Mathematical Physics Vol 1

Chapter 7. Partial differential equations

360

Reduction of parabolic equation to canonical form

In this case D = 0, and we obtain only one real solution of the characteristic equation ξ 1 ( x 1 , x 2 )= C 1 , where C 1 is an arbitrary constant , (7.116) and thus chose transformations in the form ξ 1 = ξ 1 ( x 1 , x 2 ) , ξ 2 = ϕ ( x 1 , x 2 ) , (7.117) where ϕ is an arbitrary function, independent of ξ 1 . In this case, we have ¯ a 11 = ¯ a 12 = 0 , ¯ a 22̸ = 0 , (7.118) and obtain the canonical form of the equation of parabolic type

¯ b ¯ a 22

∂ 2 u ∂ξ 2 2

(7.119)

= F = −

.

Reduction of elliptic equation to canonical form

Given that D < 0, the solutions of the characteristic equation are conjugate complex, which yields ξ 1 = C 1 , ξ 2 ≡ ξ ∗ 1 = C 2 , (7.120) where ξ 1 and ξ ∗ 1 are conjugate complex functions, i.e. ξ 1 = v + iw , ξ ∗ 1 = v − iw . (7.121) As, in this case ¯ a 11 = ¯ a 22 , ¯ a 12 = 0 , (7.122) for the canonical form we obtain

¯ b ¯ a 22

∂ 2 u ∂ v 2

∂ 2 u ∂ w 2

= F = −

(7.123)

+

.

Canonical form of second order LE with constant coefficients

Observe the equation of the form

a 11 u xx + 2 a 12 u xy + a 22 u yy + b 1 u x + b 2 u y + cu + f ( x , y )= 0 ,

(7.124)

where a 11 , a 12 , a 22 , b 1 , b 2 and c are constants. As shown before, this equation ca be transformed in one of the following forms elliptic type u ξξ + u ηη + b 1 u ξ + b 2 u η + cu + f = 0 ,

(7.125)

hyperbolic type

u ξξ − u ηη + b 1 u ξ + b 2 u η + cu + f = 0 ,

(7.126)

or

u ξη + b 1 u ξ + b 2 u η + cu + f = 0 ,

(7.127)

parabolic type