Numerical estimation of derivatives with application to radiative transfer in spherical shell

Bellman, R. E.

Kagiwada, H. H.

Kalaba, R. E.

English

NASA Technical Reports Server

. .  , 
N A S r - 2 1 (  0'7) 
* 
MEMORANDUM 
RM-4617-NASA 
J U N E  1965 
NUMERICAL ESTIMATION O F  
DERIVATIVES WITH AN APPLICATION TO 
RADIATIVE TRANSFER IN 
SPHERICAL SHELLS 
H. H. Kagiwada, R. E. Kalaba and R. E. Bellman 
P R E P A R E D  FOR: 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
https://ntrs.nasa.gov/search.jsp?R=19650023619 2020-03-17T00:51:59+00:00Z
. .  1 .  
MEMORANDUM 
RM-4617-NASA 
JUNE 1965 
NUMERICAL ESTIMATION OF 
DERIVATIVES WITH AN APPLICATION TO 
RADIATIVE TRANSFER IN 
SPHERICAL SHELLS 
H. H. Kagiwada, R. E. Kalaba and R. E. Bellman 
This research is sponsored by the National Aeronautics and Space Administration under 
Contract No. NASr-21. This report does not necessarily represent the views of the 
National Aeronautics and Space Administration. 
744 
1700 M A I N  S T  . S A N l A  M O N I C A .  C A L l F O l N I A  t O I O b  
ii 
Published by The RAND Corporation 
iii . 
This Memorandum is a result o f  RAND'S continuing study of 
Satellite Meteorology for the National Aeronautics and Space Ad- 
ministration under contract NASr-21(07). To achieve more realism, 
the authors are extending their previous study of isotropic and 
anisotropic scattering in slab geometry to shell geometry. They 
wish to be able to assess quantitatively the effects of sphericity 
on radiation fields. The ultimate aim of this study is to more 
fully exploit the mathematical and computational capabilities of 
the modern digital computer in the study of radiative transfer in 
planetary atmospheres . 
-\ 
d-- 
SUMMARY qJ-3 
The invariant imbedding equation for the scattering function 
for shell geometry is a nonlinear partial differential-integral 
equation, and its numerical solution presents difficulties. 
A simple method for integration of the above equation is 
presented. Integrals are approximated by finite sums using Gaussian 
quadrature, and partial derivatives are approximated by linear com- 
binations of functional values. The original problem is approxi- 
mated by a large system of ordinary differential equations with 
known initial conditions. 
Requisite auxiliary constants for numerical differentiation 
are given, as are the results of some trial calculations for the 
case of conservative isotropic scattering. Noteworthy differences 
between slab and shell geometry are observed, especially with grazing 
angles. 
v i  i 
CONTENTS 
PREFACE ..................................................... iii 
SUMMARY ....................................................... v 
Section 
I. INTRODUCTION ........................................... 1 
11. ESTIMATION OF DERIVATIVES .............................. 2 
111. APPLICATION TO THE SPHERICAL SHELL PROBLEM . . . . . . . . . . . . .  7 
REFERENCES ................................................... 11 
. 
1 
I. INTRODUCTION 
Of c u r r e n t  i n t e r e s t  i n  r a d i a t i v e  t r a n s f e r  i s  t h e  problem o f  
(1-3) d i f f u s e  r e f l e c t i o n  by a medium having a s p h e r i c a l  geometry. 
When c o n i c a l  f l u x  o f  n e t  f l u x  n per  u n i t  normal area i s  i n c i d e n t  on 
a s p h e r i c a l ,  hol low s h e l l  o f  i n n e r  r a d i u s  a ,  and o u t e r  r a d i u s  z ,  
t h e  equa t ion  f o r  t h e  s c a t t e r i n g  func t ion  i s  
where i s  t h e  a lbedo  f o r  s i n g l e  s c a t t e r i n g ,  and u and v a r e  t h e  
c o s i n e s  o f  t h e  i n c i d e n t  and r e f l e c t e d  a n g l e s ,  r e s p e c t i v e l y .  Th i s  i s  
a n  i n t e g r o - d i f f e r e n t i a l  equa t ion ,  i n  which p a r t i a l  d e r i v a t i v e s  w i t h  
r e s p e c t  t o  z ,  v ,  and u occur ,  and i n t e g r a l s  o v e r  v and u are p r e s e n t .  
The i n i t i a l  c o n d i t i o n s  are 
f o r  a p e r f e c t l y  abso rb ing  co re .  
arguments  u and v.  
The S f u n c t i o n  i s  symmetric i n  t h e  
To s o l v e  t h i s  problem w i t h  the  use  o f  a d i g i t a l  computer,  w e  
wish t o  r e p l a c e  Eq. (1) by an approximate system of  o r d i n a r y  d i f f e r -  
e n t i a l  e q u a t i o n s .  
Our f i r s t  t a s k  i s  t o  o b t a i n  formulas f o r  t h e  e s t i m a t i o n  o f  t h e  d e r i v -  
a t i v e s  of  S w i t h  respect to v and u.  
We choose t o  have z be t h e  independent  v a r i a b l e .  
2 
, 
11. ESTIMATION OF DERIVATIVES 
Consider the  f u n c t i o n  f ( x )  which i s  eva lua ted  a t  the N r o o t s ,  
We wish t o  x 
approximate the  f i r s t  d e r i v a t i v e  of  f eva lua ted  a t  one of t h e  r o o t s  
of the s h i f t e d  Legendre polynomial of  degree  N .  ( 4 )  
i '  
by means of a l i n e a r  e s t i m a t o r  
N 
j = l  
We r e q u i r e  t h a t  formula (3) b e  e x a c t  f o r  a l l  polynomials of degree  
N - 1  o r  l e s s ,  
N- 1 
k=O 
We then have t o  so lve  t h e  N l i n e a r  a l g e b r a i c  equa t ions  i n  t h e  N 
unknowns, cy1 ( i )  , cy2 ( i )  , ..., cy ( i )  
N '  
k- 1 N k J i )  = , k = O , l , . . ~ , N - 1 ,  
C x j  j i 
j = l  
or 
N 
k-2 
i '  k=1,2, .  . . ,N. xk-' = ( k - l ) x  1 j 
( 5 )  
( 5  ') 
j =1 
Here, i i s  a parameter which may t ake  on t h e  v a l u e s  i = 1 , 2 ,  ..., N. 
Note t h a t  the  m a t r i x  of  c o e f f i c i e n t s  { x;-'} i s  t h e  Vandermonde 
m a t r i x ,  the i n v e r s e  of which i s  g iven  i n  Appendix 6 of  Ref. 4 .  
3 
L e t  { y j k }  be  t h e  i n v e r s e  mat r ix .  Then t h e  s o l u t i o n  o f  E q .  (5’) is 
given  by 
N 
k= 1 
where 
(7) 
k- 2 
bk = (k - l )x  k=1,2 , .  . . ,N. i ’  
We r e f e r  t h e  r e a d e r  t o  Ref. 4 f o r  t h e  method of c a l c u l a t i o n  of  t h e  
e lements  o f  t h e  i n v e r s e  matrix and f o r  t h e  numer ica l  t a b l e s  of these 
e lements .  
The values of the c o e f f i c i e n t s  have been c a l c u l a t e d ,  and J 
they  are g iven  i n  Tab les  1, 2 ,  and 3 ,  f o r  N=5, 7 and 9 ,  r e s p e c t i v e l y .  
ill 
- C o l C 1 3 4 U 8 1 E  C 2  0 0 1 5 4 0 3 9 0 4 E  L 2  - 0 0 8 G 8 7 0 8 7 4 E  c?l 0 0 3 9 2 0 7 9 8 2 E  0 1  
- 0 0 1 1 2 3 5 3 3 7 E  0 1  
i s 2  
- 0 0 1 9 2 3 5 1 2 O E  0 1  - 3 0 1 5 1 6 7 0 6 4 E  0 1  0 0 4 8 0 5 5 C 1 3 E  01 - C o 1 8 5 7 1 1 6 0 E  0 1  
0 0 4 8 8 8 3 3 2 3 E - 0 0  
0 0 6 0 2 3 3 6 3 2 E  00 - 0 o 2 8 7 0 7 7 6 5 E  0 1  - 0 0 3 5 5 2 7 1 3 7 E - 1 4  0 0 2 8 7 0 7 7 6 5 E  C 1  
1 1 . 3  
- 0 0 6 8 2 3 3 6 3 2 E  0 0  
i - 4  
- 0 0 4 8 8 8 3 3 2 3 E - C O  0 o 1 8 5 7 1 1 6 C E  0 1  - C 0 4 8 0 5 5 C 1 3 E  0 1  0 0 1 5 1 6 7 0 6 4 E  GI. 
i o 1 9 2 0 5 1 2 ~ ) E  0 1  
1 5 5  
3 0 1 1 0 3 5 3 3 7 E  0 1  e G o 3 9 2 0 7 9 8 2 E  i:1 G o 8 0 8 7 a 8 7 4 E  0 1  - 0 o 1 5 4 0 3 9 0 4 E  0 2  
G o 1 0 1 3 4 C 8 1 E  0 2  
Tab le  1. The c o e f f i c i e n t s  f o r  N = 5 
J 
4 
i = l  
- 3 0 1 9 1 3 6 3 6 4 E  3 2  
- 3 0 7 3 5 5 4 C 5 4 E  0 1  
- 0 0 3 0 7 7 4 C Q l E  2 1  
0 0 2 7 7 4 3 2 6 7 E  01 
i s 2  
1 = 3  
3 . 7 3 8 7 6 6 9 1 6  0 0  
- 0 0 2 4 6 3 9 9 3 9 E  0 1  
1 - 4  
- 0 0 3 6 9 4 0 2 8 3 E - 0 0  
0 0 4 3 0 4 6 3 3 1 E  0 1  
i = 5  
0 0 2 9 6 2 1 3 5 2 E - U 0  
O o 9 7 1 7 4 7 0 3 E  00 
1 - 6  
-0  3 7 7 8 4 3 2  9E-00 
- 0 . 9 4 8 2 6 6 0 a ~  01 
i = 7  
0 0 1 3 5 3 6 2 1 0 E  3 1  
0 0 1 8 3 4 5 1 3 6 E  0 2  
0.3G166068E 0 2  
G 0 3 7 0 3 7 9 0 9 E  0 1  
- 0 . 3 2 9 4 7 3 1 3 E  0 1  
- 0 . 1 3 4 8 5 6 0 9 E  C1 
- 0 . 3 7 4 3 3 7 4 0 E  0 1  
0 . 1 0 9 5 1 9 2 9 6  G 1  
0 . 1 4 8 3 3 1 3 7 E  0 1  
- 5 . 1 4 8 0 3 1 3 7 E  0 1  
- 3 . 1 C 9 5 1 5 2 9 E  Cl 
0 . 3 7 4 3 3 7 4 3 E  01 
G o 1 3 4 8 5 6 0 9 E  '$1 
O o 3 2 9 4 7 3 1 3 E  0 1  
- 0 . 3 7 0 3 7 9 0 9 E  0 1  
- 0 . 3 0 1 6 6 0 6 8 E  3 2  
- G * 9 7 1 7 4 7 0 3 E  ?0 0 . 5 6 4 1 3 4 8 8 5  0 1  
-0 2 5 6  2 1 ? 5 2 E-CO 
-0 4 3 3  48  3 3  1 E ? 1 -0 9 9 4 7  5 9 8  3 E-.13 
0 . 3 6 9 4 0 2 8 3 E - C 0  
0 0 2 4 6 3 9 9 3 9 E  C,1 - 0 . 5 6 4 1 3 4 6 8 E  0 1  
- 0 . 7 3 8 7 a m i ~  00  
- 8 0 2 7 7 4 3 2 5 7 E  C'l C o 4 9 1 4 1 3 8 4 E  0 1  
0 . 3 0 7 7 4 0 q l E  C 1  
0 0 7 3 5 5 4 C 5 4 E  5 1  -0 .12020668E 0 2  
0 0 1 9 1 3 6 3 6 4 E  e 2  
Table  2.  The c o e f f i c i e n t s  cy!i) for N = 7 
3 
. 
5 
i = 1  
- 0 o 3 C 8 9 9 1 8 3 E  0 2  0 0 4 9 4 6 2 6 C 2 E  0 2  - 3 0 3 1 8 4 7 7 2 2 E  0 2  O o 2 3 0 0 9 7 1 3 E  0 2  
- 0 0 1 6 6 3 4 3 2 5 E  0 2  6 0 1 1 4 6 3 9 0 8 E  9 2  - C 0 7 1 4 4 4 7 5 2 E  C 1  0 0 3 6 2 2 3 7 1 1 E  0 1  
- 3 o l L 3 2 8 8 6 9 E  0 1  
i = 2  
- 3 o 4 6 3 2 1 8 4 7 E  0 1  - d o 5 5 5 4 C S 4 7 E  01 0 0 1 5 5 2 9 6 3 2 E  C Z  - C 0 6 8 5 9 4 0 1 5 E  C 1  
0 0 5 8 9 5 S b 8 7 E  0 1  - 0 0 3 9 0 7 7 2 6 6 E  0 1  2 0 2 3 8 5 6 E 8 4 E  C 1  - 0 0 1 1 9 6 1 2 7 7 E  C 1  
0 0 3 3 9 2 3 5 9 4 E - 0 0  
i = 3  
0 0 9 9 7 7 9 6 0 8 E  O G  - 0 0 5 1 9 5 3 6 0 4 E  0 1  - 0 0 1 9 6 6 6 4 1 7 E  01 O o 9 0 7 0 6 9 9 6 E  0 1  
- 0 o 4 6 4 7 4 2 5 7 E  31 O o 2 7 9 6 9 6 3 6 E  0 1  - 0 e 1 6 3 0 3 3 3 5 E  0 1  0 0 7 9 8 1 2 0 0 6 E  0 3  
- 0 0 2 2 3 8 3 8 0 L F - 0 0  
i = 4  
- 0 o 4 1 9 2 7 8 6 5 E - 2 2  O o 1 7 2 3 8 1 2 3 E  1)l - 0 o 5 2 7 5 5 6 4 3 E  C 1  - 0 o 7 2 4 7 0 2 2 4 E  C O  
17067p44574E Q l  - 0 o 3 3 8 4 0 0 7 5 E  5 1  0 e 1 6 2 6 7 2 8 0 E  Pl - 0 o 7 6 0 3 3 8 2 0 E  00  
0 0 2 C 8 8 9 3 1 6 E - 3 0  
0 0 2 5 6 5 4 3 C a E - 0 0  - 0 0 9 7 0 8 0 2 0 0 E  00 C e 2 2 8 7 7 1 7 0 E  0 1  - 0 o 5 6 7 4 4 9 4 9 E  0 1  
0 0 5 6 8 4 3 4 1 9 E - 1 1  0 0 5 6 7 4 4 9 4 9 E  0 1  - 0 0 2 2 8 7 7 1 7 0 E  0 1  O o 9 7 0 8 0 2 0 0 E  09  
i = 5  
- 3 0 2 5 6 5 4 3 0 6 E - 0 3  
i = 6  
- o 0 2 o 8 e 9 3 i 6 ~ - o o  0 . 7 6 0 3 3 8 ~ 0 ~  0 3  - o O i 6 z 6 7 2 e 3 ~  01 ~ 3 m 4 0 0 7 5 ~  0 1  
- 0 0 6 7 6 4 4 5 7 4 E  0 1  3 0 7 2 4 7 0 2 2 4 E  G O  O e 5 2 7 5 5 6 4 3 E  0 1  - 0 o 1 7 2 3 8 1 2 3 E  0 1  
0 4 1 9 2 7 8 6  5 E-3 0 
1 5 7  
C o 2 2 3 8 3 8 0 0 E - 0 0  - 0 0 7 9 8 1 2 0 G 6 E  0 0  0 0 1 6 3 0 3 3 3 5 E  0 1  - 0 0 2 7 9 6 9 6 3 6 E  0 1  
0 0 4 6 4 7 4 C 5 7 E  0 1  - 0 0 9 0 7 0 6 9 9 6 E  2 1  0 0 1 9 6 6 4 4 1 7 E  0 1  0 0 5 1 9 5 3 6 0 4 E  0 1  
- 0 r 9 9 7 7 9 6 0 8 E  00 
- 0 e 3 3 9 2 3 5 9 4 E - 0 6  0 o 1 1 9 6 1 2 7 7 E  0 1  - 0 0 2 3 8 5 6 5 8 4 E  0 1  O o 3 9 0 7 7 2 6 6 E  0 1  
- 0 0 5 8 9 5 0 U 8 7 E  0 1  0 0 8 8 5 9 4 6 1 5 E  0 1  - 0 0 1 5 5 2 9 6 3 2 E  0 2  0 0 5 5 5 4 0 6 4 7 E  0 1  
i - = 8  
0 0 4 6 3 2 1 8 4 7 E  3 1  
i = 9  
0 0 1 0 3 2 8 8 6 9 E  0 1  - 0 o 3 6 2 2 3 7 1 1 E  0 1  O e 7 1 4 4 4 7 6 2 E  0 1  - 0 0 1 1 4 6 3 9 0 8 E  0 2  
C o 1 6 6 3 4 3 2 5 C  0 2  - L j o 2 3 0 0 9 7 1 3 E  0 2  O e 3 1 8 4 7 7 2 2 E  ? 2  - 0 0 4 9 4 6 2 6 0 2 E  0 2  
3 0 3 C 8 9 9 1 8 3 E  3 2  
T a b l e  3. The c o e f f i c i e n t s  cy!i) f o r  N = 9 
J 
6 
T h i s  method f o r  t h e  e s t i m a t i o n  o f  d e r i v a t i v e s  w a s  t e s t e d  on t h e  
2 6 - X  - 4x 
fo l lowing  c a s e s :  f(x) = 1, x ,  x , ..., x , 1 - e , 1 - e , 
s i n  21~x13. The r e s u l t s  were e x c e l l e n t  f o r  t h e  polynomials and f o r  
t h e  c a s e  1 - e , w i t h  accuracy  to  s i x  decimal p l a c e s .  Accuracy to  
two decimal p l a c e s  was ob ta ined  f o r  t h e  f u n c t i o n  1 - e-4x. For  t h e  
f i n a l  t e s t  c a s e ,  f o u r  t o  f i v e  c o r r e c t  f i g u r e s  r e s u l t e d .  The most 
unfavorable  t r i a l  w a s  t h a t  f o r  f ( x )  = 1 - e , a f u n c t i o n  which 
r a p i d l y  dec reases ,  then l e v e l s  o f f ,  so t h a t  i t s  d e r i v a t i v e  i s  a t  
f i r s t  l a r g e ,  then n e a r l y  zero.  
-X 
- 4x 
7 
111. APPLICATION TO THE SPHERICAL SHELL PROBLEM 
We r e t u r n  t o  t h e  computat ional  s o l u t i o n  o f  t h e  S f u n c t i o n  o f  
Eqs. (1) and ( 2 ) .  We a l low t h e  v a r i a b l e s  v and u t o  t a k e  on on ly  
t h e  v a l u e s  [ v . ] ,  where vi i s  t h e  ith r o o t  o f  t h e  Legendre polynomial 
1 
o f  deg ree  N ,  s h i f t e d  t o  t h e  i n t e r v a l  ( 0 , l ) .  ( 4 )  We then e x p r e s s  S 
as a f u n c t i o n  o f  one argument, z ,  
t h e  s u b s c r i p t s  i and j i n d i c a t i n g  t h e  a n g u l a r  parameters .  The d e r i v a -  
t i v e s  o f  S w i t h  r e s p e c t  t o  v and u a r e  c a l c u l a t e d  u s i n g  t h e  formulas  
r 1 N 
The d e f i n i t e  i n t e g r a l s  i n  Eq. (1) a r e  approximated t o  a high degree  
o f  accuracy by t h e  u s e  o f  Gaussian quadra tu re .  ( 5 )  
e v a l u a t e d  a t  the  r o o t s  o f  t h e  Nth- s h i f t e d  Legendre polynomial,  
v i = 1 , 2 , . .  . ,N.  The corresponding w e i g h t s  are w i = 1 , 2 , . .  . , N .  
We have o u r  d e s i r e d  system of approximating o r d i n a r y  d i f f e r e n t i a l  
e q u a t i o n s  , 
The i n t e g r a n d s  a re  
( 4 )  
i’ i’ 
8 
A k= 1 k= 1 
vi 2 + v2 sij 
2 2  z +(+ ++) si j  v v  i j  
N N 
[ k = l  k = l  
w i t h  i n i t i a l  c o n d i t i o n s  
S. . ( a )  = 0 ,  
1 3  
f o r  i = 1 , 2 , . . . , N ,  j=l ,2 , . . . ,NY and z 2 a. 
We produced v a l u e s  o f  r e f l e c t e d  i n t e n s i t i e s  , 
s.  . ( z )  
r (2) = -LL- , 
i j  4 vi 
f o r  v a r i o u s  v a l u e s  o f  t h e  a lbedo  1, and f o r  v a r i o u s  i n n e r  r a d i i  a ,  
and s h e l l  t h i c k n e s s e s  
x = z - a .  (14) 
The computations were c a r r i e d  o u t  on an  IBM 7044 w i t h  a FORTRAN I V  
s o u r c e  program. 
For i n t e r n a l  checking pu rposes ,  w e  compared o u r  r e s u l t s  f o r  N = 7 
a g a i n s t  t h e  r e s u l t s  f o r  N = 9. 
i n t e g r a t i o n  s t e p  s i ze  o f  0.005 a g a i n s t  t h o s e  u s i n g  one-half  t h i s  s i z e ,  
o r  0.0025. 
We a l s o  compared r e s u l t s  u s i n g  a n  
We found complete  a g r e m e n t  among t h e  c a l c u l a t i o n s .  
9 
We v a r i e d  t h e  i n n e r  r a d i u s  of t h e  s h e l l ,  a = 100, 500, 1000, and 
w e  compared t h e  i n t e n s i t i e s ,  r ,  a g a i n s t  t h e  cor responding  i n t e n s i t i e s  
f o r  t h e  p l a n e - p a r a l l e l  s l a b , ( 6 )  which should be ob ta ined  as a -, a. 
The r e s u l t s  are  shown i n  Fig. 1. The f u n c t i o n  r i s  shown f o r  t h e  c a s e  
A = 1, x = 3, f o r  t h r e e  a n g l e s  of  i nc idence ,  13.0,  6.00, and 88.5 
deg rees .  We s e e  immediately t h a t  the cu rves  f o r  t h e  s h e l l  geometry 
always l i e  on o r  above t h e  cu rves  f o r  t h e  s l a b .  I n  p a r t i c u l a r ,  t h e  
cu rve  f o r  88.5 d e g r e e s ,  w i t h  a = 100, l ies  as much as 50 p e r  cent 
above t h e  cu rve  f o r  t h e  s l a b .  A s  t h e  i n n e r  r a d i u s  a i s  i n c r e a s e d ,  
t h e  f u n c t i o n  r f o r  t h e  shel l  approaches t h a t  f o r  t h e  s l a b .  The two 
cases are g r a p h i c a l l y  i n d i s t i n g u i s h a b l e  f o r  a = 1000. 
of i nc idence  60 , we have  drawn i n  a dashed cu rve  f o r  a = 50. 
produced from a c a l c u l a t i o n  w i t h  N = 5 ,  s i n c e  t h e  c a l c u l a t i o n s  f o r  
N = 7 "blew up." 
For t h e  a n g l e  
0 It w a s  
Th i s  po in t  r e q u i r e s  f u r t h e r  i n v e s t i g a t i o n .  
We f e e l  t h a t  t h i s  method f o r  the  numer ica l  e s t i m a t i o n  o f  de-  
r i v a t i v e s  i s  a u s e f u l  one f o r  many a p p l i c a t i o n s ,  due t o  i t s  s i m p l i c i t y  
and accuracy .  I n  t h e  n e a r  f u t u r e ,  w e  s h a l l  produce S and r f u n c t i o n s  
f o r  t h e  s h e l l  problem by means of a p e r t u r b a t i o n  technique.  A com- 
p a r i s o n  of t h e  r e s u l t s  w i l l  r e s u l t  i n  a b e t t e r  e v a l u a t i o n  of t h e  
p r e s e n t  method. 
10 
i 
1 
0. E 
0.7 
0.6 
0.5 
r 
0.4 
0.3 
0.2 
0.1 
0 
I.-.-inn I I I I I I I 
90" 60" 30 O 0" 
Angle of reflection 
Fig.1 - Some reflected intensity patterns for shells with albedo 
A =  1 and thickness x = 3, for various angles of incidence 
11 
REFERENCES 
1. Lenoble, J . ,  and Z. Sekera,  "Equation of  R a d i a t i v e  T r a n s f e r  i n  a 
P l a n e t a r y  S p h e r i c a l  Atmosphere,'' Proc.  N a t .  Acad. S c i . ,  Vol. 47, 
1961, pp. 372-378. 
2. Sobolev, V. V . ,  and I. N .  Minin, "L igh t  S c a t t e r i n g  i n  a S p h e r i c a l  
Atmosphere - I," P l a n e t .  Space S c i . ,  Vol. 11, 1963, pp. 657-662. 
3. B a i l e y ,  P. B . ,  "A Rigorous Der iva t ion  of Some I n v a r i a n t  Imbedding 
Equat ions o f  T ranspor t  Theory," J. Math. Anal. and Appl., Vol. 8 ,  
1964, pp. 144-169. 
4. Bellman, R.  E . ,  H. H. Kagiwada, R. E .  Kalaba,  and M .  C. P r e s t r u d ,  
I n v a r i a n t  Imbedding and Time-dependent T ranspor t  P r o c e s s e s ,  Ameri- 
can E l s e v i e r  P u b l i s h i n g  Company, New York, 1964. 
5. Chandrasekhar,  S . ,  R a d i a t i v e  T r a n s f e r ,  Oxford U n i v e r s i t y  P r e s s ,  1950. 
6. Bellman, R. E . ,  R. E .  Kalaba,  and M. C. P r e s t r u d ,  I n v a r i a n t  Imbed- 
d ing  and R a d i a t i v e  T r a n s f e r  i n  S l a b s  o f  F i n i t e  Thickness ,  American 
E l s e v i e r  P u b l i s h i n g  Company) 1963. 


Numerical estimation of derivatives with an application to radiative transfer in spherical shells

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server