Chapter 3
Linear Second-Order Equations
In this chapter, we derive the heat equation in general dimension, the equation of wave propa-
gation in an elastic membrane, and the Poisson's equation. We present the properties of each
equation and then classify the linear second-order partial differential equations into elliptic,
parabolic, and hyperbolic equations. The main reason for t his classification is that ea ch class
of equation has distinct solutions that behave differently. For instance, elliptic equations
have s o luti o ns that are smooth and have no singularities, whereas hyperbolic equations have
solutions that have s ingularities and can exhibit wave-like b eh avior. Understanding these
differences is crucial for solving and interpreting the solutions of partial differential equations
in various fields of science and engineering.
3.1 Heat equation
The heat equation is a fundamental equation in physics, engineering, and applied mathe-
matics that is used to model a wide range of phenomena, inclu ding heat transfer, thermal
diffusion, and mass diffusion. In this section, we will first de rive the heat equation for solid
conductive continua, and then i ntroduce the advection-diffusion equation for fluids.
3.1.1 Derivation of heat eq u ation
In the first chapter, we demonstrated the derivation of the equation for heat flow in a
conductive rod. In this chapter, we pres ent the derivation in three-dimensional space (R
3
)
to gain a deeper understanding of it s physics.
Consider a solid conductive continuum R
n
, which can be a bounded or unbounded set,
with or without a boundary. If is a bounded domain, we suppose that the only possibility
that exchange heat with the ambient space is through its boundary bnd(Ω). Let u(x; t)
denote th e temperature of an arbitrary point x2 at time t. For any bounded region D ,
the thermal energy inside D at time t is defined by the integral:
Q
D
(t) =
ZZZ
D
q(x; t) dV ;
1
where q is the density function of the heat energy. Therefore, the increme nt in energy can
be expressed as:
Q
D
(t + δt) ¡Q
D
(t) =
ZZZ
D
fq(x; t + δt) ¡ q(x; t)gdV :
In the limiting case a s δt!0, we obtain the fo llowing relation:
d
dt
Q
D
(t) =
ZZZ
D
ρc@
t
u(x; t) dV ;
where ρ is the density and c is the specific heat capacity of the continu um.
On the other hand, we can relate the rate o f change of thermal energy inside D to the heat
exchange through its boundary bnd(D). The rate of heat exch a nge through the boundary
of D is proportional to the temperature gradient, according to Fourier's law , which states
that th e heat flux J(x; t) at a point x in D is given by J(x; t)=¡ α(x)ru(x; t), where α(x)
is the thermal conductivity at x. Note that the negative sign arises because the direction
of the tempe rature gradient is towards increasing temperature, whereas the direction of the
heat flux is from hotter to colder regions. T he ne t ux leaving D through bnd(D) is equal
to the surface integral
ZZ
bnd(D)
J(z; t) ·ν(z) dS = ¡
ZZ
bnd(D)
α(z) ru(z; t) ·ν(z) dS ;
where ν(z) is the outward-pointing unit normal vector at z 2 bnd(D). By the conservation
of energy principle, we have
d
dt
Q
D
(t) = ¡
ZZ
bnd(D)
J(z; t) ·ν(z) dS ;
where the negative sign accounts for the fact that the integral measures the flux leaving D
at time t. Substituting
dQ
D
dt
by the triple integral and rearranging, we obtain
ZZZ
D
ρc@
t
u(x; t) dV =
ZZ
bnd(D)
α(z) ru(z; t) ·ν(z) dS:
Using Gauss's theorem, we can convert the surface integral in the right-hand side of the
previous relation to a volume integral. Applying the theorem yields:
ZZZ
D
div (α (x)ru(x; t))dV =
ZZ
bnd(D)
α(x)ru(x; t) ·ν(x)dS;
and y rearranging the terms, we arrive at the integral form of the heat equation:
ZZZ
[ρc@
t
u(x; t) ¡div (α(x) ru(x; t))] dV = 0:
Since this equation holds for arbitrary , we obtain the follow ing differential form of the
heat equation
u
t
(x; t) =
1
ρc
div (α(x) ru(x; t)):
2 Linear Second-Order Equations
If α is constant, then we have
u
t
(x; t) = k u;
where k =
α
ρc
> 0.
Remark 3.1. The he a t equation that we derived invo lves a first-order partial derivative of u
with respect to time. Therefore, to fully describe the heat dynamics, we need to specify the
initial condition u(x; 0) for x 2 . This is called the initial condition o f the heat equation.
In fact, the initial condition provides the initial thermal energy distribution for the sy stem,
which drives its subsequent behav ior. Additi o nally, in some physical systems, there may be
an internal source or sink of thermal energy that is not accounted for in our de rivation. In
the presence of such a source or sink, the heat equation takes the fo rm:
u
t
=
1
ρc
div (αru) + h;
where the function h represents the rate at which the internal source or sink produces or
absorbs energy.
3.1.2 Convection-diffusion equation
The convection-diffusion equation is a partial differential equation that is widely used to
model heat and mass transfer in fluids, such as in chemical engineering and environmental
science. The equation takes into account both convection, which is the transport of heat
or mass by the fluid flow, and diffusion, which is the transpo rt of heat or mass due to a
concentration gradient.
In the modeling of the heat problem, we assumed th a t heat is transferred only by conduc-
tivity. This assumption make s sense if the medium is solid. If the medium is a fluid or gas,
the heat can be transferred in the form of convection i n addition to diffusion. In this scenario,
the heat flux J consists of two terms, the conductivity term and the convection term:
J = J
cond
+ J
conv
:
If the fluid moves with velo city V , then J
conv
= uV , and thus
ZZ
bnd(D)
J ·ν dS = ¡
ZZ
bnd(D)
αru ·ν dS +
ZZ
bnd(D)
uV ·ν dS:
Therefore, the heat equation reads
ρcu
t
+div (uV ) =div (αru):
The above equation is called the convection-diffusion equation. The first term on the left-
hand side of the equation represents the change in temp erature with respect to time. The
second term represents the convective transport of heat by the uid flow, while the third
term represents the diffusive transpo rt due to a co nc entration gradient. In the presence of
a source or sink, the heat equation takes the form:
ρcu
t
+div (uV ) =div (αru) + h:
3.1 Heat equation 3
The convection-diffusion is a fundamental equation in many areas of science and engineering
and has many applications, such as modeling air and water pollution, heat transfer in pipes,
and drug delivery in biological systems.
3.1.3 Boundary conditions for bounded domains
Continuing with o ur discussion of the heat equation on bounded domain, it is worth noting
that the solution to a partial d ifferential equation, and especially to the heat equation,
depends heavily on the boundary conditions imposed on the bounded domain .
Recall that the Newton cooling law relates the te mperature inside and outside a region
. Speci fically, if T denotes the tempe rature outside , the law states that the heat flux
through bnd(Ω), is proportional to the difference T ¡u, where u is the temperature function
inside . This relationship is expressed mathematically by the Robin boundary condition
@u
@n
= κ(T ¡u);
where ν denotes the outward normal to the boundary, and κ is the conductivity factor of
the boundary. When κ = 0, the bounda ry is perfectly insulated, and the condition reduces
to the hom o g en eous Neumann boundary condition
@u
@n
= 0. More generally, the Neumann
boundary condition takes the form
@u
@n
= g on bnd(Ω), where g is a given function. In the
limiting case where κ is very large, the Ro bin condition can be rewritten as
T ¡u =
1
κ
@u
@n
;
and taking the limit κ!1 yields the condition u =T on bnd(Ω), which is called the Dirichlet
boundary condition.
Dirichlet boundary condition. The Dirichlet boundary condition is used when the
temperature at the boundary of the domain is known, such that u(z; t)= g(z; t) for
z 2bnd(Ω). This condition is named after the German mathematician Peter Gustav
Lejeune Dirichlet. For a heat equation, this means that the temperature at bnd(Ω) is
kept a t the known value g(z; t). If g is identically zero, then the condition is call ed
the homogeneous Dirichlet condition.
u = g
PDE inside
In one-dimensional problems, such as a conductive rod extending from x = 0 to
x = L, the condition becomes u(0; t) = g
0
and u(L; t) = g
L
, where g
0
and g
L
can be
constants or functions of tim e.
4 Linear Second-Order Equations
Neumann boundary condition. This condition is named after the German math-
ematician Carl Neumann. Mathematically, the Neumann bou ndary c ondition is
expressed as
@u
@n
:= ru ·ν = g;
where
@u
@n
is the direct ional derivative of u along the unit normal direction ν on
bnd(Ω). In the context of a heat problem, this condition specifies the heat flux through
the boundary bnd(Ω) in terms of function g. In part icular, if g is identically zero,
the boundary is perfectly insulated.
@u
@n
= g
ν
PDE inside
For a one-dimensional problem where the co nductive rod extends from x = 0 to
x = L, the condition reads:
u
x
(0; t ) = g
0
; u
x
(L; t) = g
L
:
Robin's or mixed boundary condition. As we observed above, the Robin boundary
condition is a straightforward derivation of the Newton cooling law in the context of
a heat problem. Mathematically, it can be expressed as
αu (z; t) + β
@u
@n
(z; t) = g(z; t):
where α and β are some coefficients, and g is a given function. In particular, if g
is identically zero, the condition is called the homogeneous Robin boundary condi-
tion. The Robin boundary condition is commonly used in problems that involve heat
transfer between a solid and a fluid, where the heat transfer coefficient (i.e., α / β)
depends on the properties of the fluid and the surface of the solid. It is also used
to model heat tran sfer in biological tissues, where the heat transfer coefficient can
depend on the blood flow rate a nd other physiological factors. The Robin boundary
condition is named after the French mathematician Victor Gustave Robin.
αu + β
@u
@n
= g
PDE inside
For a one-dimensional problem, the condition reads
α
1
u(0; t) + β
1
u
x
(0; t ) = g
0
α
2
u(L; t) + β
2
u
x
(L; t) = g
L
:
3.1 Heat equation 5
3.1.4 Fundamental solution of heat equatio n
Consider a con ductive rod of diffusivity factor k > 0, extended over the entire real line (¡1;
1). Suppose that an infinitely small portion of the rod at x = 0 has a temperature of u = 1,
and the temperature i s zero everywhere else. As time progresses, the heat diffuses across the
rod, but the total thermal energy remains constant as it was at t = 0. It can be shown that
the heat profile evolves according to the function:
Φ(x; t) =
1
4πkt
p
e
¡
x
2
4kt
;
for t > 0. In fact, one can verify that Φ
tt
= k Φ
xx
through di rect calculation. The graph of
Φ(x;t) demon strates the evolution of heat profile for t > 0. The following figure shows Φ(x;t)
for t = 0.1; 0.5 a nd 1 where k = 1:
-4 -2 0 2 4
0
0.2
0.4
0.6
0.8
That the total thermal energy remains constant over time follows from the fact that the
integral of Φ for a ny fixed value o f t is constant:
Z
¡1
1
Φ(x; t)dx = 1:
In fact, using the change of variable v =
x
4kt
p
, we have:
Z
¡1
1
Φ(x; t)dx =
1
π
p
Z
¡1
1
e
¡v
2
dv = 1:
This confirms the conservation of thermal energy since the initial heat sou rce is a unit
pointwise source at x = 0. The function Φ is called the Gaussian or heat kernel of a heat
equation.
Exercise 3.1. By direct calculation, verify that Φ(x; t) satisfies the heat equation u
t
= ku
xx
.
Now, consider the heat problem given by the partial differential equation
u
t
= ku
xx
; x 2(¡1; 1); t > 0;
with initial condition u(x; 0) = f(x), where f(x) is a continuous and integrable fun ction in
(¡1; 1). The follow ing theorem gives the solution to this problem:
6 Linear Second-Order Equations
Theorem 3.1. The solution to the given problem is given by the convolution
u(x; t) = f Φ :=
Z
¡1
1
Φ(x ¡ y; t) f(y) dy:
Proof. We rst show that u defined as above satisfies the partial differential equation. We
have
u
t
(x; t) =
Z
¡1
1
Φ
t
(x ¡ y; t) f (y) dy;
and u
xx
is
u
x x
(x; t) =
Z
¡1
1
Φ
xx
(x ¡ y; t) f(y) dy:
The above representations of the partial derivatives are justified by the assumption of inte-
grability of f, th a t is ,
Z
¡1
1
jf(x)jdx < 1;
and the Dominated Convergence Theorem which allows us to take the limits from outsi de
the integrals to inside the integrals. Accordingly, we have
u
t
¡ku
xx
=
Z
¡1
1
t
(x ¡ y; t) ¡kΦ
x x
(x ¡ y; t)] f(y) dy:
By the above exercise, we have Φ
t
¡k Φ
x x
= 0, and this completes the claim. It remains to
show that the convolution satisfies the initial condition for t !0. We have
lim
t!0
u(x; t) = lim
t!0
1
4πkt
p
Z
¡1
1
e
¡
(x¡y)
2
4kt
f(y)dy = lim
t!0
1
4πkt
p
Z
¡1
1
e
¡
y
2
4kt
f(x ¡ y)dy:
Using the substitution v =
y
4kt
p
, we can write
lim
t!0
u(x; t) = lim
t!0
1
π
p
Z
¡1
1
e
¡v
2
f(x ¡ 4kt
p
v)dv:
If we pass the limit t!0 inside the integral (which is justified by the Dominant Convergence
Theorem), we obtain
lim
t!0
u(x; t) =
1
π
p
Z
¡1
1
e
¡v
2
lim
t!0
f(x ¡ 4kt
p
v)dv:
Since f is continuous, we ob tain
lim
t!0
u(x; t) =
1
π
p
Z
¡1
1
e
¡v
2
f(x)dv =
f(x)
π
p
Z
¡1
1
e
¡v
2
dv = f (x);
which completes the proof.
Example 3.1. Consider the following problem:
8
<
:
u
t
= u
x x
u(x; 0) =
1 ¡1 < x < 1
0 otherwise
:
3.1 Heat equation 7