Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V
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698 Chapter 10
10.12. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod has the input heat flux u
x
(0,t)= q(t)
given following. The initial temperature distribution in the rod is 0, there is no
source term in the system, and the diffusivity has the magnitude k = 1/4.
Boundary conditions:
u
x
(0,t)=−e
−t
and u(∞,t)<∞
Initial condition:
f(x) = 0
10.13. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod has the input heat flux u
x
(0,t)= q(t)
given following. The initial temperature distribution in the rod is 0, there is no
source term in the system, and the diffusivity has the magnitude k = 1/4.
Boundary conditions:
u
x
(0,t)=−t e
−t
and u(∞,t)<∞
Initial condition:
f(x) = 0
10.14. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod has the input heat flux u
x
(0,t)= q(t)
given following. The initial temperature distribution in the rod is 0, there is no
source term in the system, and the diffusivity has the magnitude k = 1/4.
Boundary conditions:
u
x
(0,t)=−sin(t) and u(∞,t)<∞
Initial condition:
f(x) = 0
10.15. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod has an input heat flux u
x
(0,t)= q(t)
given following. The initial temperature distribution in the rod is 0, there is no
source term in the system, and the diffusivity has the magnitude k = 1/4.
Boundary conditions:
u
x
(0,t)=−sin(t) e
−t
and u(∞,t)<∞
Initial condition:
f(x) = 0
Laplace Transform Methods for Partial Diff erential Equations 699
10.16. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod is held at the fixed temperature 0, and
the initial temperature distribution in the rod is 0. There is a heat source term h(x, t)
(dimension of degrees per second) in the system and the diffusivity has the
magnitude k = 1/4.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial condition:
f(x) = 0
Heat source:
h(x, t) = e
−x
10.17. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod is held at the fixed temperature 0, and
the initial temperature distribution in the rod is 0. There is a heat source term h(x, t)
in the system, and the diffusivity has the magnitude k = 1/4.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial condition:
f(x) = 0
Heat source:
h(x, t) = e
−x
e
−t
10.18. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod is held at the fixed temperature 0, and
the initial temperature distribution in the rod is 0. There is a heat source term h(x, t)
in the system and the diffusivity has the magnitude k = 1/4.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial condition:
f(x) = 0
Heat source:
h(x, t) = Dirac(x −1) e
−t
700 Chapter 10
10.19. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod is held at the fixed temperature 0, and
the initial temperature distribution in the rod is 0. There is a heat source term h(x, t)
in the system, and the diffusivity has the magnitude k = 1/4.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial condition:
f(x) = 0
Heat source:
h(x, t) = e
−x
sin(t)
10.20. Consider a thin rod whose lateral surface is insulated over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the rod is held at the fixed temperature 0, and
the initial temperature distribution in the rod is 0. There is a heat source term h(x, t)
in the system, and the diffusivity has the magnitude k = 1/4.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial condition:
f(x) = 0
Heat source:
h(x, t) = sin(x) sin(t)
Laplace Transform Methods on the Wave Equation
We now consider transverse wave propagation in taut strings and longitudinal wave
propagation in rigid bars over the one-dimensional semi-infinite interval I ={x |0 <x<∞}.
We allow for an external force to act and we assume no damping in the system. The
nonhomogeneous partial differential equation in rectangular coordinates reads
∂
2
∂t
2
u(x, t) = c
2
∂
2
∂x
2
u(x, t)
+h(x, t)
with the two initial conditions
u(x, 0) = f(x) and u
t
(x, 0) = g(x)
Laplace Transform Methods for Partial Diff erential Equations 701
For Exercises 10.21 through 10.46, we seek solutions to the preceding equation for various
initial and boundary conditions. Evaluate the wave distribution u(x, t) using the Laplace
transform method with respect to the variable t. Evaluate the inverse if possible; otherwise, use
the convolution method and leave the answer in the form of a convolution integral. When
possible, generate the animated solution for u(x, t), and plot the animated sequence for
0 <t<5.
10.21. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure. The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = e
−
x
2
and g(x) = 0
10.22. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure. The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = x e
−
x
2
and g(x) = 0
10.23. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure. The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = e
−x
702 Chapter 10
10.24. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure. The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = sin(x) e
−x
and g(x) = 0
10.25. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure. The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = Dirac(x −1)
10.26. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is constrained to move in accordance
with u(0,t)= q(t) given following. The initial displacement distribution f(x) and
initial speed distribution g(x) are given following. No external force is acting and
c = 1/2.
Boundary conditions:
u(0,t)= e
−t
and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
10.27. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is constrained to move in accordance
with u(0,t)= q(t) given following. The initial displacement distribution f(x) and
initial speed distribution g(x) are given following. No external force is acting and
c = 1/2.
Boundary conditions:
u(0,t)= sin(t) e
−t
and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
Laplace Transform Methods for Partial Diff erential Equations 703
10.28. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is constrained to move in accordance
with u(0,t)= q(t) given following. The initial displacement distribution f(x) and
initial speed distribution g(x) are given following. No external force is acting and
c = 1/2.
Boundary conditions:
u(0,t)= t e
−t
and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
10.29. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is constrained to move in accordance
with u(0,t)= q(t) given following. The initial displacement distribution f(x) and
initial speed distribution g(x) are given following. There is no external force acting
and c = 1/2.
Boundary conditions:
u(0,t)= Dirac(t) and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
10.30. Consider transverse motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is constrained to move in accordance
with u(0,t)= q(t) given following. The initial displacement distribution f(x) and
initial speed distribution g(x) are given following. No external force is acting and
c = 1/2.
Boundary conditions:
u(0,t)=−t cos(t) and u(∞,t)<∞
Initial conditions:
f(x) = 0 and
g(x) = 0
10.31. Consider longitudinal wave motion in a rigid bar over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the bar experiences the applied stress force
u
x
(0,t)= q(t) given following (Young’s modulus E = 1). The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
704 Chapter 10
Boundary conditions:
u
x
(0,t)=−sin(t) and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
10.32. Consider longitudinal wave motion in a rigid bar over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the bar experiences the applied stress force
u
x
(0,t)= q(t) given following (Young’s modulus E = 1). The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u
x
(0,t)=−e
−t
and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
10.33. Consider longitudinal wave motion in a rigid bar over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the bar experiences the applied stress force
u
x
(0,t)= q(t) given following (Young’s modulus E = 1). The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u
x
(0,t)=−t e
−t
and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
10.34. Consider longitudinal wave motion in a rigid bar over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the bar experiences the applied stress force
u
x
(0,t)= q(t) given following (Young’s modulus E = 1). The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u
x
(0,t)=−sin(t) e
−t
and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
Laplace Transform Methods for Partial Diff erential Equations 705
10.35. Consider longitudinal wave motion in a rigid bar over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the bar experiences the applied stress force
u
x
(0,t)= q(t) given following (Young’s modulus E = 1). The initial displacement
distribution f(x) and initial speed distribution g(x) are given following. No external
force is acting and c = 1/2.
Boundary conditions:
u
x
(0,t)=−Dirac(t −1) e
−t
and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
10.36. Consider transverse wave motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure, and the initial
displacement distribution f(x) and initial speed distribution g(x) are given following.
There is an external source term h(x, t) (dimension of force per unit mass) acting and
c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
External force:
h(x, t) = e
−x
e
−t
10.37. Consider transverse wave motion over a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure, and the initial
displacement distribution f(x) and initial speed distribution g(x) are given following.
There is an external source term h(x, t) (dimension of force per unit mass) acting and
c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
External force:
h(x, t) = sin(x) e
−t
706 Chapter 10
10.38. Consider transverse wave motion in a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure, and the initial
displacement distribution f(x) and initial speed distribution g(x) are given following.
There is an external source term h(x, t) (dimension of force per unit mass) acting and
c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
External force:
h(x, t) = x e
−x
e
−t
10.39. Consider transverse wave motion in a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure, and the initial
displacement distribution f(x) and initial speed distribution g(x) are given following.
There is an external source term h(x, t) (dimension of force per unit mass) acting and
c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = 0
External force:
h(x, t) = Dirac(x −1) e
−t
10.40. Consider transverse wave motion in a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure, and the initial
displacement distribution f(x) and initial speed distribution g(x) are given following.
There is an external source term h(x, t) (dimension of force per unit mass) acting and
c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = e
−x
and g(x) = 0
External force:
h(x, t) = e
−t
Laplace Transform Methods for Partial Diff erential Equations 707
10.41. Consider transverse wave motion in a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure, and the initial
displacement distribution f(x) and initial speed distribution g(x) are given following.
There is an external source term h(x, t) (dimension of force per unit mass) acting and
c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = e
−x
External force:
h(x, t) = e
−t
10.42. Consider transverse wave motion in a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure, and the initial
displacement distribution f(x) and initial speed distribution g(x) are given following.
There is an external source term h(x, t) (dimension of force per unit mass) acting and
c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = x e
−x
and g(x) = 0
External force:
h(x, t) = e
−t
10.43. Consider transverse wave motion in a taut string over the semi-infinite interval
I ={x |0 <x<∞}. The left end of the string is held secure, and the initial
displacement distribution f(x) and initial speed distribution g(x) are given following.
There is an external source term h(x, t) (dimension of force per unit mass) acting and
c = 1/2.
Boundary conditions:
u(0,t)= 0 and u(∞,t)<∞
Initial conditions:
f(x) = 0 and g(x) = x e
−x
External force:
h(x, t) = e
−t