Intuition The one core ideaHeat spreads because every point drifts toward the average of its neighbours, and the more sharply the temperature bends at a point, the faster that point changes. Separation of variables trades one hard equation about space-and-time-together for two easy equations — one purely about shape in space , one purely about fading in time .
Before you can read a single line of the parent derivation, you need to own every squiggle in it. This page walks through them one at a time, in the order they build on each other. Nothing is used before it is drawn.
u ( x , t ) u temperature . It depends on two inputs:
x position along the rod (how far from the left end), a number between 0 L t time (how many seconds have passed), a number ≥ 0 u ( x , t ) "How hot is the point x t
The picture is not a curve — it is a whole landscape . Imagine the rod laid on the ground (x t
Intuition Why two variables at all?A single hot rod already needs x moves , so the picture changes as t
Definition Derivative — the steepness of a curvePick a curve. Its derivative at a point is the slope of the line that just kisses the curve there (the tangent): how fast the height rises per step to the right.
Slope > 0
Slope < 0
Slope = 0
Why do we need slope? Because heat flows down slopes of temperature — from hot to cold. The slope is literally the "steepness of the hill of heat," and physics says heat rolls downhill.
Because u two inputs, "the slope" is ambiguous — slope in which direction? So we invent slopes that hold one input frozen.
∂ partial derivative with a curly-d, ∂ d d ∂ this input changes while the others are held still."
∂ t ∂ u only time and freezing x u t ∂ x ∂ u only position , freezing t u x Do it twice in x ∂ x 2 ∂ 2 u u xx u t u x u xx ∂ u / ∂ t exactly the same thing — the subscript is just shorthand.
u t u t = ∂ t ∂ u stand still at one position x It is the slope of the temperature-versus-time graph at that spot. Positive u t
u xx bump (concave-down) the point is hotter than the average of its two neighbours , so heat leaks out to both sides — it cools. At a dip (concave-up) it is colder than its neighbours, so heat flows in — it warms. Curvature u xx u xx u x
u xx Why it's wrong: u x direction heat flows; but a straight tilted line (u x = 0 u xx = 0 bending (u xx = 0
α 2 α 2 positive number that says how good this material is at conducting heat . Big α 2 α 2 length 2 / time α 2
α 2 α square guarantees it is positive (a real number squared is never negative). Physically, diffusion is always positive, so the notation bakes that in.
The equation alone has infinitely many solutions. We pin down one with extra facts.
Definition The rod's lengthL L length of the rod . Position ranges over x ∈ [ 0 , L ] 0 L included — we will need them to state the rules at the ends). Time is t ≥ 0
Definition Boundary conditions (BCs) — what happens at the endsRules fixed at x = 0 x = L for all time . Here:
u ( 0 , t ) = 0 , u ( L , t ) = 0. 0 ∘ Dirichlet type: value fixed. Contrast: fixing the slope u x = 0 Neumann type. See Boundary Conditions — Dirichlet vs Neumann .
Definition Initial condition (IC) — the starting snapshotOne rule fixed at t = 0 for all positions :
u ( x , 0 ) = f ( x ) . f ( x ) starting temperature profile — the shape of the rod's heat before the clock starts.
Intuition BC vs IC — read the figureThe two vertical red edges of the strip are the boundary conditions (glued at 0 bottom horizontal edge is the initial condition (the snapshot at t = 0
X ( x ) T ( t ) guess about the form of the answer:
u ( x , t ) = X ( x ) T ( t ) .
X ( x ) position only — a fixed shape of the rod (e.g. a hump). It does not know about time.T ( t ) time only — a single dial that scales that shape up or down as the clock runs. It does not know about position.
So the guess says: the rod keeps the same shape and merely brightens or dims it over time.
Intuition Why are we allowed to guess this?We are not claiming every solution looks like one shape × one dial — most don't. But it costs nothing to try the simplest possible form. If it works, a hard two-variable equation splits into two easy one-variable equations (one for X T Superposition Principle lets us add many such simple pieces to build any starting profile. The picture: freeze the shape X ( x ) T ( t )
sin sin θ 0 1 0 − 1 cos θ cos 0 = 1
Why sines here? Because the ends must be zero . sin 0 x = 0 0 x = L
Definition The mode numbern sin L nπ x X n ( x ) = sin L nπ x , n = 1 , 2 , 3 , …
n = 1 n = 2 n = 3 L nπ sin ( 0 ) = 0 and sin L nπ L = sin ( nπ ) = 0 π L X ( x )
Intuition Why cosines are banned herecos 0 = 1 = 0 u ( 0 , t ) = 0 hand-pick sines. (Flip to insulated ends and cosines return — the eigenfunction always matches the boundary rule.)
∑ ∑ n = 1 ∞ n = 1 n = 2 n = 3 X n ( x ) = sin L nπ x f ( x ) Fourier Series .
e − λ t T ( t ) e ≈ 2.718 e − λ t 1 t = 0 0 λ is the time-dial T ( t )
λ n = ( nπ / L ) 2 λ n = ( nπ / L ) 2 n 2 3 9 × 1 u xx n = 1
Definition Superposition — sums of solutions are solutionsBecause the heat equation is linear (no squares of u homogeneous (the "= 0 Superposition Principle : it lets us solve one hump-times-dial at a time, then add. The space equation for X Sturm-Liouville Theory .
B n f ( x ) = ∑ n = 1 ∞ B n sin L nπ x , B n number telling how much of hump n — the amount of the n f ( x ) B n = L 2 ∫ 0 L f ( x ) sin L nπ x d x , f n B n e − λ n α 2 t
Self-test: can you say each in one breath before reading the parent note?
What does u ( x , t ) The temperature at position x t
What does the curly ∂ ∂ u / ∂ t u t It marks a partial derivative — the slope when only one input changes and the rest are frozen; u t ∂ u / ∂ t
What is u t The rate a fixed point's temperature changes in time (slope of temp-vs-time at frozen x
What is u xx The curvature of the snapshot; u xx > 0 u xx < 0
Why is α 2 To force it positive (diffusivity is always positive) and keep units length 2 / time
State the difference between a boundary condition and an initial condition. BC fixes the ends for all time (x = 0 , L t = 0
What is the separation guess, and what do X ( x ) T ( t ) u ( x , t ) = X ( x ) T ( t ) X T
Why are we allowed to make the separation guess? It costs nothing to try the simplest form; if it works the PDE splits into two easy ODEs, and superposition lets us add many such pieces for any profile.
Why do zero-value (Dirichlet) ends force sine eigenfunctions, not cosines? sin 0 = 0 cos 0 = 1 = 0
Why must L nπ So that sin L nπ x x = 0 x = L sin nπ = 0
Why does mode n n 2 Its decay constant is λ n = ( nπ / L ) 2
What does the coefficient B n How much of hump n f ( x ) f sin L nπ x
What property lets us add single-hump solutions together? Linearity + homogeneity of the heat equation → the superposition principle.
