Taylor-series
method for solving an initial value problem
Ching-Tang
Tseng
Hamilton,
New Zealand
16
March 2018
http://forthfornight.blogspot.com
This topic
belongs to the level of the university student. Maybe it is the
simplest method could be applied to solve an initial value problem of
ODE (Ordinary Differential Equation). The detailed explanation of
this method could be found from every numerical analysis textbooks. I
cited it from the following book and used it to test my ABC FORTH
system features.
David Kincaid
and Ward Cheney, “Numerical analysis: mathematics of scientific
computing”, Brooks/Cole Publishing Company, 1990, p.491
There was an
example of initial-value problem supplied in section 8.2, on page
496. I followed this example to design out my program in ABC FORTH
system. For the convenient purpose, some related information in this
textbook must be re-summarized here first.
Example
question:
Ordinary
differential equation and its initial condition expressed as equation
(1) and (2).
X’
= cos t – sin x + t ^ 2 ---------- (1)
X(-1)
= 3 ---------------------------- (2)
Try to use
Taylor series method order 4 to solve this initial value problem.
Assume 200
steps are progressing over the range from t = -1 to t = +1, i.e. let
h =( ( 1 - (-1) ) / 200 ) = 2 / 200 = 0.01. Then x =?
Fig. 1 is a
diagram to describe this question.
Taylor-series
for x can be expressed as (3)
x(t+h) = x(t)
+ h* x’(t) + (h^2/2!)*x’’(t) + (h^3/3!)*x’’’(t) +
(h^4/4!)*x’’’’(t) + … (3)
The
derivatives can be obtained from the differential equation in (1).
They are
x’’ = -
sin t – x’ cos x + 2*t
x’’’ =
-cos t – x’’ cos x + ( x’ ) ^2 * sin x + 2
x’’’’
= sint – x’’’ cos x +3*x’*x’’*sin x + ( x’)^3*cos x
Then, this is
an algorithm to solve this initial-value problem in (1), starting at
t = -1 and progressing in steps of h = 0.01. We desire a solution in
the t-interval [-1,1] and thus must take 200 steps.
A pseudo
programming code can be expressed as following:
Input: M <
= = 200 ; h < = = 0.01 ; t < = = -1.0 ; x < = = 3.0
Output: t, x
For k = 1, 2,
… M do
x’ < = =
cos t – sin x + t^2
x’’ <
= = - sin t – x’ cos x + 2t
x’’’ <
= = - cos t – x’’ cos x + (x’)^2 * sin x = 2
x’’’’
< = = sin t + ((x’)^3 – x’’’) cos x + 3x’x’’ sin
x
x < = =
x + h ( x’ + h/2(x’’+ h/3(x’’’ + h/4(x’’’’)))
x < = =
t+h
Output t, x
End
My real
program in ABC FORTH is
\ Taylor
series method to solve ODE (order 4) in Lina64
2 integers k m
7 reals h t x
x1 x2 x3 x4
: taylor
basic
10 let m = 200
20 let { h =
0.01 e 0 }
30 let { t =
-1.0 e 0 }
40 let { x =
3.0 e 0 }
50 print "
" ; { t , x }
100 for k = 1
to m
110 let { x1 =
cos ( t ) - sin ( x ) + t ** 2.0 e 0 }
120 let { x2 =
negate ( sin ( t ) ) - x1 * cos ( x ) + 2.0 e 0 * t }
130 let { x3 =
negate ( cos ( t ) ) - x2 * cos ( x ) + ( x1 * x1 ) * sin ( x ) + 2.0
e 0 }
140 let { x4 =
sin ( t ) + ( x3 ** 3.0 e 0 ) * cos ( x ) + 3.0 e 0 * x1 * x2 * sin (
x ) }
150 let { x =
x + h * ( x1 + ( h / 2.0 e 0 ) * ( x2 + ( h / 6.0 e 0 ) * ( x3 + ( h
/ 24.0 e 0 ) * x4 ) ) ) }
160 let { t =
t + h }
170 print "
" ; { t , x }
200 next k
210 end ;
Instruction
“taylor” will resulted in a huge amount of printing output for
total 200 sets data. I reserved it in the real video demonstration at
the end of this article. Besides, to express this result as an x
versus t diagram could let me easier to understand the tendency of
curve variation with respect to the ordinary differential equation
(1). Another graphic generating program executed in Win32Forth system
under Ubuntu V16.04 OS was showing in the video as well. But total
needed data for plotting was transferred via a pure text file. In
this demonstration, I am still using highlight, copy then post
operations to create such a file. It could be implemented
automatically by program. For the purpose of simplifying my demo
program as above, I didn’t do it.
Fig. 2 shows
the tendency of solution.
Since 1990, I
knew how to extend FORTH to ABC FORTH, other mathematics programming
languages were no longer to be used by me. This article shows out all
the reasons. ABC FORTH is my researching tool. This system has been
used by me for many years already. It had helped me done a lot of
things that other people was hard to do.
My pure high
level defined floating point system in Lina64 was built based on 64
bits integer operation. All functions inside were gotten based on
calculation of Taylor series expansion equation. 18 digits +, -, *, /
operations could supplied 15 digits precision out only. As to the
hardware co-processor, all functions were built based on 80 bits
operation. Its precision, of course, should be better than my pure
software results.
Taylor series
method for solving initial value problem is simple and easy to
understand. And it can be thought as a first step for doing the
research of initial value problem. I was able to create other kinds
of ordinary differential equation, to use them as other examples, but
I didn’t. All executing results in this demonstration let us have
the opportunity to do a detailed comparison with the data to be
listed in the book. As regards the other topics, such as error
analysis, advantages and disadvantages, using restrictions…etc,
please refer to the book. I am not going to discuss it here. ABC
FORTH could be applied to all numerical analysis textbooks and to
solve all mathematical problems.
Lina64 can do
the music sound playing control as well. A public domain breezy music
“Little-waltz” downloads from website:
http://www.orangefreesounds.com/
I am using
Lina64 to play it as background music in this video.
My personal
testing remarks: intrinsic functions sin, cos, ** approved.
