The method of spherical harmonics in a problem of critical parameters

TEE VETHOD OF SPHERICAL BARMONICS IN A PROBLEM OF CRITICAL PARAMETERS* 1. A. ADAMSKAYAand ELK. UODUNOV (Hoscoa) (Received

1.

The

13

!fay

1963)

systemof equations

In this work we examine the problem of determining critical parameters of spherical reactors in multigroup approximation by the method of spherical harmonics. We shall not be cancerned here with the relation of the method of spherical harmonics to the system of differential equations, since this has been described in detail in El]- [43. Following the idea described in this work a few programs were drawn up and a series of calculations carried out. Much of the work of writing the program, without which this work could not have been written, was carried out by I.F. Sharov. Considering the problem for 2n harmonics and m groups, we arrive at a system of 2 mn differential equations for 2 mn IU&IIOIVII functions Yij, i = 0, 1, . . . . (2n - l), j = 1, 2, ..., m. Here also and from now on we shall denote by the index i the number of the harmonic and by the index j - the number of the group. This system of differential

equations has the form

ai i=O,l,...,

*

Zh. vych.

sat.

4.

No.

(2n-1);

i=i,2

3, 473-484. 1964. 104

,...,

m.

Method

of

105

harmonics

spherical

Here i+l 4 = 2i + 1 I

ri = (i + i) (i + 2) ,

hi=&,

21 + 1

Vj is the speed of the reactions of the j-th group, time constant of the system) and p the density.

di=

i (i -11)

_-z-q-

A a parameter

Considering the values yij as the components of the vector dimensional space, we can rewrite this system in the form

(the

y in 2 mn-

(1.2) where P,

Q, V and 0 are matrices.

For a description of the matrices P, Q and V we shall of the form vector y as composed of m sections Y.01

l

Ylj i

consider

the

1. ,..., tn.

j=1,2

Y?n-l. j I

Then the matrices P, Q and V have the following cell construction: along the main diagonal stand the cells Kj Cj = 1, 2, . . . , ml. which are and all the remaining elements are equal to zero. 2n x 2n-matrices, It is obvious that the operation of such a matrix on the vector y = {yij) results in the 2n-dimensional vector {yir) being transformed by the matrix K1, the 2n-dimensional vector {yiz) being transformed by the matrix K2 etc. The cells Kj which enter into the composition of the matrix P, we shall from now on denote by Kp;, the cells entering into the matrix Q by KQ, and those matrix

V by h’vj.

entering

into’the

It turns out that KpI = Kp,

h’el = KQ, = . . . = KQ = KQ.

composition

= . . . = Kp

Ill

of the

= Kp.

Similarly

m

The element of the matrix Kp, which is situated at the intersection of the (i + 1) -th row and (i + 2)-th column, is equal to ai (i = 0, 1, of the . . . . 2n - 2); the element which is situated at the intersection (i + 2)-th row and (i + l)-th column is equal to bit1 (i = 0, 1, . . . . 2n - 2); all the remaining elements are zero. The element

of the matrix Kg,

situated

at the intersection

of the

I.A.

Adarrkaya

and S.K.

Godunov

(i + 1)-th row and (i + 2)-th CO~UIUI, IS equal to yi (i = 0, 1, . . ., the element situated at the intersection of the (i + 2)-th row and (i + 1)-th column, is equal to 6i+l (i = 0, 1, . . . , 2n - 2); all the remaining elements are zero.

2n - 2);

Formulae for the elements oi, The cells Kvj are different

b i,

yi and 6i have heen given ahove

for various j.

Each such cell

represents

a diagonal matrix, in which all the elements of the main diagonal will be equal to each other, i.e. equal to l/Vj, where Vj is the speed of the neutrons of the j-th group. The structure of the matrix D differs from that of the matrices P, and V. In order to describe the matrix D we write the vector y in the form of a column, consisting of 2n sections of the form

Q

Y.IL

1:I Yi, .

\

1

i=O,

I,.

.., 2n--1.

yimI

With such an order of distribution of the components of the vector y the matrix D will have a cell construction such that only the cells Di (i = 0, 1, . . . , 2n - l), situated along the diagonal are different from zero. Each cell Di is en m x m-matrix, in which at the intersection of the j-th row and k-th column there stands the element r~:~. At the end of the section we give formulae from which, as we know the components yij of the vector y, we can calculate the components of this vector transformed with the help of the matrices P-‘, Q, V and D. The components of the transformed vector will be denoted here by Xii. Formulae for the matrices Q, V and D follow directly from the above description of these matrices. Formulae for P-’ rnrwbe deduced without difficulty from the description of the matrix f’. We omit this deduction leaving it to the reader. The USC of

the

matrix

P- ‘:

if

{tij} =

P-'{yij}, then p=1.2 ,...,

zlj = Yf)jP

p=(k-I),

(n-2)

,...,

1.

11-l.

Method

The use of the

;Lbj =

of

spherical

matrix 0: if

2Ylj,

Q

(Xijl=

107

harmonics

IYijl

+ 2) y xp ,j = b i- 1) (CL p+l, 2p + 1

then i -

‘w

Yp-l,jt

p = 1, 2, . . . ) (2n - 2), Sm._,,

(2R-1)(2n-2)y j =

4n--

The use of

%e

the

matrix

V: if

use of the matrix &

if

_

2n z,j

1

l

{xii) = V (yij), then

(Xsj)

==I D

{TJ,,),

then

m

Xij

=

2

akjy,,.

k=l

2. Tbe general

idea

of

the

method

described

We shall consider the solution of the system (1.2) fo,

in the region

RI.

On the outer boundary of the region r =R we are given boundary conditions of the form Cy(Rj = 0, where C is some rectangular matrix. From the regularity of the solution at the centre some boundary conditions are obtained with F = 0. Howto obtain these boundary conditions will be described below in Section 4. The problem of looking for the critical consist of the following:

parameters of the system must

(1) considering the given me~urements of the reactor as fixed, to find the least value of the parameter A, for which equations (1.2) have a nontrivial solution, which satisfies the given boundary conditions, or (2) to determine with what least value of the parameter p. if h = 0, the system (1.2) has a non-trivial solution in the region EO,@I, which satisfies the given boundary conditions (ths problem of the critical measurementof the reactor) etc. The method by which we give a solution

of the given problem la the

108

I.A. Adarrkaya

and S.K.

Codunov

method of trial and error. Given successive values of a definite parameter (p or A) we solve the system of equations (1.2) and calculate each time some quantity A - the “discrepancy”, which, roughly speaking, shows to what extent the solutions fail to satisfy one boundary condition while they satisfy another. The trials are continued until it turns out that with the selected values of the definite parameter (p or A) the value of A is practically equal to zero. We shall describe in more detail how to obtain the “discrepencies”. Let us take a complete system of linearly independent vectors which satisfy the boundary condition at the centre. For fixed values of the required parameter (p or h) we shall extend (see 151) each of the vectors through the region [O, PRI (if we seek the parameter A then p = l!). On the outer boundary r = PR we obtain mn linearly independent vectors, each of which is the value of one of the solutions which are regular at the centre. On the other hand any solution of system (1.21 with r = PA, which satisfies the boundary condition on the outer boundary of the region, will be a linear combination of the system of mn basic vectors which satisfy this condition. In order that there should exist at least one non-trivial solution which is regular at the centre and satisfies the right-hand boundary condition, It Is necessary that the determinant A of the matrix composed of m vectors, obtained as a result of the extension, and the mn basic vectors of the right-hand boundary condition should be equal to zero. It is obvious that A = A(A) (or A = AQ311. Testing the calculations with various values of A (or p) we find the root of the equation A(A) = 0 CA(P) = 0) by the method of trial and error or by interpolation. We carry out the extension of the vectors, which satisfy the boundary condition at the centre, through the region [O, pR1 by the method given in [51. This method requires the division of the region [O, by the points 0 = r. < r1 < ... < r,,= R.

@I into n parts

We can make this division so small that, on each of the parts [rk, rk+Il the quantities p and atj may be considered as constants, not permitting any essential errors. Such an assumption about the sectional constancy of the coefficients p and a’ is convenient for us and will be

Method

of

spherical

harnonics

io9

used from now on. The integration of the system of equations on the part bk, rk+ll is carried out by means of the expansion of the solution in a Taylor series (see Section 3). After finding the critical parameter. to find the corresponding eigenvector y the reverse extension is carried out, which is described in paper t51 mentioned above.

3. The presentation of the solution sf system (1.2) in the fora of a Taylor series Suppose that we know the solution of the system (1.2): (gijj = {Gj} for some P = 3, Our problem will be the calculation of the value t?i+j

t'

+

'11.

In this paragraph we shall use the notation

(yij (z f 6)) = {Zij).

To calculate (xii) we makeuse of the expansion of the solution series of powers of s=y(R$

in a

(3.4)

For the calculation of successive give simple recurrence formulae.

derivatives

y’ (R), y” (Ti)

From equation (I.21 we find that

Introducing the notation

and using equation d” (yf r)

r-=__L.-dr”

d”rl dr”

n r C

p d”” (a / r) dr”-”

3

’

etc.

we

110

Z.A.

we obtain

the following

of the series

(3.1)

points

y.

{YIP} =

within

r = A,

y a8 solutions points

for the convergence Is necessary

of

eigenvalues.

= 0

such convergence

If

partial

sums

will

8 circle

coincides

on the plane of the

which there

lies

differential

rapid,

point equa-

of the equation). with any initial

Ib/RI < 1.

that

condition, if

however,

the convergence

the matrix

CIof the equation

P-‘(pD

-

Av)

det IIp-l(pD -

modulus, then for the series

be approximately

no singu-

y has only one singular

of the process_!3.3)

uo is the root

with

As is well known,

of the coefficients

this

with the largest

(3.2)

(3.1).

of the linear

and sufficient

Even with the fulfilment

w)-_lrq

successive

{Zij};

inside

The solution

may not turn out to be sufficiently has large

for

of the process

converges

with singular

Therefore, z it

formulae

of the series

r with centre

of the solution

coincide

vector

convergence

series

r = 0 (singular tion

Codunov

{Zfj};

of convergence

of

complex variable

{ulf’) =

{zij};

The conditions the conditions the Taylor

recurrence

and S.K.

(3.1):

{w$‘} =

lar point

Adanrkaya

(3.1)

the 8811188s for the Taylor

series

of the function

Therefore

If

method described (at the centre).

pj

is

large

it

Is necessary

for 6 to be small.

for obtaining the expansion is not applicable The method of Obtaining the expansion In this

be given in the following

The of E = Q case will

paragraph.

4. llm espamioa, in a merim, of lolutionm which ue rcgulu in the neighhourhood of the centrt We shall the centre,

look for a solution after

wrftlng

of equation (1.3) which lit? regular the latter in the following form:

at

Method

in the form of a series

spherical

glr

-k y2r2+

and yi = {yj:)} Here Y = {Yfjl given in Section I, 1. we shall first

111

harmonics

in powers of r:

?/o+

y =

of

’

l

l

+&f-g+ *

l

’

(4.2)

l

are vectors which have the structure

consider the case where yo # 0.

After substituting the expression for y and its derivative in equstion (4.1) and equating to zero the coefficients of various powers of r, we shall obtain a series of equations:

(4.3)

The matrices (1P + (2) have the same cell structure as the matrices P and Q, i.e. the cells (ZKp + KQ) different from zero are situated only along the diagonal. The square 2n X hmatrfx (El(p + KQ~ has the following construction: at the intersection of the (i + If-th row and the {i + 2)-th column there stands an element ni“I (i = 0, 1, . . . , %I - 1). at the intersection of the (i + 2)-th row and (i + I)-th column there stands an element rni$ (i CO, 1, ,.., 31 - 11, and all the remaining elements of zero. Here m(l)= l

q

IL-._-_ 2s + I

= I_ s-t-i 2s+

1 -I-

s(s1) 2s+l ’ (St-l)(s-t-2)

r=i,2,,..,(2n-1);

s=o,1,2 zs+i 3 1=0,¶, 2,..**R.

,...,

2n-2;

In order to study the operation of the matrix (lf’ + q) on the vector for us to consider the operation of the matrix CY$‘>, it is sufficient

112

I.A.

(&J + KQ) on the vector

Adasrkaya

{z#>

and

S.K.

with fixed

Godunov

j.

We note also that the determinant of the matrix (u(p t %I to the product of determinants of the second order 0

n:J

rn$)+

0

I

Let us consider the first

I

s=O,

,

is equal

n-i.

f,...,

equation of the system (4.3):

W’ + Q)Y,

=

0.

(*)

We first consider the case where yo # 0, and consequently the equation just written can hold only when the matrix (OP + Q) is singular. The element mi”) of the matrix (OKp + KQ) is equal to zero and in addition the determinant of the second order, which stands in the lefthand upper corner of the matrix, also vanishes; in view of the above remark the determinant of the matrix (OKp + KQ) is then zero and consequently also that of the matrix UP + Q). Thus the equation (‘1 may be valid also if the vector Yo # 0, but with definite conditions, imposed on the coordinates of this vector by the structure of the matrix (OKp + KQ). Applying the matrix (OKp + KQ) to the vector Ci =l, 2, . . . , “1. we obtain the equations nf)y$)

{YE)}

with fixed j

= 0,

mp)y$) + nf)y$) = 0, mr)y$)

+ nr)y$) = 0,

...... ........... m$Qff~_3,

j

+

(4.4)

n(g_,Y!$-l,j = Oy mpd__lypd_2j = 0.

Note that all the coefficients are different from zero.

of the equations (4.4).

except rnr’,

From the first equation it follows that y$) = 0, from the third etc. going through each equation from above downwardswe find that all odd components of the vector Yo are zero. ~$1 = 0

From.the last equation of the system (4.4)

it follows that ys+ j = 0.

Method

Going consecutively that all zero.

through

is satisfied consequently

the fact

with j = I,

y$)

next that

yg) = 1,

tain m linearly equations (4.3), any

harmonics

each equation

that

0

ntj") =

113

from below upwards we find YO up to

inclusive

are

the second of the equations

(4.41

2, ..., m). that

y$) = 1

and all

and that all

the remaining

the remaining

#,y) = 0

one of these

,The matrix

etc.,

y,!,;) = 0,

we shall

ob-

independent vectors, which satisfy the first of the each of which may be taken as the vector y. We shall vectors,

e.g.

that

for

(P + (8 is non-singular

determined

by the vector

y(O) o1 = 1

which

the remaining y!?) = 0 and show how to determine cients of the ser%es (4.2).

uniquely

y$

with any value of y$. Also since j = I, 2, . . . , m, then m components of the vector yo may be chosen arbitrarily

Assuming initially

take

spherical

the even components of the vector

In view of

(viz.

of

the remaining

and therefore

yO: yi = (P +

and all

Q)-’

the vector (pD -

From now on it is important for us to know the structure vector. We shall assume that (pD - hlr) y,, = zo.

coeffi-

y1 is

AT/‘)yO. of this

Bearing in mind the structure of the matrices D and C’ (see Section and also of the vector Yo, we can sav that the components zo of the vector

22)

Cj = I,

all the remaining . . . ) m) are zero.

2, . . .,

ml are generally

components

z$)

Applying the matrix (P + Q) to the vector thus obtained to ~0, we obtain the following tion of the components of the vector yl:

m(l) an-1

The only coefficient

of this

different

from zero,

and

(i = 1, 2, . . . , 2n - 1; j = 1, 2,

yl and equating the vector system for the determina-

y(1) 2n-2.j

=

system to vanish

0. is

mp).

The first

of

1)

I.A.

114

Adarrkaya

and S.K.

Codunov

the equations makes it possible to determine generally be numbers different from zero.

y$

for all j.

Let us consider the third equation of the system (4.51. rnp, = 0,

this equation is satisfied

if

y$) = 0

These will

Since

for all j.

Going also

through each equation from above downwards, we find that all odd components if the vector yi beginning with From the last equation it follows thnt

92)

are zero.

Y"'_ an 2.j = 0.

Going through

each equation from below upwards we find that all even components of the vector yi are zero. Thus the vector yl has the following structure: as a rule y$ # 0 if j = 1, 3, . . . , m, but the remaining components are zero. The vector has the seme structure. 21 = (PD - W Yl To determine the vector y2 we encounter the same difficulty as with the determination of the vector yo; the matrix (2?’ + @ is singular. Considering the structure of the matrix (Z’ + 0) and also the vector zi, we find the components of the vector y2 in the same way as we found the components of the vector yo. Here it happens that the vector @# # 0

y2

has the following

structure:

if j = 1, 3, . . . , m, and all the remaining components are zero.

Since the matrix (3p + Qj is non-singular the vector y3 is determined by y2 uniquely, whilst it turns out that the components of the vector @) differ Ya, y$' and yaj

from zero and the remaining components are

equal to zero.

It is easy to show by the method of mathematical induction that the odd vectors ?&l-1 (1 = 1, 2, . . . , (n - 1)) have all their even components equal to zero; the odd components up to and including the (21 - I)-th, for all j, are equal to numbers which in general differ from zero; from the (31 + I)-th up to the (3n - I)-th all the odd components are zero.

The even vectors ~21 (I = 1, 2, . . . , (n - 1)) have all their odd components equal to zero; the even components UP to the (31 - 3)-th are equal to numbers; the even components from the 31-th UP to the (3n-3)-th are equal to zero.

Method

of

rphcrical

harmonicr

115

In carrying out the proof we actually construct vectors ~22-1 and yzl (1 = I, 2, 3, . . . ) (n - 1)) possessing the properties given above and satisfying the system (4.3). For all I > 2n - 3 the matrices (IP + 0, are non-singular and therefore the vector Yl+i is determined uniquely from the vector yl by the equation

yI+l =

[(l + i)P + Ql”

(pD - AV) yl.

It was shown above that there exist m linearly independent vectors which satisfy the first equation of the system (4.31, each of which consequently may be taken as the vector ye. Taking one of them we have shown that despite the singularity of the matrices (Ip + @ for even 1, we may uniquely determine from the system (4.3) the remaining coefficients of the serfes (4.2). Similarly it can be shown that if we take any of these m vectors as yo we can determine all the remaining coefficients of the series (4.3). If we take successively each of these vectors as ye we shall obtain independent solutions of the equations (4. I) which satisfy the condition of regularity at the centre.

m linearly

3. We note here that the series which is a solution cannot start with odd powers of r.

of equation (4.1)

Let us assume that this is not so and that the series starts with an odd power: y = yILk_+r*-I + y,rak + . . . . Substituting this series in equation (4. I), equation:

we obtain for the determination of

[@k -

1) p +

y#_1 the following

QI?&s&1 = 0.

Since the matrices (1P + @ with odd I are non-singular, from this that the vector y2r-l is zero.

it follows

The series (4.3) cannot also start with powers of r greater than because in this case all the matrices (IP + Q) are non-singular. (2n - 3,

Let us consider the series beginning with the second degree: Y = yc)rr + yf)r*

+

. . . + yF)p

+

... .

(4.6)

BY a method similar to that described above it can be shown that there exist m linearly independent vectors which msy be taken ss the

116

Z.A.

and S.K.

Adoarkoyo

Godunov

vector yy and, for each of them we can determine successively the coefficients of the eerles (4.6) in spite of the singularity matrices

all of the

(2lP + 0).

Thus we obtain B more linearly independent (4.1) which are regular with r = 0.

solutions

of

equations

Considering the series which begin with the fourth power of r, with the sixth, the eighth etc., ending with the (2n - 2)-th power of r we obtain all mn linearly Independent solutions of the equations (4. I) which are regular in the neighbourhood of the centre. Each of the initial beginning

vectors

with the power

ponents of vector

= 1, ,$;-OL

Here

m$z;2) and

-1

vectors, . . . . n).

structure.

y(2'1-2)= 29 2.1.

n$;;z)

All odd com-

The even component 8 from the zero for some j = jo,

,(2cl-2)

are calculated

by the

m;w-2’mfQ-2)

(29-2)= ' y&i.

(-q-l

42.2-2),p-2)

' *.

l

(2P-2),t22cr-2) . . . #&22' ml (2P-2),f?-W . . . ng?-+2J ' 3

are the elements

The even components, beginning components with i f io are zero. ASSUming

has a definite

inclusive,

qq-2)

***’

2,

1Q-2) are zero. y’W_o

one to the (29 - 2)-th formulae y3-2’

r2‘+1,

. . . , n) of the series

y$"_;") (q = 1,

of the matrix

with the Wth,

that j = 1, 2, . . . * m, we obtain

which may be taken as the initial

all

vector

[(29-

and also

all

m linearly

2)p +Ql. the even

independent

y&9p-;2)(9 = 1, 2,

In the conclusion of this paragraph we give a summary of the formulae by which we can find all the mn linearly independent solutions for some r = 6 in the neighbourhood of the centre. we shall

denote

the initial

vectors

y:‘,“)

by

zi:),

(q

= 0,

1,

. . . ,

(n - 1); t = 1, 2, . . . , m). The index q shows with what power the series begins, which is a solution of the equation being studied. It has been shown that with any 9 we may take as the initial Independent vectors. The second index vector which is taken with fixed 9.

vector

tp snows the order

(?rl) m linearly

y2q

of the original

Method

of

117

horronics

spherical

By {Y#‘)t, we shall denote the I-th with degree 2q, in which as the initial

term of the series,

possible linearly independent vectors. l-th partial sum of this series.

By

tq = I, 2,. . . , m; In the case of even values application

{z$)}~

above. We formulate with regard to this Let

a$) = -

b . $2’

bij

m-2. j =

i =

character of

in the

The components

the matrix

ai:)

4p + 1 o(l) ap+l,j = (2P + 1) (1+ 2P + 2) p = 1, 2,. . . , (n-l);

(ZP + Q)-’

are determined

1) (I-

2n + 2) k-1,

sb2p.j - 2P(1L

from

4p--1 &-l, 1) (1 - 2P + 2) 0 p=n

34 + 1)=(I)

4P + 1

2p-1.

j

1 *

with 1- 2n + 2 # 0,

j

with l--n+2

0

((2P -

a(‘) 2p-2,

(2n -

by

as follows:

4n--1 a(‘)

and also

the

about which we have spoken

the application

Q)-’ {bij}.

we shal 1 denote

= {z$p))tpl

1 is some specific

a rule for specific.

{o$)} = (IP $

the components

Q

of m

2q + 1, 2q + 2,. . . .

I=

(ZP + Q)-‘,

of the matrices

we take the tq-th

vector

We assume that {&y)}rq = {z{?)}~~, {~if)}~, recurrence determine

beginning

2P(:;J,.i)

j-

U$) j]with(l-2p

-0;

+2)#0,

with(l-2p+2)=0; -1,

n-2

,...,

1.

5. A substitution which tramforms tbe solutions of a system of equations with constant coefficients into solutions of system (1.2) In this paragraph we wish to give without detailed substitution which brings the spherical system Pg+;Qy+Wy=pDy

proof

one curious

(5.1)

118

Z.A. Adarrkaya

and S.K.

Godunov

to the “plane” form (5.2) This substitution may be useful both for analytical investigation and for carrying out calculations. In some of our programs it has been used for eelecting solutiona which are regular at the centre. The substitution has the form y = T(r)z, where T tr)

=

+‘+$

E, [(pD -

Al!)‘‘PI +

l

l

l

+-&Em-,

[(PD -

km-‘P1’“-‘.

The matrix Ei have a structure similar to the structure of the matrices P and Q, viz. only along the diagonal are there cells KE which L are different from zero. The 2n x In-matrices intersection an element

uik represent

w = +, PI

are obtained

(r) =

that

the coefficients

+-&.

i+,(r)

. - ., clp+1 (4

are connected

P pp y

by means of the follow-

(r) -

h-1

-

q+P

P)

0.

Let us consider

&El+&E2+ . . .

By means of the formulae for +,(r) the identity PF+(PM

=

ui(r):

by the relation

Pp+d-

The scheme of the proof is as follows.

M(ks +fEO+

of the polynomials

by recurrence

3 + k (PP-

fies

at the

of the j-th row and j-th column, with j > i, there stands and all other elements are zero.

These polynomials ing equations:

We can verify

construction:

pj-l,i+l,

The elements

PO

Kgi have the following

(k-

the matrix Parameter)

we can prove that l+j(k, r)

-MP)k+fQM=O.

satis-

Method

of

spherical

119

hataonics

Let us consider the particular solution of the plane system (5.2) of the form z = ekrze, where k is the root of the characteristic equation detI(kP + hV -

pDll=

0.

For simplicits of reasoning we shall assume that this equation has no multiple roots although in the final result there would be no such roots. We can show that the function y = 1% will be some particular solution of the system (5.1). In order to convince ourselves of this we carry out the following transformation of the expression P$+;Qy+(W-pD)y=P$z+PM~+fQMz+(W-pD)Mt= ZZ Pdgz -

+ (PM -MP)$+fQMz+

pD)l z + M[P g

M (AV -

l(hV-pD)M+ (hV -

pD) z].

Since dz/dr = kz and the matrix M is commutative with the matrices V and D we arrive at the equation

Since

and t satisfies to prove:

equation (5.2)

we obtain the equation which we intended

P$+;Qy+(bV-pD)y=O. Since z = @q, l/k

(PD -

vector matrix

T =

is a solution of system (5.21,

is the ssme as applying

to z ~J-Ioperator

given

then multiplying z by by the matrix

,rhus we see that the operation of the matrix V on the z is equivalent to the operation on this vector of the following 1’ which does not depend on k:

W)_lP.

+E + -;l;-El [(pD

-

W)-‘PI

+ . . = + f&n_1

t(pD -

J.VY-lpl~“-‘.

Thus we have shown that if z is a Particular solution of system (5.2) of the form ekrzO, then y = 7’~.is a Particular solution of the

120

I.A.

Adaaskaya

and

S.K.

Godunov

system (5. I). If we take solutions of the system (5.2) of the form ekrz,,, corresponding to all the 2 mn roots k of the characteristic equation, then on transforming them by means of the matrix T, we shall obtain 2 mn linearly independent solutions of the system (5.1). Their linear independence follows from Liouville’s theorem for the gronskian and from the fact that T =2:(l/r) E as t + M. Since any solution of the system (5.2) may be obtained as a linear combination of the Darticular solutions considered, we see that the matrix T transforms every set of solutions of the system (5.2) into every set solutions of (5. I), i.e. if in system 15.1) we make the substitution y = Tz, where the matrix T is defined by the equation (5.41, we shall obtain for L a system of equations with constant coefficients (5.2) *

Translated

by

W.F. Cleaves

REFERENCES

The tbeory of neutron transfer,

1.

Davison.

2.

Vladimirov, V. S. , Mathematical problems

B.,

of particle 3.

Marchuk, raschcta

transfer.

0. I.,

Methods

yadcrnykh

Tr. Matern. of reaktorov).

calculation

Interscience,

in the

in-ta for

single

Akad.

SSSR,

theory 1961.

reactors

(Metody

Nauk

nuclear

Gostomizdat,

1956. speed

1961.

4.

Adamskaya. I. A., A description of the kinetic equation in spherical harmonics (curvilinear coordinates - the axial-symmetric case). 3, 5, 927-941, 1963. Zh. uych. mat.,

5.

Godunov, S. K. , On the numerical solution of for systems of ordinary linear differential natcm. nauk, 16, 3(9§), 171-174, 1961.

boundary value problems equations. Uspckhi