1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 431

14.The analytical solution of the problem in Exercise 13 is u(x,y)=
( 2 /π )tan−^1 (y/x). Compare your numerical results with the exact so-
lution.

15.Using x= y= 1 /4andr= 1 /4, find a numerical solution for this
problem:

∇^2 u=∂∂ut, 0 <x< 1 , 0 <y< 1 , 0 <t,

u(x, 0 ,t)=u(x, 1 ,t)= 0 , 0 <x< 1 , 0 <t,

u( 0 ,y,t)=u( 1 ,y,t)= 0 , 0 <y< 1 , 0 <t,

u(x,y, 0 )= 1 , 0 <x< 1 , 0 <y< 1.

16.TheanalyticalsolutionoftheprobleminExercise15is

u(x,y,t)=

∑∞

n= 1

∑∞

n= 1

4 ( 1 −cos(nπ ))( 1 −cos(mπ))

π^2 mn

×sin(nπx)sin(nπy)e−(m^2 +n^2 )π^2 t.

Using just the termm=n=1 of this solution, compare the ratio

R=

u

( 1

2 ,

1

2 ,tm+^1

)

u

( 1

2 ,

1

2 ,tm

)

and the ratio of the correspondingu’s computed in Exercise 15.