Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 7

1.1.5 Canonical Forms

To facilitate resolving Eq. (1.2), we may transform it into a canonical form by
a transformation of independent variables.
Consider the transformation

ξ = ξ ( x , y ) , η = η ( x , y ) . (1.3)

Assume that ξ and η are twice continuously differentiable and the Jacobian

ξ x ξ y

J =

η x η y

is nonzero in the region under consideration so that x and y can be determined
uniquely from the system (1.3). By the chain rule, we have

u x = u

ξ ξ x +

u η η x , u y = u

ξ ξ y +

u η η y

2
x

2
x

u xx = u ξξ ξ + 2 u ξη ξ x η x + u ηη η + u ξ ξ xx + u η η xx ,

u xy = u

ξξ ξ x ξ y +

u

ξη ( ξ x η y + ξ y η x )+

u ηη η x η y + u

ξ ξ xy +

u η η xy ,

2
y

2
y

u yy = u

ξξ ξ

+ 2 u

ξη ξ y η y +

u ηη η + u

ξ ξ yy +

u η η yy .

Substituting these into Eq. (1.2) leads to

A 11 u ξξ + 2 A 12 u ξη + A 22 u ηη + B 1 u ξ + B 2 u η + Cu + F = 0 , (1.4)

where

2
x

2
y

A 11 = a 11 ξ + 2 a 12 ξ x ξ y + a 22 ξ ,

A 12 = a 11 ξ x η x + a 12 ( ξ x η y + ξ y η x )+ a 22 ξ y η y ,

2
x

2
y

A 22 = a 11 η + 2 a 12 η x η y + a 22 η ,

B 1 = a 11 ξ xx + 2 a 12 ξ xy + a 22 ξ yy + b 1 ξ x + b 2 ξ y ,

B 2 = a 11 η xx + 2 a 12 η xy + a 22 η yy + b 1 η x + b 2 η y ,

C = c , F = f .

The resulting equation (1.4) is in the same form as the original equation (1.2)
under the general transformation (1.3). The nature of the equation remains invariant
under such a transformation if the Jacobian does not vanish. This can be inferred
from the fact that the sign of the discriminant

Δ does not vary under the transforma-

tion, that is,

J 2 a 12 − a 11 a 22 ,

A 12 − A 11 A 22 =