AIMS Lecture Notes 2006
Peter J. Olver
10. Numerical Solution of
Ordinary Differential Equations
This part is concerned with the numerical solution of initial value problems for systems
of ordinary differential equations.
We will introduce the most basic onestep methods,
beginning with the most basic Euler scheme, and working up to the extremely popular
Runge–Kutta fourth order method that can be successfully employed in most situations.
We end with a brief discussion of stiff differential equations, which present a more serious
challenge to numerical analysts.
10.1.
First Order Systems of Ordinary Differential Equations.
Let us begin by reviewing the theory of ordinary differential equations. Many physical
applications lead to higher order systems of ordinary differential equations, but there is a
simple reformulation that will convert them into equivalent first order systems. Thus, we
do not lose any generality by restricting our attention to the first order case throughout.
Moreover, numerical solution schemes for higher order initial value problems are entirely
based on their reformulation as first order systems.
First Order Systems
A
first order system of ordinary differential equations
has the general form
du
1
dt
=
F
1
(
t, u
1
, . . . , u
n
)
,
· · ·
du
n
dt
=
F
n
(
t, u
1
, . . . , u
n
)
.
(10
.
1)
The unknowns
u
1
(
t
)
, . . . , u
n
(
t
) are scalar functions of the real variable
t
, which usually
represents time. We shall write the system more compactly in vector form
d
u
dt
=
F
(
t,
u
)
,
(10
.
2)
where
u
(
t
) = (
u
1
(
t
)
, . . . , u
n
(
t
) )
T
, and
F
(
t,
u
) = (
F
1
(
t, u
1
, . . . , u
n
)
, . . . , F
n
(
t, u
1
, . . . , u
n
) )
T
is a vectorvalued function of
n
+ 1 variables. By a
solution
to the differential equation,
we mean a vectorvalued function
u
(
t
) that is defined and continuously differentiable on
an interval
a < t < b
, and, moreover, satisfies the differential equation on its interval of
definition.
Each solution
u
(
t
) serves to parametrize a curve
C
⊂
R
n
, also known as a
trajectory
or
orbit
of the system.
2/25/07
152
c
circlecopyrt
2006
Peter J. Olver
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In this part, we shall concentrate on initial value problems for such first order systems.
The general initial conditions are
u
1
(
t
0
) =
a
1
,
u
2
(
t
0
) =
a
2
,
· · ·
u
n
(
t
0
) =
a
n
,
(10
.
3)
or, in vectorial form,
u
(
t
0
) =
a
(10
.
4)
Here
t
0
is a prescribed initial time, while the vector
a
= (
a
1
, a
2
, . . . , a
n
)
T
fixes the initial
position of the desired solution.
In favorable situations, as described below, the initial
conditions serve to uniquely specify a solution to the differential equations — at least for
nearby times. The general issues of existence and uniquenss of solutions will be addressed
in the following section.
A system of differential equations is called
autonomous
if the right hand side does not
explicitly depend upon the time
t
, and so takes the form
d
u
dt
=
F
(
u
)
.
(10
.
5)
One important class of autonomous first order systems are the steady state fluid flows.
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 Fall '09
 Olver
 Differential Equations, Numerical Analysis, Equations, Derivative, Partial differential equation, Peter J. Olver

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