Approximate solution of two-point boundary value proble

Lancaster, E. R.

English

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X- 643- 68-79 
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APPROXIMATE SOLUTION 
OF A TWO-POINT BOUNDARY VALUE PROBLEM 
E. R.  Lancaster 
March 1968 
GODDARD SPACE FLIGHT CENTER 
Greenbelt, Maryland 
https://ntrs.nasa.gov/search.jsp?R=19680009980 2020-03-24T00:24:00+00:00Z
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APPROXIMATE SOLUTION OF A TWO-POINT 
BOUNDARY VALUE PROBLEM 
E. R. Lancaster 
ABSTRACT 
Approximation formulas are found for k (0 )  and G( 1),  where 
x( t ) satisfies 
iii 
APPROMMATE SOLUTION OF A TWO-POINT 
BOUNDARY VALUE PROBLEM 
INTRODUCTION 
Consider the following boundary-value problem : 
x = f(x, t) + A ( t ) x  , 
x(0) = xo, x(1) = x1 , 
where t is a scalar, x is a column matrix, A( t ) is a matrix function of t ,  and 
the overdots indicate differentiation with respect to t . 
Formulas will be found for 2, and G1 such that 
xO - = &O), x1 = &(I) I 
where +( t satisfies 
+(O) = x(O), 4(0) = X(O), &O) = X(0) 1 
$(1) = x(l), +(1) = X(l), &1) = %(I) 
It will not be necessary to find $( t ) but only to make certain assumptions as to 
its form. 
The author has found the method useful in the calculation of transfer tra- 
jectories for space vehicles and in preliminary orbit determination from 
observations of the position of a spacecraft at two times. 
AN APPROXIMATE SOLUTION 
Let a be any component of X, ~ ( 0 )  =ao,a(l) = al, and assume approxi- 
mations of the form 
1 
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The scalars ao ,  bo ,  a 2  , b, , a3 ,  and b, are determined by assuming (2) to be 
exact whena = 4i(t) ,  i = 0, ..- , 5, where the cPi1s are linearly independent 
over the interval [0, 11 with derivatives through the third order at t = 0 and 
t = 1. The coefficients in (3) are determined in a similar way. If we use the 
same set of 4;s for each component of x, we can write 
(4) 
a -  
xo - aOxO t boxl t a2xo + b,xl + a,xo + b3xl , 
We eliminate the third derivatives in (4) and (5) by using 
(6) 
... 
x = ( P t A ) 2 + & t w ,  
obtained from (l), where P is a matrix with element in the i th row and j th 
column equal to the value of a f  '/axJ, and w is a column matrix with i th  ele- 
ment equal to the value of 3 f ' /at ,  f 
component of f and the j th component of x . Substituting (6) into (4) and (5), 
we obtain 
and X J  being respectively the i t h  
where I is the unit matrix and 
Q = P + A ,  
2 
A POLYNOMIAL APPROXIMATION 
Subs ti tuting suc ce s sivel y 
into equation (2), we obtain 
a. bo = 0 ,  
bo = 1 ,  
bo t 2a2 + 2b, = 0 ,  
bo + 6b2 + 6a3 + 6b3 = 0 ,  
bo + 12b2 t 24b3 = 0 ,  
bo t 20b2 t 60b3 = 0. 
3 
. -  
The solution of this set of equations is 
1 - -  1 - 3 20, a3 - - -  20 9 b, - 30 * - 7 - - a. - -1, bo = 1, a ,  - -20, b, - - -  
In a similar way we find 
4 


Approximate solution of a two-point boundary value problem

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