Solve the following initial value problem: u t + u x = v , v...

klipbodok6

klipbodok6

Answered

2022-07-02

Solve the following initial value problem:
u t + u x = v , v t + v x = u , u ( 0 , x ) = u 0 ( x ) , v ( 0 , x ) = v 0 ( x ) .

Answer & Explanation

trantegisis

trantegisis

Expert

2022-07-03Added 20 answers

u t t + u x t = v t
u x t + u x x = v x
v t + v x = u t t + 2 u x t + u x x = u
u t t + 2 u x t + u x x + u = 0
Let { p = x + t q = x t
Then u x = u p p x + u q q x = u p + u q
2 u x 2 = x ( u p + u q ) = p ( u p + u q ) p x + q ( u p + u q ) q x = 2 u p 2 + 2 u p q + 2 u p q + 2 u q 2 = 2 u p 2 + 2 2 u p q + 2 u q 2
u t = u p p t + u q q t = u p u q
2 u x t = x ( u p u q ) = p ( u p u q ) p x + q ( u p u q ) q x = 2 u p 2 2 u p q + 2 u p q 2 u q 2 = 2 u p 2 2 u q 2
2 u p 2 2 2 u p q + 2 u q 2 + 2 ( 2 u p 2 2 u q 2 ) + 2 u p 2 + 2 2 u p q + 2 u q 2 + u = 0
4 2 u p 2 + u = 0
u ( p , q ) = f ( q ) sin p 2 + g ( q ) cos p 2
u ( t , x ) = f ( x t ) sin x + t 2 + g ( x t ) cos x + t 2
u t ( t , x ) = f t ( x t ) sin x + t 2 + f ( x t ) 2 cos x + t 2 g t ( x t ) cos x + t 2 g ( x t ) 2 sin x + t 2
u x ( t , x ) = f x ( x t ) sin x + t 2 + f ( x t ) 2 cos x + t 2 + g x ( x t ) cos x + t 2 g ( x t ) 2 sin x + t 2
v ( t , x ) = f x t ( x t ) sin x + t 2 + g x t ( x t ) cos x + t 2 + f ( x t ) cos x + t 2 g ( x t ) sin x + t 2
Hence
{ u ( t , x ) = f ( x t ) sin x + t 2 + g ( x t ) cos x + t 2 v ( t , x ) = f x t ( x t ) sin x + t 2 + g x t ( x t ) cos x + t 2 + f ( x t ) cos x + t 2 g ( x t ) sin x + t 2

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