Mathematical Physics Vol 1

Chapter 7. Partial differential equations

358

n (= 2 ) ∑ i , j = 1

∂ 2 ξ i ∂ x k ∂ x l

k = l = 1 , k = 1 , l = 2 k = l = 2

∂ 2 u ∂ x k ∂ x l

∂ 2 u ∂ξ i ∂ξ j

∂ξ j ∂ x l

∂ξ i ∂ x k

∂ u ∂ξ i

(7.98)

=

+

,

Now the transformed equation (7.83) becomes

n (= 2 ) ∑ i , j = 1

n (= 2 ) ∑ i = 1

∂ 2 u ∂ξ i ∂ξ j

∂ u ∂ξ i

+ ¯ bu = 0 .

¯ L ≡

¯ a i j

¯ a i

(7.99)

+

The relation between new and old coefficients is given by the following equations

n (= 2 ) ∑ i , j = 1

n (= 2 ) ∑ i = 1

n (= 2 ) ∑ i , j = 1

∂ 2 ξ k ∂ x i ∂ x j

∂ξ k ∂ x i

∂ξ l ∂ x j

∂ξ k ∂ x i

¯ a kl =

a i j

¯ a k =

a i

a i j

(7.100)

,

+

.

The general form of equation (7.83) is an equation that is linear only with respect to second derivatives n (= 2 ) ∑ i , j = 1 a i j ∂ 2 u ∂ x i ∂ x j + b = 0 , (7.101) where coefficients b and a i j are function of the form

b = b ( x 1 , x 2 , u , ∂ u / ∂ x 1 , ∂ u / ∂ x 2 ) ; a i j = a i j ( x 1 , x 2 ) .

(7.102)

The transformed form is

n (= 2 ) ∑ i , j = 1

∂ 2 u ∂ξ i ∂ξ j

+ ¯ b = 0 .

¯ a i j

(7.103)

We choose new variables so that the transformed equation is as simple as possible, for example, that some of the coefficients ¯ a i j are equal to zero. To this end, and considering the relations between the new and old coefficients (7.100), we observe the following first order partial equation n (= 2 ) ∑ i , j = 1 a i j ∂ z ∂ x i ∂ z ∂ x j = 0 , (7.104) which, given that a 12 = a 21 , can be represented also as the following set of two linear partial equations ∂ z ∂ x 1 = − a 12 ∓ q a 2 12 − a 11 a 22 a 11 ∂ z ∂ x 2 . (7.105) To these equations corresponds the system of ordinary differential equations

a 12 ∓ q a 2 a 11

12 − a 11 a 22

= −

d x 2 d x 1

(7.106)

,

or

a 22 · d x (7.107) Equations (7.106), that is, (7.107), are called characteristic equations of the partial differ ential equation (7.101). 2 1 − 2 a 12 · d x 1 d x 2 + a 11 · d x 2 2 = 0 .