Mathematical Physics Vol 1

7.4 Linear second order PDE

361

u ξξ + b 1 u ξ + b 2 u η + cu + f = 0 .

(7.128)

For further simplifications, let us introduce a new function v , instead of u , defined by the following relation u = e λξ + µη · v , (7.129) where λ and µ are undefined constants, which will later be chosen to make the transformed form as simple as possible. From (7.129) we obtain the following relations

∂ u ∂ξ ≡

λξ + µη · v λξ + µη · λξ + µη · v λξ + µη · v λξ + µη · v

ξ + λ · v , ( v η + µ · v ) ,

u ξ = e

(7.130)

u η = e u ξξ = e u ξη = e u ηη = e

(7.131) (7.132) (7.133) (7.134)

2 · v , 2 · v ,

ξξ + 2 λ · v ξ + λ ξη + λ · v η + µ · v ξ + λµ · v , ηη + 2 µ · v η + µ

and thus, for equations of elliptic type, we further obtain

v ξξ + v ηη +( b 1 + 2 λ ) v ξ +( b 2 + 2 µ ) v η + + λ 2 + µ 2 + b

1 λ + b 2 µ + c v + f 1 = 0 .

(7.135)

Let us now determine the coefficients λ and µ so that expressions in the first two brackets are annulled λ = − 1 2 b 1 µ = − 1 2 b 2 , (7.136)

which thus, according to (7.135), yields

v ξξ + v ηη + γ · v + f 1 = 0

(7.137)

for the elliptic type. In the previous relation we have introduced the following notation

λξ − µη

γ = λ 2 + µ 2 + b

1 λ + b 2 µ + c , f 1 = f · e −

(7.138)

.

Similarly, for the remaining two cases we obtain - for the hyperbolic type v ξη + γ · v + f 1 = 0 ,

(7.139)

or

v ξξ − v ηη + γ · v + f 1 = 0 ,

(7.140)

- for the parabolic type

v ξξ + b 2 · v η + f 1 = 0 .

(7.141)