2.3. HEAT EQUATION 45

2.3.1. Fundamental solution.

a. Derivation of the fundamental solution. As noted in §2.2.1 an

important ﬁrst step in studying any PDE is often to come up with some

speciﬁc solutions.

We observe that the heat equation involves one derivative with respect

to the time variable t, but two derivatives with respect to the space vari-

ables xi (i = 1,... , n). Consequently we see that if u solves (1), then so

does u(λx,

λ2t)

for λ ∈ R. This scaling indicates the ratio

r2

t

(r = |x|) is

important for the heat equation and suggests that we search for a solution

of (1) having the form u(x, t) = v(

r2

t

) = v(

|x|2

t

) (t 0, x ∈

Rn),

for some

function v as yet undetermined.

Although this approach eventually leads to what we want (see Problem

13), it is quicker to seek a solution u having the special structure

(4) u(x, t) =

1

tα

v

x

tβ

(x ∈

Rn,

t 0),

where the constants α, β and the function v :

Rn

→ R must be found. We

come to (4) if we look for a solution u of the heat equation invariant under

the dilation scaling

u(x, t) →

λαu(λβx,

λt).

That is, we ask that

u(x, t) =

λαu(λβx,

λt)

for all λ 0, x ∈ Rn, t 0. Setting λ = t−1, we derive (4) for v(y) :=

u(y, 1).

Let us insert (4) into (1) and thereafter compute

(5)

αt−(α+1)v(y)

+

βt−(α+1)y

· Dv(y) +

t−(α+2β)Δv(y)

= 0

for y :=

t−βx.

In order to transform (5) into an expression involving the

variable y alone, we take β =

1

2

. Then the terms with t are identical, and

so (5) reduces to

(6) αv +

1

2

y · Dv + Δv = 0.

We simplify further by guessing v to be radial; that is, v(y) = w(|y|) for

some w : R → R. Thereupon (6) becomes

αw +

1

2

rw + w +

n − 1

r

w = 0,