Mathematical Physics Vol 1

7.4 Linear second order PDE

359

Classification

In region S , in which coefficients a i j and b are defined, observe a point M ( x 1 , x 2 ) forwhich the following conditions are fulfilled: 1 ◦ discriminant D ≡ a 2 12 − a 11 · a 22 > 0, i.e. the equation (7.107) has two different real solutions. In that case the equation (7.101) is said to be of hyperbolic type at point M ; 2 ◦ if the D < 0, then the respective equation has conjugate complex solutions, and it is said that the equation is of elliptic type at point M ; 3 ◦ if discriminant D = 0, then the respective equation has a double real solution, and it is said that the equation is of parabolic type at point M . R Note that the type of the equation (7.101), in a region S or at a point M in that region, is invariant with respect to the transformation ξ i = ξ i ( x 1 , x 2 ) , J̸ = 0 . (7.108)

Namely, starting from relations (7.100), we can obtain

¯ D = D · J 2 .

(7.109)

Here, ¯ D ≡ ¯ a 2

12 − ¯ a 11 · ¯ a 22 , and the discriminants (new-old) are thus of the same sign.

7.4.3 Reduction to canonical form

Reduction of hyperbolic equation to canonical form

In this case D > 0, which yields two solutions of the characteristic equation

ξ i ( x 1 , x 2 )= C i , ( i = 1 , 2 ) ,

(7.110)

where C i are arbitrary constants. We chose transformations in the form

ξ i = ξ i ( x 1 , x 2 ) , ( i = 1 , 2 ) ,

(7.111)

and thus, in this case

¯ a 11 = ¯ a 22 = 0 .

(7.112)

The transformed equation (7.103) becomes

∂ 2 u ∂ξ 1 ∂ξ 2

= F ( ξ 1 , ξ 2 , u , ∂ u / ∂ξ 1 , ∂ u / ∂ξ 2 ) ,

(7.113)

¯ b 2¯ a 12 . This is the canonical form of the equation of hyperbolic type.

where F = −

By substitution

ξ 1 = u 1 + v , ξ 2 = u 1 − v ,

(7.114)

another canonical form is obtained

∂ 2 u ∂ u 2

∂ 2 u ∂ v 2

= F 1 ≡ 4 · F .

(7.115)

1 −