430 Chapter 7 Numerical Methods
with the solution of the problem consisting of the equation
∂^2 u
∂x^2− 16 u=∂u
∂t, 0 <x< 1 , 0 <t,with the same initial and boundary conditions. Use x= 1 /4, t=
1 /48 in both cases.
8.In Exercise 7, what is the longest stable time step for each of the two
problems?
9.Solve for several time levels using x= 1 /5andr= 1 /2. What is t?
∂^2 u
∂x^2=∂u
∂t, 0 <x< 1 , 0 <t,
u( 0 ,t)= 25 t, u( 1 ,t)= 0 , 0 <t,
u(x, 0 )= 0 , 0 <x< 1.10.Same as Exercise 9, except the second boundary condition is∂∂ux( 1 ,t)=0.
11.This problem describes the displacement of a string whose end is jerked:
∂^2 u
∂x^2 =∂^2 u
∂t^2 ,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)= 1 , 0 <t,
u(x, 0 )= 0 ,∂u
∂t(x,^0 )=^0 ,^0 <x<^1.
Solve numerically through one period (untilt=2) with x= t=
1 /4.
12.Same problems as Exercise 11, except the right-hand boundary condi-
tion isu( 1 ,t)=h(t),0<t,whereh(t)={ 1 , 0 <t≤1,
0 , 1 <t≤2,
andh(t+ 2 )=h(t). Solve numerically with x= t= 1 /4 for enough
values oftso that resonance becomes noticeable.
13.Using x= y= 1 /4, find a numerical solution of this problem:
∂^2 u
∂x^2+∂
(^2) u
∂y^2
= 0 , 0 <x< 1 , 0 <y< 1 ,
u(x, 0 )= 0 , u(x, 1 )=^2
π
tan−^1
(
1
x)
, 0 <x< 1 ,u( 0 ,y)= 1 , u( 1 ,y)=π^2 tan−^1 (y), 0 <y< 1.