Separation of Variables: Heat, Wave, and Laplace Equations

Three second-order linear partial differential equations organize most of mathematical physics, one for each way a field can evolve in space and time.

• Heat equation, ${\alpha }^2{u}_{xx}=u_t$: diffusive processes that smooth and decay.
• Wave equation, $a^2{u}_{xx}=u_{tt}$: oscillatory processes that propagate without loss.
• Laplace's equation, $u_{xx}+u_{yy}=0$: steady states, the time-independent limit of the other two.

One method solves all three. Separation of variables, used by d'Alembert, Daniel Bernoulli, and Euler around 1750, replaces the partial differential equation with ordinary differential equations whose solutions a Fourier series reassembles.

Heat conduction in a rod

Take a uniform bar of length $L$ with insulated sides, so temperature $u(x,t)$ depends only on position and time. It satisfies

where the thermal diffusivity ${\alpha }^2=\kappa /(\rho s)$ collects the conductivity, density, and specific heat.¹ The ends are held at zero temperature and the initial profile is prescribed:

This is an initial value problem in $t$ and a boundary value problem in $x$: one condition at each end of the bar, one condition at the start of time.

The separation argument

Look for solutions that are a product of a function of $x$ alone and a function of $t$ alone,

Substituting into ${\alpha }^2{u}_{xx}=u_t$ gives ${\alpha }^2{X}^{\prime \prime }T=X{T}^{\prime }$. Divide by ${\alpha }^{2}XT$:

The left side depends only on $x$, the right only on $t$. For two functions of independent variables to be equal everywhere, both must equal the same constant. Call it $-\lambda$. The single equation splits into two.

Solving the pieces

Substituting $u=XT$ into the boundary conditions forces $X(0)=0$ and $X(L)=0$. The problem for $X$,

is the eigenvalue problem with zero endpoints. Its eigenvalues and eigenfunctions are

For each ${\lambda }_n$ the time equation $T^{\prime }+{\alpha }^2{\lambda }_{n}T=0$ gives exponential decay, $T_n(t)\propto \exp (-n^2{\pi }^2{\alpha }^{2}t/L^2)$. Multiplying the two factors yields the fundamental solutions

each satisfying the equation and both boundary conditions. Because the problem is linear and homogeneous, any sum of them does too. Superpose them and choose the weights to match the initial profile:

At $t=0$ the exponentials are $1$, so $u(x,0)=\sum c_n\sin (n\pi x/L)=f(x)$. This is the Fourier sine series of $f$, and its coefficients are

Each harmonic decays at its own rate: the exponent grows like $n^2$, so high modes vanish almost immediately and the profile smooths toward the fundamental, then to zero. Diffusion erases the finest detail first.

Nonhomogeneous ends and insulation

Two variations widen the method's reach, both reducing to the case just solved.

Insulated ends. No heat flows through an end when $u_x=0$ there. The boundary conditions $u_x(0,t)=u_x(L,t)=0$ change the eigenvalue problem to $X^{\prime }(0)=X^{\prime }(L)=0$, whose eigenfunctions are cosines: $X_n(x)=\cos (n\pi x/L)$ for $n=0,1,2,\dots$, including the constant $X_0=1$. The solution is a Fourier cosine series,

and the constant term $c_0/2=\frac{1}{L}{\int }_0^{L}f\mathbf{d}x$ is the average initial temperature, the value the insulated bar settles to since no heat escapes.

The wave equation

The vibrating string carries the same eigenvalue problem but a different time factor. Let $u(x,t)$ be the transverse displacement of a string of length $L$ stretched to tension $T$ with mass density $\rho$. Small motions obey

with fixed ends $u(0,t)=u(L,t)=0$. Being second order in time, the equation needs two initial conditions, the shape and the velocity: $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$.³ Released from rest ($g=0$), separation $u=XT$ gives the same $X^{\prime \prime }+\lambda X=0$ with the same sine eigenfunctions, but now $T^{\prime \prime }+a^2\lambda T=0$, whose solutions oscillate rather than decay. With $T^{\prime }(0)=0$ the time factor is $\cos (n\pi at/L)$, and the fundamental solutions are

The general free motion superposes the modes,

with $c_n$ the Fourier sine coefficients of the initial shape $f$. These are the harmonics of a plucked string; the fundamental sets the pitch and the higher modes the timbre, the same structure met in mechanical vibrations.

Unlike the heat series, the wave series has no decaying factor, so it cannot be differentiated term by term without care. An equivalent closed form resolves this. Writing $h$ for the odd periodic extension of $f$,

d'Alembert's form of 1746. It shows the solution as two copies of the initial shape traveling in opposite directions at speed $a$. In contrast to diffusion, any corner or jump in the initial data is preserved for all time rather than smoothed. A nonzero initial velocity $g$ contributes a companion series with $\sin (n\pi at/L)$ in place of the cosine, and the general problem is the sum of the two, by superposition.

Laplace's equation

Remove the time dependence and the heat equation's steady state satisfies Laplace's equation, $u_{xx}+u_{yy}=0$. The same equation governs electrostatic potential in a charge-free region, gravitational potential in free space, and the velocity potential of an ideal fluid, so it is also called the potential equation. With no time variable there are no initial conditions; instead one condition is imposed at every point of the boundary of the region.

Dirichlet problem for a rectangle

Solve $u_{xx}+u_{yy}=0$ on $0<x<a$, $0<y<b$ with $u=0$ on three sides and $u(a,y)=f(y)$ on the fourth. Separation $u=X(x)Y(y)$ gives, with separation constant $\lambda$,

The homogeneous conditions act on $Y$: $Y(0)=Y(b)=0$ forces the familiar eigenvalues $\lambda =(n\pi /b{)}^2$ and eigenfunctions $\sin (n\pi y/b)$. The $X$-equation is then $X^{\prime \prime }-(n\pi /b{)}^{2}X=0$ with $X(0)=0$, solved by $\sinh (n\pi x/b)$ rather than an exponential, so that the endpoint condition is easy to impose. The fundamental solutions are $u_n=\sinh (n\pi x/b)\sin (n\pi y/b)$, and

with $c_n\sinh (n\pi a/b)$ the Fourier sine coefficients of $f$ on $[0,b]$.

Dirichlet problem for a disk

On a circular region $r<a$ with boundary data $u(a,\theta )=f(\theta )$, Laplace's equation in polar coordinates is

Separation $u=R(r)\Theta (\theta )$ yields $r^2{R}^{\prime \prime }+r{R}^{\prime }-\lambda R=0$, an Euler equation, and ${\Theta }^{\prime \prime }+\lambda \Theta =0$. There are no homogeneous boundary conditions here; instead $\Theta$ must be periodic with period $2\pi$ and $u$ must stay bounded as $r\to 0$. Periodicity forces $\lambda =n^2$ with $\Theta$ a combination of $\cos n\theta$ and $\sin n\theta$; boundedness discards the $r^{-n}$ solution of the Euler equation, leaving $R=r^n$. The solution is a full Fourier series in $\theta$ with radial weights,

where $a^n{c}_n$ and $a^n{k}_n$ are the Fourier coefficients of $f$ on $[0,2\pi )$. Both sine and cosine terms appear because the boundary data spans a full period.

The three equations side by side

The method never changes; only the time factor and the region do.

In every case the spatial eigenvalue problem $X^{\prime \prime }+\lambda X=0$ underlies the solution, and its eigenfunctions form the basis for the series. Whether a given boundary value problem has such a real, orthogonal, complete set of eigenfunctions is a property of the self-adjoint Sturm-Liouville form.