Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 10

may introduce the new real variables

1
2 ( ξ + η ) ,

1
2i ( ξ − η )

α = β =

so that ξ = α + i β , η = α − i β . Eq. (1.2) thus reduces to

u αα + u ββ + ψ 5 ( α , β , u , u α , u β )= 0 . (1.14)

This is called the canonical form of the elliptic equation .

Remark 1. If the coefficients in the canonical forms are constants, a further function
transformation u e k 1 ξ + k 2 η v can be used to vanish terms of the first

derivatives. Here k 1 and k 2 are undetermined coefficients.
Example. Simplify the second-order linear PDE with constant coefficients

( ξ , η )= ( ξ , η )

u xx + u xy + u yy + u x = 0 .

1
2

2

3
4 <

Solution .Since Δ = − 1 = − 0, the equation is elliptic everywhere. The

characteristic equation is
d y
d x

2

i √ 3
2

dy
dx +

d y
d x =

1 ±

− 1 = 0 r .

Its integration gives

i √ 3
2

i √ 3
2

1 + 1 −

y − x = c 1 , y − x = c 2 .

√ 3
2

x
2 ,

From a transformation of independent variables ξ = y − η = − x , we obtain

the canonical form

2 √ 3
3

2
3 u ξ −

u ξξ + u ηη − u η = 0 .

To simplify further, we introduce the new dependent variable v ( ξ , η )=

e − ( k 1 ξ + k 2 η ) ,where k 1 and k 2 are undetermined coefficients. Substituting v
into the canonical form yields

u ( ξ , η )

2 k 2 − v η + k 1 + v

2 √ 3
3

2 √ 3
3

2 k 1 − v ξ +

2
3

2
3 k 1 −

k 2 − v ξξ + v ηη + k 2 = 0 .

√ 3
3

1
3 and k 2 =

Set k 1 = so that the terms involving the first derivatives vanish.

4
9 v

Thus, the above equation is reduced to v

ξξ +

v ηη − = 0.