Every equation so far has had one independent variable. But heat spreads through
space and time, a wave depends on position and time, and an electric
potential varies across a whole region. Their unknown is a function of several variables, so
the equation involves
partial derivatives
— it is a partial differential equation (PDE).
Three linear second-order PDEs underpin an astonishing share of physics. Writing
u_t = \partial u/\partial t,
u_{xx} = \partial^2 u/\partial x^2, they are:
\underbrace{u_t = \alpha\, u_{xx}}_{\text{heat}}, \qquad \underbrace{u_{tt} = c^2\, u_{xx}}_{\text{wave}}, \qquad \underbrace{u_{xx} + u_{yy} = 0}_{\text{Laplace}}.
Heat diffuses (smooths out), waves propagate (travel undamped), and
Laplace describes the steady state when nothing changes in time. One technique cracks
all three: separation of variables.
Separating the heat equation, line by line
Solve u_t = \alpha\, u_{xx} by hunting for solutions that split
into a function of x times a function of
t.
Step 1 — assume a product.
u(x, t) = X(x)\,T(t).
Step 2 — take the derivatives the equation needs. Only
T depends on t, only
X on x:
u_t = X\,T', \qquad u_{xx} = X''\,T.
Step 3 — substitute into u_t = \alpha\, u_{xx}.
X\,T' = \alpha\,X''\,T.
Step 4 — separate the variables. Divide by
\alpha\,X\,T so each side depends on one variable only:
\frac{T'}{\alpha\,T} = \frac{X''}{X}.
Step 5 — the separation constant. The left side is a function of
t alone, the right of x alone, yet they
are equal for all x, t. A function of
t can equal a function of x only if both
equal the same constant — call it -\lambda (the sign chosen for
convenience):
\frac{T'}{\alpha\,T} = \frac{X''}{X} = -\lambda.
Step 6 — two ordinary equations fall out.
X'' = -\lambda X, \qquad T' = -\alpha\lambda\,T.
Step 7 — solve each. The space equation is our old friend with
oscillating solutions; the time equation decays exponentially:
X(x) = \sin\!\big(\sqrt{\lambda}\,x\big),\ \cos\!\big(\sqrt{\lambda}\,x\big), \qquad T(t) = e^{-\alpha\lambda t}.
Step 8 — a product solution (a "mode"). With boundary conditions
u(0,t) = u(L,t) = 0 forcing
\lambda_n = (n\pi/L)^2,
u_n(x, t) = e^{-\alpha\lambda_n t}\,\sin\!\Big(\frac{n\pi x}{L}\Big).
Step 9 — superpose into a Fourier series. The equation is linear, so sums of
modes still solve it. The general solution matching an initial temperature
u(x, 0) = f(x) is
u(x, t) = \sum_{n=1}^{\infty} b_n\, e^{-\alpha\lambda_n t}\,\sin\!\Big(\frac{n\pi x}{L}\Big),
with the b_n chosen so the t = 0 sum
equals f(x) — the Fourier coefficients. Each mode decays at its own
rate; higher modes (larger n) vanish fastest, which is why sharp
features smooth out first.
-
Heat (parabolic) u_t = \alpha\,u_{xx} —
diffusion; data smooths and decays. Modes
e^{-\alpha\lambda_n t}\sin(n\pi x/L).
-
Wave (hyperbolic) u_{tt} = c^2\,u_{xx} —
propagation at speed c; modes oscillate
\cos(\omega_n t)\sin(n\pi x/L), no decay.
-
Laplace (elliptic) u_{xx} + u_{yy} = 0 —
steady state; solutions are harmonic, set entirely by their boundary values.
-
Separation of variables: seek
u = X(x)T(t); a separation constant
-\lambda splits the PDE into ODEs, whose product solutions
superpose (a Fourier series) to fit the boundary and initial data.
Separation only ever produces the building blocks
\sin(n\pi x/L) and
\cos(n\pi x/L). The miracle — Fourier's 1822 claim, doubted at
the time — is that any reasonable initial profile f(x)
is a sum of these:
f(x) = \sum_{n=1}^{\infty} b_n \sin\!\Big(\frac{n\pi x}{L}\Big), \qquad b_n = \frac{2}{L}\int_0^L f(x)\sin\!\Big(\frac{n\pi x}{L}\Big)\,dx.
The sines are orthogonal — like perpendicular axes — so the coefficient
b_n is just the projection of f onto
the nth mode, computed by that integral. This is what makes
separation a complete method and not a lucky guess: the
boundary conditions pick which modes \lambda_n
are allowed, and the initial condition sets their weights
b_n. Specify both, and the solution is determined.