math 10Q fast help (partial diff questions)

I NEED VERY FAST HELP IN 10 QUESTIONS IN PARTIAL MATHS ON 11TH MAY , ARIZONA TIME 10 AM . I WILL ONLY HAVE 1 HR TO SOLVE 10Q.

please only top math experts message me and i will fix the deal

i will pay $6 each question

here are sample questions attached

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 ct +

show that

( ) ( )( )2 , , ,

0

u x t

=

.

(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 = + . (20) 8. Consider the following Sturm-Liouville problem: ( ) ( )2 0, 0 1 d xd x k x dx dx + = < < . 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.

August 19, 2022

August 19, 2022

August 19, 2022

August 19, 2022