet, Neumann, or mixed) are essential for specifying a unique solution and reflect the physical setup
• PDEs classify into three types based on the discriminant $B^2 - 4AC$: parabolic (heat), hyperbolic (wave), and elliptic (Laplace)
• Separation of variables transforms PDEs into ODEs, allowing us to apply familiar techniques combined with Fourier series