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 Ω⊂R

n

, which can be a bounded or unbounded set,

with or without a boundary. If Ω is a bounded domain, we suppose that the only possibility

that Ω exchange heat with the ambient space is through its boundary bnd(Ω). Let u(x; t)

denote th e temperature of an arbitrary point x2Ω at time t. For any bounded region D ⊂Ω,

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 ﬂux 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

Ω

[ρc@

t

u(x; t) ¡div (α(x) ru(x; t))] dV = 0:

Since this equation holds for arbitrary Ω, 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 Ω. 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 ﬂuid 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 Ω.

Recall that the Newton cooling law relates the te mperature inside and outside a region

Ω. Speci ﬁcally, if T denotes the tempe rature outside Ω, the law states that the heat ﬂux

through bnd(Ω), is proportional to the diﬀerence T ¡u, where u is the temperature function

inside Ω . 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 Ω 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 Ω

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 Ω

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 Ω

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 ﬁrst 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

[Φ

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