Mathematical Physics Vol 1

Chapter 7. Partial differential equations

392

The corresponding partial derivatives are ∂ξ ∂ x = − 2 + cos x , ∂ξ ∂ y = 1 ,

∂η ∂ x

∂η ∂ y

= 2 + cos x ,

= 1 ,

∂ 2 ξ ∂ x 2 ∂ 2 η ∂ x 2

∂ 2 ξ ∂ y 2 ∂ 2 ξ ∂ y 2

∂ 2 ξ ∂ x ∂ y ∂ 2 ξ ∂ x ∂ y

= − sin x , = − sin x ,

= 0 ,

= 0 ,

= 0 ,

= 0 .

Check of the Jacobian:

2 + cos x 1 2 + cos x 1

J = −

= − 4̸ = 0 .

Calculation of new coefficients:

¯ a 11 = a 11

∂ξ ∂ x

+ a 22

∂ξ ∂ y

2

2

∂ξ ∂ x

∂ξ ∂ y

+ 2 a 12

=

=( − 2 + cos x ) 2 + 2 ( − cos x )( − 2 + cos x )+( − 3 − sin 2 x )= 1 − ( cos 2 x + sin 2 x )= = 0 , ¯ a 22 = a 11 ∂η ∂ x 2 + 2 a 12 ∂η ∂ x ∂η ∂ y + a 22 ∂η ∂ y 2 = =( 2 + cos x ) 2 + 2 ( − cos x )( 2 + cos x )+( − 3 − sin 2 x )= 1 − ( cos 2 x + sin 2 x )= = 0 ,

+ a 12

∂η ∂ x

∂ξ ∂ x

∂ξ ∂ y

∂η ∂ x

∂ξ ∂ y

∂η ∂ x

∂ξ ∂ y

∂η ∂ y

¯ a 12 = a 11

+ a 22

+

=

=( − 2 + cos x )( 2 + cos x )+( − cos x )( − 2 + cos x + 2 + cos x )+( − 3 − sin 2 x )= = − 8 .

∂ 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 22

=

= − y − sin x .

From transformations

ξ = y + sin x − 2 x , η = y + sin x + 2 x ,

it follows that y + sin x = 1 / 2 ( ξ + η ) , and thus

1 2

( ξ + η ) .

¯ a 1 =

∂ 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 22

= − y − sin x =

1 2

( ξ + η ) .

=