1.1. EXERCISES 3
arises in the study of surfaces of constant negative curvature [28], and in the study of crystal
dislocations [29].
12. Equilibrium equations
@
xx
@x
C
@
xy
@y
C F
x
D 0
@
xy
@x
C
@
yy
@y
C F
y
D 0
(1.12)
arise in elasticity. Here,
xx
;
xy
and
yy
are normal and shear stresses, and F
x
and F
y
are body
forces [30]. ese have been used by Cox, Hill, and amwattana [31] (see also [32]) to model
highly frictional granular materials.
13. e Navier–Stokes equations
r u D 0
u
t
C u ru D
rP
C r
2
u
(1.13)
describe the velocity field and pressure of incompressible fluids. Here is the kinematic
viscosity,
u
is the velocity of the fluid parcel,
P
is the pressure, and
is the fluid density [33].
1.1 EXERCISES
1.1. Show solutions exist for the nonlinear diffusion equation
u
t
D
.
u
m
u
x
/
x
; m 2 R (1.14)
of the form u D kt
p
x
q
for suitable constants k; p; and q. Use these to obtain solutions
to
u
t
D
.
uu
x
/
x
and u
t
D
u
x
u
x
: (1.15)
1.2. Show that Fisher’s equation
u
t
D u
xx
C u.1 u/ (1.16)
admit solutions of the form u D f .x ct / where f satisfies the ordinary differential
equation (ODE)
f
00
C cf
0
C f f
2
D 0: (1.17)
Further, show exact solutions can be obtained in the form
f
D
1
a
C
be
kz
2
(1.18)