ORDINARY
DIFFRENTIAL EQUATIONS
By :
Dewi Rachmatin
Euler's Method
The first method we shall study for solving
differential equations is called Euler's method, it serves to illustrate the concepts involved in the advanced methods. It has limited use because of the larger error that is accumulated with each successive step. However, it is important to
study Euler's method because the remainder term and error analysis is easier to understand.
Theorem (Euler's Method)
Assume that f(t,y) is continuous and satisfies a Lipschits condition in the
variable y, and consider the I. V. P.
(initial value problem) with
over the interval .
Euler's method uses the formulas and
for
as an approximate solution to the
differential equation using the discrete set
of points .
Algorithm (Euler's Method)
To approximate the solution of the initial value problem with
over at a discrete set of points using the formulas
and
for .
Mathematica Subroutine
(Euler's Method)
Algorithm
(Modified Euler's Method)
To approximate the solution of the initial value problem with
over at a discrete set of points using the formulas
for
where
Mathematica Subroutine
(Modified Euler's Method)
Example 1.
Solve the I.V.P. .
Runge Kutta Method
Theorem
(Runge-Kutta Method of order 4)
Assume that f(t,y) is continuous and satisfies a Lipschits condition in the variable y, and
consider the I. V. P. (initial value problem) with ,
over the interval .
The Runge-Kutta method uses the formulas
and for
where
as an approximate solution to the
differential equation using the discrete set
of points .
Algorithm (Runge-Kutta Method)
To approximate the solution of the initial value problem with
over at a discrete set of points using the formulas
and
for .