388 Chapter 6 Laplace Transform
EXERCISES
1.Find the persistent part of the solution of the heat problem∂^2 u
∂x^2 =∂u
∂t,^0 <x<^1 ,^0 <t,
∂u
∂x(^0 ,t)=^0 ,∂u
∂x(^1 ,t)=^1 ,^0 <t,
u(x, 0 )= 0 , 0 <x< 1.2.Verify that the persistent part of the solution to Example 2 actually satisfies
the heat equation. What boundary condition does it satisfy?
3.Find the functionv(x,t)whose transform iscosh(^12 s)−cosh(
s(^12 −x))
s^2 cosh(^12 s).
What boundary value–initial value problem doesv(x,t)satisfy?
4.Solve∂^2 u
∂x^2 =∂^2 u
∂t^2 ,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)= 0 , 0 <t,u(x, 0 )= 0 ,∂u
∂t(x,^0 )=^1 ,^0 <x<^1.- a. Solve forω=π:
∂^2 u
∂x^2=∂
(^2) u
∂t^2
−sin(πx)sin(ωt), 0 <x< 1 , 0 <t,
u( 0 ,t)= 0 , u( 1 ,t)= 0 , 0 <t,
u(x, 0 )= 0 , ∂u
∂t
(x, 0 )= 0 , 0 <x< 1.
b. Examine the special caseω=π.
6.Obtain the complete solution of Example 1 and verify that it satisfies the
boundary conditions and the heat equation.