Chapter 53
An introduction to partial
differential equations
53.1 Introduction
A partial differential equation is an equation that
contains one or more partial derivatives. Examples
include:
(i) a
∂u
∂x
+ b
∂u
∂ y
= c
(ii)
∂ 2 u
∂x 2 =
1
c 2
∂u
∂t
(known as the heat conduction equation)
(iii)
∂ 2 u
∂x 2 +
∂ 2 u
∂ y 2 = 0
(known as Laplace’s equation)
Equation (i) is a first order partial differential equation, and equations (ii) and (iii) are second order
partial differential equations since the highest power
of the differential is 2.
Partial differential equations occur in many areas of
engineering and technology; electrostatics, heat conduction, magnetism, wave motion, hydrodynamics and
aerodynamics all use models that involve partial differential equations. Such equations are difficult to solve,
but techniques have been developed for the simpler
types. In fact, for all but for the simplest cases, there are
a number of numerical methods of solutions of partial
differential equations available.
To be able to solve simple partial differential equations knowledge of the following is required:
(a) partial integration,
(b) first and second order partial differentiation — as
explained in Chapter 34, and
(c) the solution of ordinary differential equations —
as explained in Chapters 46–51.
It should be appreciated that whole books have been
written on partial differential equations and their solutions. This chapter does no more than introduce the
topic.
53.2 Partial integration
Integration is the reverse process of differentiation.
Thus, if, for example,
∂u
∂t
= 5 cos x sin t is integrated partially with respect to t , then the 5 cosx term is considered
as a constant,
and u =
5 cos x sin t dt = (5 cos x)
sin t dt
= (5 cos x)(−cos t ) + c
= −5 cos x cos t + f (x)
Similarly, if
∂ 2 u
∂x∂ y
= 6x 2 cos 2y is integrated partially
with respect to y,
then
∂u
∂x
=
6x
2 cos 2y d y =
6x
2
cos 2y d y
=
6x
2
1
2
sin 2y
+ f (x)
= 3x
2 sin 2y + f (x)
An introduction to partial
differential equations
53.1 Introduction
A partial differential equation is an equation that
contains one or more partial derivatives. Examples
include:
(i) a
∂u
∂x
+ b
∂u
∂ y
= c
(ii)
∂ 2 u
∂x 2 =
1
c 2
∂u
∂t
(known as the heat conduction equation)
(iii)
∂ 2 u
∂x 2 +
∂ 2 u
∂ y 2 = 0
(known as Laplace’s equation)
Equation (i) is a first order partial differential equation, and equations (ii) and (iii) are second order
partial differential equations since the highest power
of the differential is 2.
Partial differential equations occur in many areas of
engineering and technology; electrostatics, heat conduction, magnetism, wave motion, hydrodynamics and
aerodynamics all use models that involve partial differential equations. Such equations are difficult to solve,
but techniques have been developed for the simpler
types. In fact, for all but for the simplest cases, there are
a number of numerical methods of solutions of partial
differential equations available.
To be able to solve simple partial differential equations knowledge of the following is required:
(a) partial integration,
(b) first and second order partial differentiation — as
explained in Chapter 34, and
(c) the solution of ordinary differential equations —
as explained in Chapters 46–51.
It should be appreciated that whole books have been
written on partial differential equations and their solutions. This chapter does no more than introduce the
topic.
53.2 Partial integration
Integration is the reverse process of differentiation.
Thus, if, for example,
∂u
∂t
= 5 cos x sin t is integrated partially with respect to t , then the 5 cosx term is considered
as a constant,
and u =
5 cos x sin t dt = (5 cos x)
sin t dt
= (5 cos x)(−cos t ) + c
= −5 cos x cos t + f (x)
Similarly, if
∂ 2 u
∂x∂ y
= 6x 2 cos 2y is integrated partially
with respect to y,
then
∂u
∂x
=
6x
2 cos 2y d y =
6x
2
cos 2y d y
=
6x
2
1
2
sin 2y
+ f (x)
= 3x
2 sin 2y + f (x)