Mathematical Physics Vol 1

Chapter 7. Partial differential equations

394

The Jacobian is

3 1 − 1 0

J = −

= 1̸ = 0 .

Calculation of new coefficients ¯ a 11 = a 11 ∂ξ ∂ x 2

+ 2 a 12

∂ξ ∂ x

∂ξ ∂ y

+ a 22

∂ξ ∂ y

2

=

=( − 3 ) 2 + 2 ( 3 )( − 3 )+ 10 = = 1 ,

∂η ∂ x

+ 2 a 12

∂η ∂ x

∂η ∂ y

+ a 22

∂η ∂ y

¯ a 22 = a 11

2

2

=

=( − 1 ) 2 + 2 ( 3 )( − 1 ) 0 + 0 = = 1 ,

¯ a 12 = a 11

∂ξ ∂ x

∂η ∂ x

+ a 12

∂η ∂ x

∂ξ ∂ x

∂η ∂ y

∂ξ ∂ y

∂ξ ∂ y

∂η ∂ y

+ a 22

+

=

=( − 3 )( − 1 )+ 3 [( − 3 ) 0 + 1 ( − 1 )]+ 10 · 0 = 3 − 3 = 0 ,

∂ 2 ξ ∂ x 2

∂ 2 ξ ∂ x ∂ y

∂ 2 ξ ∂ y 2

∂ξ ∂ x

∂ξ ∂ y

¯ a 1 = a 1

+ a 2

+ a 11

+ 2 a 12

+ a 22

=

= 1 ( − 3 )+ 3 · 1 = 0 ,

∂ 2 η ∂ x 2

∂ 2 η ∂ x ∂ y

∂ 2 η ∂ y 2

∂η ∂ x

∂η ∂ y

¯ a 2 = a 1

+ a 2

+ a 11

+ 2 a 12

+ a 22

=

= 1 ( − 1 )+ 3 · 0 = − 1 . Transformed equation u ( x , y ) → v ( ξ , η ) ¯ a 11 + 2¯ a 12 + ¯ a 22 + ¯ a 1

∂ 2 v ∂ξ 2

∂ 2 v ∂ξ∂η

∂ 2 v ∂η 2

∂ v ∂ξ

∂ v ∂η

+ ¯ bv + ¯ c = 0 ⇒

+ ¯ a 2

gains the form

∂ 2 v ∂η 2 −

∂ v ∂ξ

∂ v ∂η

= 0 .

+