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Say the solution is
1/2
x t
u(x, t) = sin .
t x 1/2
Making the change of variables ξ = x/t, f(ξ) = u(x, t), we could rewrite this equation
as

−1/2
f(ξ) = ξ sin ξ .
We see now that if we had guessed that the solution of this partial differential equation
was only dependent on powers of x/t we could have changed variables to ξ and f and
instead solved the ordinary differential equation

df
G , f, ξ = 0.
dξ

By using similarity methods one can reduce the number of independent variables in some
PDE’s.

Example 2.5 Consider the partial differential equation

∂u ∂u
x + t − u = 0.
∂t ∂x
One way to find a similarity variable is to introduce a transformation to the temporary
0
0
0
variables u , t , x , and the parameter λ.
0
u = u λ
0 m
t = t λ
0 n
x = x λ
where n and m are unknown. Rewriting the partial differential equation in terms of
the temporary variables,

∂u 0 ∂u 0
0
0 m
0 n
x λ λ 1−m + t λ λ 1−n − u λ = 0
∂t 0 ∂x 0
∂u 0 ∂u 0
0
x 0 λ −m+n + t 0 λ m−n − u = 0
∂t 0 ∂x 0
There is a similarity variable if λ can be eliminated from the equation. Equating the
coefficients of the powers of λ in each term,
−m + n = m − n = 0.

This has the solution m = n. The similarity variable, ξ, will be unchanged under the
transformation to the temporary variables. One choice is

0 n
t t λ t 0
ξ = = = .
0 m
x x λ x 0

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