390 Chapter 6 Laplace Transform
2.Find the “persistent part” of the solution of
∂^2 u
∂x^2 =∂u
∂t,^0 <x<^1 ,^0 <t,
∂u
∂x(^0 ,t)=^0 , u(^1 ,t)=t,^0 <t,
u(x, 0 )= 0 , 0 <x< 1.3.Find the complete solution of the problem in Exercise 2.
4.A solid object and a surrounding fluid exchange heat by convection. The
temperaturesu 1 andu 2 are governed by the following equations. Solve
them by means of Laplace transforms.
du 1
dt =−β^1 (u^1 −u^2 ),
du 2
dt=−β 2 (u 2 −u 1 ),
u 1 ( 0 )= 1 , u 2 ( 0 )= 0.5.Solve the following nonhomogeneous problem with transforms.
∂^2 u
∂x^2 =∂u
∂t−^1 ,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)= 0 , 0 <t,
u(x, 0 )= 0 , 0 <x< 1.6.Find the transform of the solution of the problem
∂^2 u
∂x^2 =∂u
∂t,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)= 1 , 0 <t,
u(x, 0 )= 0 , 0 <x< 1.7.Find the solution of the problem in Exercise 6 by using the extended
Heaviside formula.
8.Solve the heat problem
∂^2 u
∂x^2 =∂u
∂t,^0 <x,^0 <t,
u( 0 ,t)= 0 , 0 <t,
u(x, 0 )=sin(x), 0 <x.