PERTANIKA 14(3), 331-346 (1991)

The Group Explicit Methods for Parabolic Problemswith Cylindrical and Spherical Symmetry

MOHD SALLEH SAHIMI and DAVID J. EVANS'Department of Mathematics

Universiti Kebangsaan Malaysia43600 UKM, Bangi, Selangor Darul Ehsan, Malaysia.

Key words: GE method, cylindrical and spherical symmetry, error cancellation

ABSTRAKStrategi tak tersirat kumpulan (TK) yang telah digunakan dengan jayanya oleh Evans dan Abdullah (1983) kiniditerapkan pula ke atas masalah parabolik yang melibatkan domain sekata dan mempunyai kesimetrian silinder dansfera. Hasil pengiraan menunjukkan kaedah TK masih boleh digunakan dan mempunyai ciri-ciri kebaikan sepertisebelumnya. Walau bagaimanapun, julat kestabilannya terhad.

ABSTRACTThe group explicit (GE) strategy which has been used successfully to solve parabolic partial differential equations inregular domains involving 1 and 2 space dimensions by Evans and Abdullah (1983) is now applied to parabolicproblems involving regular domains that possess both cylindrical and spherical symmetry. The results indicate that theGE methods are still applicable possessing many of their previous advantages but with a reduced stability range.

INTRODUCTIONConsider the following parabolic equation in one-space dimension

3U 3^U a 3U ( 1 !)dt 3r2 r dr

together with the initial-boundary conditions

U ( r , 0 ) = f ( r ) , 0 < t < T

and

au o, t) = o, u(i, t) = o, o < t < T ( L 2 )

Equation (1.1) reduces to the simple diffusionequation when a = 0. Evans and Abdullah (1983)have successfully implemented the GE algorithmsfor this equation. We shall now extend the GEapplication to parabolic problems that possesscylindrical and spherical symmetry by putting a = 1and 2 respectively. The Group Explicit with RightUngrouped Point (GER), the Group Explicit withLeft Ungrouped Point (GEL), the Single Alternating

Group Explicit (SAGE) and the (Double)Alternating Group Explicit (DAGE) schemes willbe developed and the stability requirements aswell as accuracy established.

2. DERIVATION OF THE GE SCHEMESFollowing Evans and Abdullah (1983), thegeneralised formulae to approximate thederivatives in (1, 1) at the point (r, t.+1/2) = (iAr,(j+1/2) At) are given by

^r ui+l/2,jfl ~®2 ^r ui

l5rUi-i/2,j )/(Ar)2 ,

A>l/2ui+l/2,j+l -*

and

(2.1)

(2.2)

(2.3)

Present address: Parallel Algorithm Research Centre, Loughborough University of Tech, UK

MOHD SALLEH SAHIMI AND DAV1DJ. EVANS

where Ar and At are the increments with respect to ui-i,j+i =[(1 + Pi)Pi-i ui-2,j +(1 + Pi-l)ui-i.j + (\\-i

the r- and t-axes and Ar> Vf> and 6r are the usual ( l -q^u , : + q i_ i q i u i+1,jl/(l 4 p, +q, ,)forward, backward and central difference operatorswith respect to r. The finite-difference analogue of(1. 1) is therefore given by and

(2.9)

+ a 4A ru y)

where X = At/(Ar)2 the mesh ratio.

(2.4)

If we choose 0j = 94 = 1; 62 = 93 = 0; a} = a2 =1and a3 = a4 = 0, equation (2.4) gives us the followingright-left (RL) approximation,

= (l-pi)uij + P i u i _ 1 J ( 2 5 )

whilst with the choice of 0j = 94 = 0; 02 = 63 =1; a{ =a2 =0 and <x3 = a4 = 1, we get the following LRanalogue

i - l . 2, . . . ,m- l (2.6)

By applying the equation (2.5) at the point

(IUl» ^1/2) w^ find that - q M u ^ , +(l + qi_i)

Ui-l,j+l =(l-Pj-l)ui_]J + Pj-iU^j

which on coupling with (2.6) forms the system

[a+qi+i)-qi-l irui-l.j+ll= f< 1-Pi-l ) 0

[-p, a+Pi)J[uiij+I J [o a-qi)Jui,j Lqi U'+1*J J

Au = Bu + u

(2.7)

i.e.

u -

This leads to the following explicit equations forgeneral points not on the axis

P i + q M )

(2.10)

From the L' Hospital's rule, the followingrelationships hold on the axis,

[-1

Choosing Gj =G4 = 0; 02 = 03= 1; a3 = a4 = 1 a n d ^

= a2 = 0 in (2.1), (2,2) and (2,3), the approximations

- . . . du d u , duof the derivatives — , — r - and—— at the points

(0, j+1/2) on the axis can be obtained.

The substitution of these derivatives into (2.11)leads to the approximations,

uo,j+l - u 0 j - ( l + a ) f c ( u w - u O j - u 0 ( j + i

+ u - i ,> i ) (2.12)

and by utilising the boundary condition (1.2) in

which —— (0, t) = 0, we arrive at the followingar

formula for the fictitious values u_i L^ ,

u-ifj+i • uo,j+i + u i j ~UOJ (2 13)

By inserting (2.13) into (2.12), we therefore obtainthe approximations to the left boundary values.

= (l-2d)u0j (2.14)

where d = (1 + a)X. The solutions at the singleungrouped point near the right boundary can beobtained from (2.5) by taking i = m - 1 . Hence, theGER scheme is expressed by the following implicitequations

»,~0: + 2CXUU

i=2, 4, -,

M_l,j)l(m-4). (m-2)

I m even and

m > 4

332 PERTANIKAVOL. 14NO.3,1991

GROUP EXPLICIT METHODS FORPARABOIJC PROBLEMS WITH CYIJNDRKIALANDSPHERICALSYMMETRY

Pm-I

u=(uoj,uIj>...,um_l/, b | n o - ^

with v

J '

or in matrix form

(l + CM )u = (l + G 2 ) u + b

b - (I + G , ) - 1 b

(ii) The GEL Scheme

i.e. u = G,) * (I + G 2 ) u + b (2.15) By choosing e i = 0 4 = l ; e 2 = e3 = 0, in (2.1), we have~ .F1 ~j "*! as approximation to (2.1) at r = 0, the formula,

where I is the identity matrix, (l+djuQj+i -Otu i j+i= (i-oc)u0j + d U_J J

G, =

and

f °

1

r (i)

I

c (2)

1

G<l/2)(m-2))

0

(mxm)

'2 •

1 -2d -2dPi -Pi

G 2( 1 )

1 1

^(l/2)(m-4))

q»n-2 qm-2

Pm-1 -(Pm-1 + qm- l )

1

with

r ^ q . i . mX 2LPa+i -Pa+i J

From (2.6), we also obtain, with i = 1, the relation,

(2.17

PERTANIKAVOL. I4NO.3, 1991

MOHD SALLEH SAHIMI AND DAVID J. EVANS

Equations (2.16) and (2.17) form the coupled andsystem,

[ • i f , . 1 r ^ 1"u 1 u l t+

(1-KX) -a uo,j+i _ (1+a) 0 I uoj *J

-Pi d+Pi)J [ui.j+ij " 1° fl-qi^uijj

1

(1+ p , +a)

(1-qi) + (p

{Pl(l-d)uOj + (1 + d)

2i} (2.19)

which leads to

+ d)

These equations togetherwith the system (2.7) and(2.8) for i = 3,5,..., (m-1) describe the GEL schemewhich can be written in matrix form as,

(I + G2) b =1

u + b—j ~2

i.e u = + b

By substituting a ] = a = 0 and a2 = a 4 = l in (2.2) and

utilising the boundary condition | -̂ — = 0,Ao,>l/2)(du)

>n ^—

-2 -2

V or ; (oj+i/2)we obtain the following approximation

u l , j - u _ 1 j l / 2 A r = 0 a n d hence - U . J J = u1(j

With these values, the required equations for theleft boundary as well as the single ungrouped pointadjacent to it are given respectively by,

(ii) The (S) AGE and (D) AGE Schemes

The alternating group explicit schemes are formedby the application of the GER and GEL processes intheir appropriate sequences. Thus, the followingformulae constitute the (S) AGE two-step scheme,

j = 0, 2, 4,..

(2 + p1-q1)du1j+ d q i u 2 j } (2.18)

u + b

(I+G2) ?M*(1+6 .) V b,

-(2 + p!)d (2 + p 1 -q 1 )d dq ,

(1 + Pi+d) (l + p]+d) (l + p]+d)

o P 2 - P 2

-G,<2>

L_- G / 3 >

l

, - q m - i

334 PERTANIKAVOL. 14NO.3,1991

GROUP EXPLICIT METHODS FOR PARABOLIC PROBLEMS WITH CYLINDRICAL AND SPHERICAL SYMMETRY

and

1 0 0

0 0

1

c (i)

r (2)

1

2

1(mxm)

whilst the (D) AGE four-step scheme is given by

(i + G ) u = (I + G2)u + b,v ' ~>i ~j ~j

fl+G,) U u + b ,-j+1 ~jf2

(I + G ) u = (I + G ) u + b ,-j+3 ' ~j+2 ~2

and

( i + G ) u = ( i + G , ) u + b

, a i + a 2 = 5. (3.1)

j = 0, 4, 8,.. Similar expansion of (2.4) and (2.10) about thepoints (ri-1, tj+1/2 and (ri, tj+1/2) lead to thefollowing t.e at the general grouped points,

L 2 . 9

3. TRUNCATION ERROR ANALYSIS

(i) Truncation Error (L e.) for the GER Scheme

Taylor's expansion of UQ +1>U0 andUj about the

point (rQ, t+1 / 2) provides the t.e. for (2.14)

)(l + P i + q ^ X A t K — )i_i>jti/2

T - TLB '214

dr

(Ar)(At)2

12

4 d ( A r ) 3

1 .

i-i } ( A r ) 2 ( A t ){ a3u

drdt

<7irdr dt i-1

(] + 15q,)q1_1}(Ar)4 ( — ) , _ ,

dr

PERTANIKAVOL. 14NO.3,1991 335

MOHD SALLEH SAHIMI AND DAV1DJ. EVANS

(Ar)4

16L

+ O(Ar)otl(At)),a1 + a2 = 5;

and

T2.10

(3.2)

dU,

>i + q1_1)(At)(^) i )

ar

^{Pi(l+Pw)-a+qw)qi}(Ar)(At)2(^)i>>1/2

24

(1 + qi-l )qi}(Ar)4 (— T ) i ) > 1 / 2dr

or dt

ai*3 l j a

0((Ar)a'(At)a2),a2 + a2 = 5. (3.3)

Finally, from equation (2.5) with i=m-l, the t.e.of the approximation near to the right boundary isgiven by

-l.jfV2

,d3U.

(At)( )3u

33U1 9 3 U-(p i-q i)(Ar)(At)2(T-T)m_1) j+1/2

o drat

where O] + a2 = 5.

• O((Ar)a' (At)"2),

(3.4)

(n) Truncation Error for the GEL SchemeT h e Le- at t h e lef t boundary and at the ungroupedpoint near to it can be derived by expandingequations (2.18) and (2.19) about the points (rQ,

TLB "

dr)

3r2

d3U.^ ( 2 + Pi+3qI)(Ar)^(A)(^)0j+1/2

^ _ .. o ' u,

336 PERTANIKAVOL. 14NO.3,1991

GROUP EXPLICIT METHODS FOR PARABOLIC PROBLEMS WITH CYLINDRICAL AND SPHERICAL SYMMETRY

12

16

+ - (2 + P l

384

0).

3r3r

2

0((Ar)ai(At)or2)anda1 + a2 = 5;

andTL= T2.19

Al,at

24 at

ar at

arot

l,jfl/2

0((Ar)ai(At)a2)anda1 + a2 =5.

l.j+1/2

(3.6)

(3*5) The t.e. of the scheme at the remaining pointsgrouped two at a time are given by T2 and T2

respectively for i = 3, 5, ... (m-1)(Hi) Truncation Error for the (S) AGE and (D) AGESchemes

The t.e. of these alternating group explicitmethods are given by the t.e. of the GER and theGEL schemes when they are applied in theircorrect sequence.

4. STABILITY ANALYSIS

We shall show here the analysis of stability of theGERschemeforthecaseot = 2 (spherical symmetry)only. From (2.15), the explicit form of the GERscheme is

u. , • rGER u. + b,

whererG£R = ((I + G1)~'(l

matrix which is given by

), the amplification

rGER •

2a

Pl2

S12

0

q i 2

0

r12

P3,4

•

P m-3, m-2sm-3, m-2

Pl2r3,4

93,4

«3,4

P3.4

^ m-3, m-2 rm-3, m-2 sm-3, m-2rm-3, m-2 9 m-3, m-2 P m-3, m-2

Pm-I,m 9m-l,m

PERTANIKAVOL. 14NO.3,1991

(mxm)

337

MOHD SALLEH SAHIMI AND DAV1DJ. EVANS

m even, m > 4where (withV.. = 1+ p. + q. and p =0)

y j l m

/ V i j ' ^ij a n d P m " °)

For i = 3,..., (m-3), we find that

if Pi < 1 then X < —

andifq i+1 < 1 then X -—-

Sy -qjqi/Vij.Pi

d-qJ)/v8,N o w fo rX < min. —)

Wehavea = 3X, pi = -, q, = 2A,andq2= —. The Pi+PiPi+l +

characteristic equation of the matrix IQER is |D | = ,,.

det (rGER = jil) = 0 where (X are the eigen values of _ ,rGER- Now, if the expand the determinant by ther , /i ci \ l For the third row, the sum of the moduli of thefirst column, we get u = (1-6/.) as an eigen values

elements of the rows isof rGER. For stability, we require |(J,|< 1 which leadsto the conditions « n

(4.1)

For the second and subsequent even rows of rGER,we have as the sum of the moduli of the elements ofthe row

Again, if X < — as in (4.2) we have

For the subsequent odd rows pf r we get

= +-

i=l, 3,...., m-3) , m>4

Since p. and q. are non-negativeFor i =1 we obtain

Up 2 +q 1 | l -q 2 | - r -q 2 q 1bo =

If q2 < 1 then -X < 1

For these values of X,

s u

l + P2+qi)

23

= l.

(4.2)

i = 3, ..., (m-3) with m > 4

Again using (4.3), if X ' then(i+2)

si+2 = 1

An application of Brauer's the theorem to the lastrow leads to

Pm-1

d+qm-i)

i. e.,

Let = 1 " 2 P m - 1 andn2 =0+q m - i )

The requirement |^, | < 1 implies -1<

d+q m - i )

~] <1,

or -J-qm-l ^ l-2pm_, < l + qm_ r

338 PERTANIKAVOL. 14NO.3,1991

GROUP EXPLICIT METHODS FOR PARABOLIC PROBLEMS WITH CYLINDRICAL AND SPHERICAL SYMMETRY

The left - hand side inequality gives

m

(m-1)

X<

(m-1)-X<2

The second requirement

l̂ i 21 < 1 implies

all X > 0.

(m-4)

< 1 which is true for

Therefore, for overall stability, using (4.1), (4.2),

1 2 . i+ l - o , o u 2 m - 2-, - , m m ( ,1 = 3, ..., (m-3)),3 3 i + 2 m-4

for m > 4

23'

Hence the GER scheme for the sphericallysymmetric parabolic problem is conditionally

stable for

where J0((3r) is the Besselfunction of the first kind of

order 0 and p is the first root of J0(P) =0. The exactsolution is

u(r,t) =J0(pr)e-P2t (5.2)

and the values of the Bessel function at the gridpoint are generated using the NAG librarysubroutine (the first four roots of J0(P) = 0 are p} =2.405, p2 = 5.520, ps =8.654,p4 = 11.79). In Tables1-3 are displayed a comparison of the numericalsolutions of the GE schemes with the exact solutionsat the appropriate grid points in terms of theirabsolute errors for various values of the mesh ratioX. The absolute errors of the solutions of theexplicit (EXP) scheme and the Crank-Nicolson(CN) method are also included.

Experiment 2The following parabolic problem with sphericalsymmetry (Saulev 1964) is considered

3U 32U 2 3U Ly ,

Results of the stability restrictions of the otherschemes in the GE class can be obtained by applyingthe same procedure as the above and are shown tobe

Method

GER

GEL

S(AGE)

D(AGE)

a •

X<

—

X<

x<

1

0.454

0.85587

1.0006

a

X

X

X

= 2

-

< i6

3

< 13

5. NUMERICAL EXAMPLES

Experiment 1This experiment dealt with the solution of thefollowing parabolic problems with cylindricalsymmetry (Mitchell and Pearce (1963)

du 1 3u d2u ,~ . - -.— = - — + , (0 < r< 1), /f- ix

given the auxiliary condition

u(r, 0) =J0(pr), 0 < r < 1,

and

| i ( 0 , t) = 0,t > 0,dru(l, t) = 0, t > 0,

(k(r, t) =e~ t{[6+(l-r2)rc2t2-(l-r2)]cos(7trt)-

[( l-r2)r + 4rt-2t(l-r2)/r]7csin (rat)}]

subject to the initial-boundary conditions

U(r, o) = 1-r2,

^(0,0 = 0,or

and U(l, t) = 0,

with the exact solutionU(r, t) = (l-r2)e"L cos(Tirt)- (5.4)

Since our parabolic equation incorporates asource term k(r,t), some modifications on the basicequations governing the GE schemes are thereforerequired.

The numerical solutions of the above sphericalproblem using the GE schemes are obtained forvarious values of X and to indicate their accuracy,Tables 4-6 provide a comparison with the exactsolution in terms of their absolute errors.

It is observed that, presumably due to the terma au— -jT" in (5.1) and (5.3) (with a = 1 and 2respectively),the solutions of the GE schemes areslightly less accurate in the vicinity of the axis (r = 0)than in the remainder of the field. As we have

PERTANIKAVOL. 14NO.3,1991 339

s

bo

TABLE 3The absolute errors of the numerical solutions to the cylindrical

problem t = 0.6, X = 0.6, At - 6.003, Ar = 0.1

\ . r

Method\

(S)AGE

(D)AGE

ON

EXACTSOLUTION

0

1.39X10"3

2.88X10"4

9.75X10"4

0.0311041

0.1

1.39xlO"3

2.72xl0"4

9.63xlO"4

0.030656

0.2

1.3xlO"3

2.86X10"4

9.23x10^

0.0293309

0.3

1.25xlO"3

2.18X10"4

8.57X10"4

0.0271860

0.4

LOSxlO"3

2.5X10"4

7.69xlO"4

0.0243135

0.5

9.58xlO"4

1.43X10"4

6.62xlO"4

0.0208362

0.6

6.93X10"4

1.87xlO"4

5.4x10~4

0.0169017

0.7

5.82xlO"4

6.33xlO"5

4.08xl0"4

0.0126752

0.8

2.93xlO"4

1.2X10"4

2.71X10"4

0.0083321

0.9

1.84X10"4

2.35xlO"5

1.35X10"4

0.0040493

Average ofallabsoluteerrors

9.09X10"4

1.85X10"4

6.50x10"4

-

oo

I

T-

I

TABLE 4The absolute errors of the numerical solutions to the cylindrical

problem t = 0.175, X = 0.175, At = 0.00175, Ar = 0.1

\ r

Method\

GER

GEL

(S)AGE

(D)AGE

EXACTSOLUTION

0

7.86x10"4

6.23xlO"4

5.53x10"4

5.73X10"4

0.839457

0.1

7.88x10"4

6.27x10~4

5.57xlO"4

5.69X10"4

0.828068

0.2

7.9 lxl 0"4

6.01xl0"4

5.37xl0~4

5.56x10"4

0.8010120

0.3

7.64x10"4

5.89x10"4

5.15xlO~4

5.29xlO~4

0.07535391

0.4

7.5xl0"4

5.43x10"4

4.78xlO"4

4.94xlO"4

0.6881618

0.5

7.23X10"4

4.84x10"4

4.3xlO"4

4.52X10"4

0.6059549

0.6

6.41X10"4

4.45x10"4

3.91X10"4

3.88x10"4

0.5082867

0.7

6.45x10"4

2.75xlO"4

3.0xl0"4

3.42X10"4

0.3968087

0.8

4.15X10"4

2.82xlO"4

2.77xlO"4

2.31xlO"4

0.2734428

0.9

9.9x10"4

9.27x10"5

1.16X10"4

1.85xlO~4

0.1403673

Average ofallabsoluteerrors

6.79X10"4

4.56X10"4

4.16X10"4

4.32X10"4

ID

i-OaC/5s73

s

1

122

I2?

Is

O

TABLE5The absolute errors of the numerical solutions to the cylindrical

problem t = 0.6, X = 0.3, At = 0.003, Ar = 0.1

\^ r

Method\

GER

GEL

(S)AGE

(D)AGE

EXACTSOLUTION

0

1.04xl0"2

9.85xlO"3

9.71xlO~3

9.87x10"3

0.5488116

0.1

1.03x10*

9.77xl(T3

9.64xlO"3

9.7x10"3

0.5336998

0.2

9.88xlO"3

9.19X10"3

9.04x10"3

9.2X10"8

0.4898613

0.3

8.97xlO"3

8.48xlO~3

8.23xlO"3

8.3xl0":^

0.4216731

0.4

7.99xlO"3

7.32xlO"3

7.06xl0"3

7.23xl0"3

0.3360558

0.5

6.66x10"3

6.11xlO"3

5.79x10"3

5.92x10"^

0,2419376

0.6

5.24X10"3

4.74x10~3

4.44xlO"3

4.51xlO"3

0.1495505

0.7

3.91xlO"3

3.19xlO"3

S.OxlO"3

3.2xlO"3

0.0696068

0.8

2.28xl0"3

2.06xl0"4

1.92X10"3

1.8X10"4

0.0124057

0.9

1.39X10"3

5.12x10^

7.11X10"4

9.28X10"4

-0.013068

Average ofall

absoluteerrors

6.71xlO"3

6.12xlO"3

5.96xl0"3

6.07xl0"3

-

Ia

a

iD

I*0

55

0PI

3

TABLE6The absolute errors of the numerical solutions to the cylindrical

problem t - 0.6, X = 0.6, At = 0.006, Ar = 0.1

\ r

Method\

(S)AGE

(D) AGE

EXACTSOLUTION

0

7.91xlO"3

9.01xl0"3

0.5488116

0.1

8.45xlO"3

9.16X10"3

0.5336998

0.2

7.55xl0"3

8.8xlO"3

0.4898613

0.3

6.74xlO"3

7.64X10"3

0.4216731

0.4

5.28xlO"3

6.69xlO"3

0.3360558

0.5

4.01X10"3

5.24X10"3

0.2419376

0.6

2.79x10'3

3.82x10"3

0.1495505

0.7

1.37xlO"3

2.73xl0"3

0.0696068

0.8

1.17xl0"3

9.33X10"4

0.0124057

0.9

1.35X10"5

1.12X10"3

-0.013068

Average ofall

absoluteerrors

4.53x10"3

5.52X10"3

-

1os70

8

D

2

im2

S I•

MOHD SALLEH SAHIMI AND DAV1DJ. EVANS

already seen, special equations have to beformulated to cope with this difficulty at the pointof singularity. In fact, an examination of thetruncation errors of the GE schemes at r = 0

Atindicates the presence of the term—(equations

(3.1) and (3.5) and it is therefore essential that toattain consistency, At approaches 0 faster than doesAr. For our cylindrical problem, it is also found thatthe GE class of methods is more accurate than theother schemes under investigation. From Table 2,the (S) AGE and (D) AGE methods in that orderare more superior whilst in Table 3, (D) AGE hasthe edge on other different formulae. This is to beexpected since the truncation error expressions ofthe constituent GER and GEL formulae possessterms of different signs and hence the correctalternate applications of these formulae toconstitute the (S) AGE and (D) AGE schemes canlead to cancellations of error terms. The sameobservation also applies for our spherical problemalthough comparative results from other schemesare not available. However it suffices to say thatamong the GE class of methods, the (S) AGE methodgives a more satisfactory result.

We conclude that despite the limited stabilityof the GE schemes, their stability ratios are not sorestricted as to be impractical for implementationfor special geometries when compared with otherschemes. Being explicit, they are simple and incurlow computational load and above all exhibit betteraccuracy. The use of the alternating schemes inparticular, i.e. (S) AGE and (D) AGE, is highlyrecommended.

REFERENCES

EVANS, DJ. and A.R.B. ABDULLAH. 1983. Group ExplicitMethods for Parabolic Equations. Intern. J.Computer Math. 14:73-105.

MITCHELL, A.R. and R.P. PEARCE. 1963. ExplicitDifference Methods for Solving the CylindricalHeat Conduction Equation. Maths. Comp. 17:426-432.

SAHIMI, M.S. and D J.EVANS. 1987. The Group ExplicitMethods for Parabolic Problems with SpecialGeometries. Technical Report: Computer Studies LUT382.

SAULEV, V.K. 1964. Integration of Equations of ParabolicType by the Method of Nets. Oxford: Pergamon Press.

(Received 26July, 1991)

346 PERTANIKAVOL. 14NO.3,1991