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This formula is valid only for t > 0. Now use (7.3), expressed as a limit as follows.
+∞ −p 2 π
R √
If x > 0, 1 = lim t→+0 Q = c 1 e dp + c 2 = c 1 + c 2 .
0 2 √
R −∞ −p 2 π
If x < 0, 0 = lim t→+0 Q = c 1 e dp + c 2 = −c 1 + c 2 .
0 √ 2
1
This determines the coefficients c 1 = 1/ π and c 2 = . Therefore, Q is the function
2
√
Z x/ 4kt
1 1 2
Q(x, t) = + √ e −p dp (7.5)
2 π 0
for t > 0. Notice that it does indeed satisfy (7.1), (7.2), and (7.4).
Step 4. Having found Q, we now define S = ∂Q/∂x. (The explicit formula for S
will be written below.) By property 2, S is also a solution of (7.1). Given any function
φ, we also define
Z ∞
u(x, t) = S(x − y, t)φ(y)dy for t > 0. (7.6)
−∞
By property 4, u is another solution of (7.1). We claim that u is the unique solution of
(7.1), (7.2). To verify the validity of (7.2), we write

∞ ∂Q ∞ ∂
Z Z
u(x, t) = (x − y, t)φ(y)dy = − (Q(x − y, t))φ(y)dy =
−∞ ∂x −∞ ∂y
∞
Z
0
= + Q(x − y, t)φ (y)dy − Q(x − y, t)φ(y) y=+∞
y=−∞
−∞
upon integrating by parts. We assume these limits vanish. In particular, let’s temporary
assume that φ(y) itself equals zero for |y| large. Therefore,
Z ∞ Z x
0
0
u(x, 0) = Q(x − y, 0)φ (y)dy = φ (y)dy = φ x = φ(x).
−∞
−∞ −∞
because of the initial condition for Q and the assumption that φ(−∞) = 0. This is the
initial condition (7.2). We conclude that (7.6) is our solution formula, where

∂Q 1 2
S = = √ e −x /4kt for t > 0. (7.7)
∂x 2 πkt

That is,
1 Z ∞ 2
u(x, t) = √ e −(x−y) /4kt φ(y)dy. (7.8)
4πkt −∞

S(x, t) is known as the source function, Green’sfunction, fundamental solution, gaussian,
or propagator of the diffusion equation, or simply the diffusion kernel. It gives the solution
of (7.1), (7.2) with any initial datum φ. The formula only gives the solution for t > 0.
When t = 0 it makes no sense.
The source function S(x, t) is defined for all real x and for all t > 0. S(x, t) is positive
and is even in x [S(−x, t) = S(x, t)]. For large t, it is very spread out. For small t it is a
very tall thin spike (a ”delta function”) of height (4πkt) −1/2 . The area under its graph is

∞ 1 ∞ 2
Z Z
S(x, t)dx = e −q dq = 1
−∞ 4π −∞

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