Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers
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14.11 Exercises 671
8. Solve the Dirichlet problem by the explicit finite difference method
u
xx
+ u
yy
=0, 0 <x<1, 0 <y<1,
u (x , 0) = sin πx, u (x, 1) = 0 on 0 ≤ x ≤ 1,
u (x , y)=0, for x =0,x=1 and 0≤ y ≤ 1.
9. Using a square grid system with h =
1
2
, find the finite difference solution
of the Laplace equation on the quarter-disk given by
u
xx
+ u
yy
=0,x
2
+ y
2
< 1,y>0,
u (x , 0) = 0, −1 <x<1,
u (x , y)=10
2
,x
2
+ y
2
=1,y>0.
10. Find a finite difference solution of the wave problem
u
tt
− u
xx
=0, 0 <x<1,t>0,
u (0,t)=u (1,t)=0,t≥ 0,
u (x , 0) =
1
2
x (1 − x) ,u
t
(x, 0) = 0, 0 ≤ x ≤ 1.
Compare the numerical results with the exact analytical solution
u (x , t)=
2
π
3
∞
r=1
1
r
3
{1 − (−1)
r
}cos πrt sin πrx,
at various p oints.
11. Obtain a finite difference solution of the problem
u
tt
= c
2
u
xx
, 0 <x<1,t>0,
u (0,t)=sinπct, u (1,t)=0,t≥ 0,
u (x , 0) = u
t
(x, 0) = 0, 0 ≤ x ≤ 1.
12. Show that the transformation v =logu transforms the nonlinear system
v
t
= v
xx
+ v
2
x
, 0 <x<1,t>0,
v
x
(0,t)=1,v(1,t)=0,t≥ 0,
v (x, 0) = 0, 0 ≤ x ≤ 1
into the linear system
u
t
= u
xx
, 0 <x<1,t>0,
u
x
(0,t)=u (0,t) ,u(x, 1) = 1,t≥ 0,
u (x , 0) = 1, 0 ≤ x ≤ 1.
Solve th e linear system by the explicit finite difference method with the
derivative boundary condition approximated by the central difference
formula.
672 14 Numerical and Approximation Methods
13. Solve the following parabolic system by the Crank–Nicolson method
u
t
= u
xx
, 0 <x<1,t>0,
u (0,t)=u (1,t)=0,t≥ 0,
with the initial condition
(a) u (x, 0) = 1, 0 ≤ x ≤ 1,
(b) u (x, 0) = sin πx, 0 ≤ x ≤ 1.
(c) u (x, 0) = sin πx, 0 ≤ x ≤ 1
with 0 ≤ t ≤ 0.2 and in formula (14.5.2) κ =1,k = h
2
.
14. Use the Crank–Nicolson implicit method with the central difference
formula for the boundary conditions to find a numerical solution of the
differential system
u
t
= u
xx
, 0 <x<1,t>0,
u
x
(0,t)=u
x
(1,t)=−u, t ≥ 0,
u (x , 0) = 1, 0 ≤ x ≤ 1.
15. Find a numerical solution of the wave equation
u
tt
= c
2
u
xx
, 0 <x<l, t>0,
with the boundary and initial conditions
u =
1
20
u
x
at x = 0 and x = l, t > 0,
u (x , 0) = 0,u
t
(x, 0) = a sin
πx
l
0 ≤ x ≤ l.
16. Determine the function representing a curve which makes the following
functional extremum:
(a) I (y (x)) =
1
0
y
′2
+12xy
dx, y (0) = 0,y(1) = 1,
(b) I (y (x)) =
π/2
0
y
′2
− y
2
dx, y (0) = 0,y
π
2
=1,
(c) I (y (x)) =
x
1
x
0
1
x
1+y
′2
1
2
dx.
17. In the problem of tautochroneous motion, find the equation of the curve
joining the origin O and a point A in the vertical (x, y)-plane so that a
particle sliding freely from A to O under the action of gravity reaches
the origin O in the shortest time, friction and resistance of the medium
being neglected.
14.11 Exercises 673
18. In the problem of minimum surface of revolution, determine a curve
with given boundary points (x
0
,y
0
)and(x
1
,y
1
) such that rotation of
the curve about the x-axis generates a surface of revolution of minimum
area.
19. Show that the Euler equation of the variational principle
δI [u (x, y)] = δ
D
F (x, y, u, p, q, l, m, n) dx dy =0
is
F
u
−
∂
∂x
F
p
−
∂
∂y
F
q
+
∂
2
∂x
2
F
l
+
∂
2
∂x∂y
F
m
+
∂
2
∂y
2
F
n
=0,
where
p = u
x
,q= u
y
,l= u
xx
,m= u
xy
,n= u
yy
.
20. Prove that th e Euler–Lagrange equation for the functional
I =
R
F (x, y, z, u, p, q, r, l , m, n, a, b, c) dx dy dz
is
F
u
−
∂
∂x
F
p
−
∂
∂y
F
q
+
∂
2
∂z
F
r
+
∂
2
∂x
2
F
l
+
∂
2
∂y
2
F
m
+
∂
2
∂z
2
F
n
+
∂
2
∂x∂y
F
a
+
∂
2
∂y∂z
F
b
+
∂
2
∂z∂x
F
c
=0,
where ( p, q, r)=(u
x
,u
y
,u
z
), (l, m, n)=(u
xx
,u
yy
,u
zz
), and (a, b, c)=
(u
xy
,u
yz
,u
zx
).
21. In each of the following cases apply the variational principle or its sim-
ple extension with appropriate boundary conditions to derive the cor-
responding equations:
(a) F = u
2
x
+ u
2
y
+2u
2
xy
.
(b) F =
1
2
u
2
t
− α
u
2
x
+ u
2
y
− β
2
u
2
!
,
(c) F =
1
2
u
t
u
x
+ αu
2
x
− βu
2
xx
,
(d) F =
1
2
u
2
t
− α
2
u
2
xx
,
(e) F = p (x) u
′2
+
d
dx
q (x) u
2
− [r (x)+λs (x)] u
2
,
where p, q, r,ands are given functions of x,andα, β are constants.
674 14 Numerical and Approximation Methods
22. Derive the Schr¨odinger equation from the variational principle
δ
R
2
2m
ψ
2
x
+ ψ
2
y
+ ψ
2
z
+(V − E) ψ
2
dx dy dz =0,
where h =2π is the Planck constant, m is the mass of a particle moving
under the action of a force field described by the potential V (x, y, z)
and E is the total energy of the particle.
23. Derive the Poisson equation ∇
2
u = F (x, y) from the variational prin-
ciple with the functional
I (u)=
D
u
2
x
+ u
2
y
+2uF (x, y)
!
dx dy,
where u = u (x, y) is given on the boundary ∂D of D.
24. Derive th e equation of motion of a vibrating string of length l under
theactionofanexternalforceF (x, t) from the variational principle
δ
t
2
t
1
l
0
1
2
ρu
2
t
− T
∗
u
2
x
+ ρuF (x, t)
dx dt =0,
where ρ is the line density and T
∗
is the constant tension of the string.
25. The kinetic and potential energies associated with the tran sverse vibra-
tion of a thin elastic plate of constant thickness h are
T =
1
2
ρ
D
˙u
2
dx dy,
V =
1
2
µ
0
D
"
(∇u)
2
− 2(1− σ)
u
xx
u
yy
− u
2
xy
#
dx dy,
where ρ is the surface density and µ
0
=2h
3
E/3
1 − σ
2
.
Use the variational principle
δ
t
2
t
1
D
[(T − V )+fu] dx dy dt =0
to derive the equation of motion of the plate
ρ¨u + µ
0
∇
4
u = f (x, y, t) ,
where f is the transverse force per unit area acting on the plate.
26. The kinetic an d potential energies associated with the wave motion in
elastic solids are
14.11 Exercises 675
T =
1
2
D
ρ
u
2
t
+ v
2
t
+ w
2
t
dx dy dz
V =
1
2
D
"
λ (u
x
+ v
y
+ w
z
)
2
+2µ
u
2
x
+ v
2
y
+ w
2
z
+µ
(
(v
x
+ u
y
)
2
+(w
y
+ v
z
)
2
+(u
z
+ w
x
)
2
)#
dx dy dz.
Use the variational principle
δ
t
2
t
1
D
(T − V ) dx dy dz =0
to derive the equation of wave motion in an elastic medium
(λ + µ)grad divu + µ ∇
2
u = ρ u
tt
,
where u =(u, v, w) is the displacement vector.
27. From the variational principle
δ
D
Ldx dt =0 with L = −ρ
η
−h
φ
t
+
1
2
(∇φ)
2
+ gz
dz
derive the basic equation s of water waves
∇
2
φ =0, −h (x, y) <z<η(x, y, t) ,t>0,
η
t
+ ∇φ ·∇η − φ
z
=0, on z = η,
φ
z
+
1
2
(∇φ)
2
+ gz =0, on z = η,
φ
z
=0, on z = −h,
where φ (x, y, z, t) is the velocity potential, and η (x, y, t) is the free
surface displacement function in a fluid of depth h.
28. Derive the Boussinesq equation for water waves
u
tt
− c
2
u
xx
− µu
xxtt
=
1
2
u
2
xx
from the variational principle
δ
Ldxdt=0,
where L ≡
1
2
φ
2
t
−
1
2
c
2
φ
2
x
+
1
2
µφ
2
xt
−
1
6
φ
3
x
and φ is the potential for
u (u = φ
x
).
29. Determine an approximate solution of the problem of finding an ex-
tremum of the fun ctional
I (y (x)) =
1
0
y
′2
− y
2
− 2xy
dx, y (0) = y (1) = 0.
676 14 Numerical and Approximation Methods
30. Find an approximate solution of the torsion problem of a cylinder with
an elliptic base; the domain of integration D is the interior of the ellipse
with the major and minor axes 2a and 2b respectively. The associated
functional is
I (u (x, y)) =
D
∂u
∂x
− y
2
u +
∂u
∂y
+ x
2
u
dx dy.
31. Use the Rayleigh–Ritz method to find an approximate solution of the
problem
∇
2
u =0, 0 <x<1, 0 <y<1,
u (0,y)=0=u (1,y) ,
u (x , 0) = x (1 − x) .
32. Find an approximate solution of the boundary-value problem
∇
2
u =0,x>0,y>0,x+2y<2,
u (0,y)=0,u(x, 0) = x (2 − x) ,
u (2 − 2y, y)=0.
33. In the torsion problem in elasticity, the Prandtl stress function
Ψ (x, y)=ψ (x, y) −
1
2
x
2
+ y
2
satisfies the boundary value problem
∇
2
Ψ = −2inD
Ψ =0 on ∂D.
Use the Galerkin method to find an approximate solution of the problem
in a rectangular domain D = {(x, y):−a ≤ x ≤ a, −b ≤ y ≤ b}.
34. Apply the Galerkin approximation method to find the first eigenvalue
of the problem of a circular membrane of radius a governed by the
equation
∇
2
u ≡
d
2
u
dr
2
+
1
r
du
dr
= λu in 0 <r<a
u =0, on r = a.
35. Use the Rayleigh–Ritz method to find the solution in the first approxi-
mation of the problem of deformation of an elastic plate (−a ≤ x ≤ a ,
−a ≤ y ≤ a) by a parabolic distribution of tensile forces over its op-
posite sides at x =+
a. The problem is governed by th e differential
system
∇
4
u =
2α
a
2
in |x| <a and |y| <a,
u,
∂u
∂x
=(0, 0) on |x| = a,
u,
∂u
∂y
=(0, 0) on |y| = a,
14.11 Exercises 677
where U = u
0
+ u =
1
2
αy
2
1 −
1
6
y
2
+ u,and∇
4
U =0.
36. Show that the Kantorovich solution of the torsion problem in exercise
33 is
Ψ
1
(x, y)=
1
2
b
2
− y
2
1 −
cosh kx
cosh ka
,k=
1
b
5
2
.
37. (a) If the functional I in (14.6.3) depends on two functions u and v,
that is,
I (u, v)=
b
a
F (x, u, v, u
′
,v
′
) dx,
show that there are two Euler–Lagrange equations for this functional
∂F
∂u
−
d
dx
∂F
∂u
′
=0, and
∂F
∂v
−
d
dx
∂F
∂v
′
=0.
(b) Generalize the above result for the functional
I (u)=
b
a
F (x, u, u
′
) dx,
where u =(u
1
,u
2
,...,u
n
), u
i
∈ C
2
([a, b]), and u
i
(a)=a
i
and u
i
(b)=
b
i
, i =1,2,..., n.
38. Show that the Euler–Lagrange equation for the functional
I [u (x, y)] =
D
1+u
2
x
+ u
2
y
1
2
dx dy
is
1+u
2
y
u
xx
− 2u
x
u
y
u
xy
+
1+u
2
x
u
yy
=0.
39. Show that the Euler–Lagrange equation for the functional
I (u)=
D
F (x, y, u, u
x
,u
y
,u
xx
,u
xy
,u
yy
) dx dy
is
∂F
∂u
−
∂
∂x
(F
u
x
) −
∂
∂y
F
u
y
+
∂
2
∂x
2
(F
u
xx
)+
∂
2
∂x∂y
F
u
xy
+
∂
2
∂y
2
F
u
yy
=0.
678 14 Numerical and Approximation Methods
40. Derive the Euler–Lagrange equation for the f unction al
I [y (x)] =
b
a
F (x, y, y
′
) dx
where
(a) F (x, y, y
′
)=u (x, y)
1+y
′2
,
(b) F (x, y, y
′
)=
1
√
2g
1+y
′2
y
1
−y
1
2
with y (a)=y
1
and y (b)=y
2
<y
1
, (Brachisto chrone problem).
(c) F (x, y, y
′
)=y
′2
1+y
′2
2
,
(d) F (x, y, y
′
)=
xy
3
− y
′2
+3xyy
′
.
41. Show that there are an infinite number of continuous functions with
piecewise continuous first derivatives that minimize th e functional
I [y (x)] =
2
0
y
′2
(1 + y
′
)
2
dx
with y (0) = 1 and y (2) = 0.
42. The torsion of a prismatic rod of rectangular cross section of length 2a
and width 2b is governed by
∇
2
u =2 inR = {(x, y):−a<x<a, −b<y<b}
u =0 on∂R.
(a) Find an approximate solution u
1
(x, y). Hence, calculate the tor-
sional moment M for a = b and for a = b.
(b) Find the exact classical solution for u (x, y) and the torsional mo-
ment M.
43. Find an approximate solution of the biharmonic problem
∇
4
u =0,R= {(x, y):−a<x<a, −b<y<b}
with the boundary conditions
u
xy
=0,u
yy
= c
1 −
y
2
b
2
for x =+
a,
u
xy
=0,u
xx
=0 for y =+b,
where c is a constant.
14.11 Exercises 679
44. Show that the Euler–Lagrange equation (14.6.12) can be written in the
form
d
dx
F − u
′
∂F
∂u
′
−
∂F
∂x
=0.
45. Show that the extremals of the functional
I [y (x)] =
b
a
p (x) y
′2
− q (x) y
2
!
dx
subject to the constraint
J [y (x)] =
b
a
r (x) y
2
dx =1,
are solutions of the Sturm–Liouville equation
d
dx
p (x)
dy
dx
+[q (x)+λr(x)] y =0.
46. Consider the finite element method for the wave equation
u
tt
− u
xx
=0, 0 ≤ x ≤ l, t > 0,
u (0) = u (l)=0,
with given initial conditions.
(a) Show that an appropriate requirement is that
n
i=1
A
′′
i
(t)
l
0
v
i
(x) v
j
(x) dx +
n
i=1
A
i
(t)
l
0
∂v
i
∂x
·
∂v
j
∂x
dx =0,
where j =1,2,..., n and that the approximate solution is given by
u
n
(x)=A
1
(t) v
1
(x)+...+ A
n
(t) v
n
(t)=
n
i=1
A
i
(t) v
i
(x) .
(b) Show that the finite element method leads to a system of ordinary
differential equations
B
d
2
A
dt
2
+ CA(t)=0,A(0) = D
where B and C are n × n matrices, A (t)isan-vector function and D
is an n-vector.