math 10Q fast help (partial diff questions)

AME 500B Final Exam (3 Hours)

5/10/20

Before beginning any problem, read the entire exam. Do the problem that seems simplest first. Please begin each problem on a separate page of your own paper. Open book and notes, but no internet. Submit to Final Submission Box

(20) 1. Consider the following wave equation for an infinite string:

( ) 2 2

2 2 2 , 0c u x tt x

 ∂ ∂ − = ∂ ∂ 

.

Using the coordinate transformation ,x ct x ctξ η≡ + ≡ −

show that ( ) ( )( )2 , , ,

0 u x tξ η ξ η

ξ η ∂

= ∂ ∂

.

(15) 2. If ( ) ( ), 0u x f x= and ( ) ( )

0

,

t

u x t g x

t =

∂ =

∂ , show the solution to the wave

equation of Problem 1 is

( ) ( ) ( ) ( )1 1, 2 2

x ct

x ct u x t f x ct f x ct dt g t

c

+

−

′ ′= + + − +   ∫ .

(20) 3. Solve the wave equation

( ) 2 2

2 2 2 , 0c u x tt x

 ∂ ∂ − = ∂ ∂ 

subject to the boundary conditions ( ) ( )0, , 0u t u L t= =

and initial conditions

( ) ( ) ( ) ( ) 0

, 0 , 0 ,

t

u x u x f x g x

t =

∂ = =

∂

to show that the solution satisfies ( ) ( ) ( ), vu x t x ct w x ct= + + − .

(10) 4. Is the polynomial 2 2( , ) 2P x y x y ixy= + −

analytic? What two changes will make this polynomial analytic? (15) 5. If f(z) is an analytic function, show that

( ) ( ) ( ) 22

2 f z f z f z

x y

  ∂ ∂   ′+ =   ∂ ∂      .

Hint: ( ) vuf z i x x ∂ ∂

′ = + ∂ ∂

(10) 6. Criticize the following argument: Since

1

0

1

1 ; 1

k

k

k

k

z z

z z

z z

∞

=

∞ −

=

= −

= + −

∑

∑

therefore

0 1 1

z z z z + =

− − .

(20) 7. Evaluate the following integral for k < 1:

( ) 2

0

1 1 cos

I d k

π

θ θ

= +  

∫ .

(20) 8. Consider the following Sturm-Liouville problem:

( ) ( )2 0, 0 1 d xd

x k x dx dx

β φ φ β  

+ = < <   

.

satisfying homogeneous BC. State at least 7 features of the solution that are known without having to solve the equation.

(25) 9.a. Directly from the solution of the following 2-D heat transfer equation:

( ) 2 2

2 2 , , 0, 0 , 0u x y t x a y bt x y  ∂ ∂ ∂

− − = < < < < ∂ ∂ ∂ 

with homogeneous BC ( ) ( ) ( ) ( ) 0, , , , 0

, 0, , , 0

u y t u a y t

u x t u x b t

= =

= =

and IC ( ) ( ) ( ), , 0u x y f x g y= ,

show that ( ) ( ) ( )1 2, , , ,u x y t u x t u y t= ,

where

( )

( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )

2

12

1 1

1

2

22

2 2

2

, 0, 0 ;

0, 0, , 0

, 0

, 0, 0

0, 0, , 0

, 0 .

u x t x a t x

u t u a t

u x f x

u y t y b t y

u t u b t

u y g y

 ∂ ∂ − = < < ∂ ∂ 

= =

=

 ∂ ∂ − = < < ∂ ∂ 

= =

=

b. Predict what the solution will be for 3D with the same BC in the third

dimension and a corresponding product IC.