Important partial differential equations

PDES: GENERAL AND PARTICULAR SOLUTIONS
20.1 Important partial differential equations
Most of the important PDEs of physics are second-order and linear. In order to
gain familiarity with their general form, some of the more important ones will
now be briefly discussed. These equations apply to a wide variety of different
physical systems.
Since, in general, the PDEs listed below describe three-dimensional situations,
the independent variables are r and t, where r is the position vector and t is
time. The actual variables used to specify the position vector r are dictated by the
coordinate system in use. For example, in Cartesian coordinates the independent
variables of position are x, y and z, whereas in spherical polar coordinates they
are r, θ and φ. The equations may be written in a coordinate-independent manner,
however, by the use of the Laplacian operator ∇2 .
20.1.1 The wave equation
The wave equation
∇2 u =
1 ∂2 u
c2 ∂t2
(20.1)
describes as a function of position and time the displacement from equilibrium,
u(r, t), of a vibrating string or membrane or a vibrating solid, gas or liquid. The
equation also occurs in electromagnetism, where u may be a component of the
electric or magnetic field in an elecromagnetic wave or the current or voltage
along a transmission line. The quantity c is the speed of propagation of the waves.
Find the equation satisfied by small transverse displacements u(x, t) of a uniform string of
mass per unit length ρ held under a uniform tension T , assuming that the string is initially
located along the x-axis in a Cartesian coordinate system.
Figure 20.1 shows the forces acting on an elemental length ∆s of the string. If the tension
T in the string is uniform along its length then the net upward vertical force on the
element is
∆F = T sin θ2 − T sin θ1 .
Assuming that the angles θ1 and θ2 are both small, we may make the approximation
sin θ ≈ tan θ. Since at any point on the string the slope tan θ = ∂u/∂x, the force can be
written
∂2 u(x, t)
∂u(x + ∆x, t) ∂u(x, t)
≈T
∆F = T
∆x,
−
∂x
∂x
∂x2
where we have used the definition of the partial derivative to simplify the RHS.
This upward force may be equated, by Newton’s second law, to the product of the
mass of the element and its upward acceleration. The element has a mass ρ ∆s, which is
approximately equal to ρ ∆x if the vibrations of the string are small, and so we have
ρ ∆x
∂2 u(x, t)
∂2 u(x, t)
=T
∆x.
∂t2
∂x2
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20.1 IMPORTANT PARTIAL DIFFERENTIAL EQUATIONS
u
T
θ2
∆s
θ1
T
x
x + ∆x
x
Figure 20.1 The forces acting on an element of a string under uniform
tension T .
Dividing both sides by ∆x we obtain, for the vibrations of the string, the one-dimensional
wave equation
1 ∂2 u
∂2 u
= 2 2,
2
∂x
c ∂t
where c2 = T /ρ. The longitudinal vibrations of an elastic rod obey a very similar equation to
that derived in the above example, namely
∂2 u
ρ ∂2 u
=
;
∂x2
E ∂t2
here ρ is the mass per unit volume and E is Young’s modulus.
The wave equation can be generalised slightly. For example, in the case of the
vibrating string, there could also be an external upward vertical force f(x, t) per
unit length acting on the string at time t. The transverse vibrations would then
satisfy the equation
T
∂2 u
∂2 u
+ f(x, t) = ρ 2 ,
∂x2
∂t
which is clearly of the form ‘upward force per unit length = mass per unit length
× upward acceleration’.
Similar examples, but involving two or three spatial dimensions rather than one,
are provided by the equation governing the transverse vibrations of a stretched
membrane subject to an external vertical force density f(x, y, t),
2
∂2 u
∂2 u
∂ u
+
+ f(x, y, t) = ρ(x, y) 2 ,
T
∂x2
∂y 2
∂t
where ρ is the mass per unit area of the membrane and T is the tension.
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PDES: GENERAL AND PARTICULAR SOLUTIONS
20.1.2 The diffusion equation
The diffusion equation
κ∇2 u =
∂u
∂t
(20.2)
describes the temperature u in a region containing no heat sources or sinks; it
also applies to the diffusion of a chemical that has a concentration u(r, t). The
constant κ is called the diffusivity. The equation is clearly second order in the
three spatial variables, but first order in time.
Derive the equation satisfied by the temperature u(r, t) at time t for a material of uniform
thermal conductivity k, specific heat capacity s and density ρ. Express the equation in
Cartesian coordinates.
Let us consider an arbitrary volume V lying within the solid and bounded by a surface S
(this may coincide with the surface of the solid if so desired). At any point in the solid
the rate of heat flow per unit area in any given direction r̂ is proportional to minus the
component of the temperature gradient in that direction and so is given by (−k∇u) · r̂. The
total flux of heat out of the volume V per unit time is given by
dQ
−
(−k∇u) · n̂ dS
=
dt
S
=
∇ · (−k∇u) dV ,
(20.3)
V
where Q is the total heat energy in V at time t and n̂ is the outward-pointing unit normal
to S; note that we have used the divergence theorem to convert the surface integral into
a volume integral.
We can also express Q as a volume integral over V ,
Q=
sρu dV ,
V
and its rate of change is then given by
dQ
=
dt
sρ
V
∂u
dV ,
∂t
(20.4)
where we have taken the derivative with respect to time inside the integral (see section 5.12).
Comparing (20.3) and (20.4), and remembering that the volume V is arbitrary, we obtain
the three-dimensional diffusion equation
∂u
,
∂t
where the diffusion coefficient κ = k/(sρ). To express this equation in Cartesian coordinates,
we simply write ∇2 in terms of x, y and z to obtain
2
∂u
∂2 u
∂2 u
∂ u
+ 2 + 2 =
κ
.
2
∂x
∂y
∂z
∂t
κ∇2 u =
The diffusion equation just derived can be generalised to
k∇2 u + f(r, t) = sρ
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∂u
.
∂t
20.1 IMPORTANT PARTIAL DIFFERENTIAL EQUATIONS
The second term, f(r, t), represents a varying density of heat sources throughout
the material but is often not required in physical applications. In the most general
case, k, s and ρ may depend on position r, in which case the first term becomes
∇ · (k∇u). However, in the simplest application the heat flow is one-dimensional
with no heat sources, and the equation becomes (in Cartesian coordinates)
∂2 u
sρ ∂u
.
=
∂x2
k ∂t
20.1.3 Laplace’s equation
Laplace’s equation,
∇2 u = 0,
(20.5)
may be obtained by setting ∂u/∂t = 0 in the diffusion equation (20.2), and
describes (for example) the steady-state temperature distribution in a solid in
which there are no heat sources – i.e. the temperature distribution after a long
time has elapsed.
Laplace’s equation also describes the gravitational potential in a region containing no matter or the electrostatic potential in a charge-free region. Further, it
applies to the flow of an incompressible fluid with no sources, sinks or vortices;
in this case u is the velocity potential, from which the velocity is given by v = ∇u.
20.1.4 Poisson’s equation
Poisson’s equation,
∇2 u = ρ(r),
(20.6)
describes the same physical situations as Laplace’s equation, but in regions
containing matter, charges or sources of heat or fluid. The function ρ(r) is
called the source density and in physical applications usually contains some
multiplicative physical constants. For example, if u is the electrostatic potential
in some region of space, in which case ρ is the density of electric charge, then
∇2 u = −ρ(r)/0 , where 0 is the permittivity of free space. Alternatively, u might
represent the gravitational potential in some region where the matter density is
given by ρ; then ∇2 u = 4πGρ(r), where G is the gravitational constant.
20.1.5 Schr-odinger’s equation
The Schrödinger equation
−
2 2
∂u
∇ u + V (r)u = i ,
2m
∂t
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(20.7)