1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

382 Chapter 6 Laplace Transform

EXERCISES

1.Find all values ofs, real and complex, for which the following functions are

zero.

a. cosh(√s);

c. sinh(s);

e.cosh(s)+ssinh(s).

b.cosh(s);

d.cosh(s)−ssinh(s);

2.Find the inverse transforms of the following functions in terms of an infi-

nite series.

a.^1

s

tanh(s); b. sinh(sx)

scosh(s)

.

3.Find the transformU(x,s)of the solution of each of the following prob-

lems.

a. ∂

(^2) u
∂x^2 =
∂u
∂t,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)=t, 0 <t,
u(x, 0 )= 0 , 0 <x< 1 ;
b. ∂
(^2) u
∂x^2
=∂u
∂t
, 0 <x< 1 , 0 <t,
u( 0 ,t)= 0 , u( 1 ,t)=e−t, 0 <t,
u(x, 0 )= 1 , 0 <x< 1.
4.Solve each of the problems in Exercise 3, inverting the transform by means
of the extended Heaviside formula.
5.Solve each of the following problems by Laplace transform methods.
a. ∂
(^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 ;
b. ∂
(^2) u
∂x^2 =
∂u
∂t,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)= 0 , 0 <t,
u(x, 0 )= 1 , 0 <x< 1.