N-wave solution of modified Burgers equation

Applied Mathematics Letters

Applied M a t h e m a t i c s L e t t e r s 13 (2000) 1 6

PERGAMON

www.elsevier.nl/locate/anti

N - W a v e Solution of Modified Burgers Equation P.

L . S A C H D E V AND C I t . S R [ N I V A S A R A O Department of Mathematics Indian Institute of Science, Bangalore, india sachdev~math, i i s c . e r n e t , in

(Received August 19.99: accepted September 1999) Communicated by J. 1~. Ockendon A b s t r a c t - - T h e modified Burgers equation ut + ~L'~u, = 6 x:r/2, where 'n _> 2 is even, is trea.ted m~alytically for N-wave initial conditions. An exact asymptotic solution is presented, extending the validity of the linear solution far back in time. @ 2000 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - M o d i f i e d Burgers equation.

1.

INTRODUCTION

In this p a p e r , we consider t h e Modified B u r g e r s E q u a t i o n ( M B E ) 5

(1.s)

w i t h N - w a v e initial c o n d i t i o n f

>t

t 0,

otherwise,

_< 1,

(1 2)

a n a l y t i c a l l y . E q u a t i o n (1.1) has varied a p p l i c a t i o n s (see for e x a m p l e , [1,2]). L e e - B a p t y and C r i g h t o n [3] a n d H a r r i s [4] s t u d i e d e q u a t i o n (1.1) for 5 sufficiently small, using s i n g u l a r p e r t u r b a t i o n techniques. Here, we a d o p t an a p p r o a c h which is an e x t e n s i o n of t h e m e t h o d of S a c h d e v a n d his c o l l a b o r a t o r s [5-7], w h e r e i n we mimic B E and e x t e n d tile linear (old age) s o l u t i o n so t h a t tile n o n l i n e a r effects are t a k e n into account. This a p p r o a c h is e x a c t in t h e sense t h a t we can, in principle, include as m a n y t e r m s as we m a y to improve t h e a c c u r a c y of t h e sohltion. T h i s s o l u t i o n e x t e n d s t h e v a l i d i t y of t h e linear solution far back in time. By using an i n d u c t i o n a r g u m e n t , we show t h a t this s o l u t i o n t e n d s to t h e old age s o l u t i o n as t -+ .~c (see Section 2). W h i l e we are u n a b l e to cover a n a l y t i c a l l y t h e c o m p l e t e t i m e d o m a i n of t h e solution, our s o l u t i o n s p a n s t h e entire n o n l i n e a r regime e x c e p t some initial time. T h e a u t h o r s wish to express their sincere t h a n k s I,o t h e referees for their criticM review a n d valuable suggestions. T h e first a u t h o r (PLS) was s u p p o r t e d by C S I R (India) Project No. 2 1 ( 0 4 0 0 ) / 9 7 / E M R - I L 0 8 9 3 - 9 6 5 9 / 0 0 / $ - see front m a t t e r @ 2000 Elsevier Science Ltd. All rights re.served. PII: S0893-9659(99)00199-8

T y p e s e t by A,~d~';-~I~X

2

P.L.

SACHDEV AND CH. SRINIVASA RAO

2. N O N L I N E A R I S A T I O N

OF

THE

LINEAR

SOLUTION

We attempt an asymptotic analytic solution of the N-wave problem for MBE (1.1) through nonlinearisation of the linear solution following an approach due to Sachdev and his coinvestigators [5-7]. Observe that the old age (linear) form of MBE (1.1) is

ut

=

5 7Ux~.

(2.1)

It has the antisymmetric solution about an arbitrary point x0,

~t ~ C

(x - Xo) e_(Z_xo)2125t '

(2.2)

t312

where c is the so-called old age constant. We choose x0 to be the point where the node of the evolving N-wave finally comes to rest. This is strictly true when the N-wave retains its essential character, namely, it has both positive and negative lobes. When (1.1) is solved with N-wave initial conditions (1.2) numerically, it is found that the node shifts from its original position during its evolution; however, the movement of the zero of the Nwave becomes very small after some time. We assume that this has come about. This motivates the choice (2.2) of the linear solution. We now modify the linear solution (2.2) to satisfy (1.1) 'exactly'. This solution extends the validity of the latter far back in time. We set _ (x X0) (25) 1/2 (25t)1/2, T = t 1/2, u -- vl/-------E--, (2.3) -

so that (1.1) changes to 2~vv~ + ~ [ ~ V ~ - 2~:v ~ - 2~-~v~] + 2n~%v~ + -4n(25)(~-W2r~+lv~

4~(2~)('~-~)/:~Cv

(n + 1)~v~

-

=

0.

(2.4)

Motivated by the exact solution of the plane Burgers equation, namely ~(x,t) =

t(1 + ctl/2 exp(x2/26t)) '

we attempt a solution of the form (2.3) where

(2.5)

v = Z fi(~)~. i=O

Substituting (2.5) into (2.4) and equating coefficients of various powers of ~ to zero, we get

2n E

i

k=O i-1

fk

fi+l-k

k! ~--V)~ + ~ Zk=O k! (i-7--k)!

_(n + 1) E fk+, k=O

i--1

fk fi+l-k

k!

i-1

fi-k (i-l-k)!

_2n2Efkk! ( i -~7,~_ k - )!

(2.~)

k=O

i-1

--2nTE f£ fi l - k _ -I-2n~ fk D~k_k k=O

+4n2(25)(n 11127(i -

k! ( i - 7 - - ) , -

"

•

k=O

k! ( : -

(25)(n n l 2 w ( i - n - 1 ) ! -

:,-,,

-

)!

:0,

i>O.

Modified Burgers Equation For

:~;

= 1,3, (2.6) r e s p e c t i v e l y b e c o m e s - 2 n . f o -- '2r./o' = O.

3f.,

5'n.toJ~ + 3 ( n - 2 ) f ~ + [2,,,(1 - n ) f o f 2 For

- (in.r(foj~)'

(2.r)

O.

= 0 , 2 , we have f l ( r ) = 0,

I') ~)

f3(r)=-d(25)l/2r,

t,,

=0,

=

o

tz > 2.

{12.9)

Here we are i n t e r e s t e d in f i n d i n g a s o l u t i o n of (1.1) of t h e form 1)

.n

u)

P

:

-k,

.n

(7-):

t)

E

.r,

:

k=0

k=O

(2.t0)

E k=o

w h e r e t h e l e a d i n g t e r m comes from t h e c o n t r i b u t i o n of t h e l i n e a r s o l u t i o n (2.2). T h e limit p ..... w o u l d give t h e final form. For p = 2, 3, a n d 4, .to, f2, a n d .f4 c a n be f o u n d explicitly. S u b s t i t u t i n g (2.10) in (2.7) a n d e q u a t i n g coefficients of like powers of 7 to zero, we g(,t ~t s y s t e m of a l g e b r a i c e q u a t i o n s for as, bi, a n d c,/. S o l v i n g this s y s t e m we get t h e c o e i ~ c i e n t s b,s a n d (:is in t e r m s of t h e first a n d last coefficients in f0 (see (2.10)), n a m e l y a0 a n d ap. W e give ~ll(' explicit s o l u t i o n for p = 2. 3, a n d 4. p=2 (~ =2(aoa2)

b0=2na0,

U2,

c , , : 12.,,.%.

~:, : 8

bl = 4

n,-

((loo2) "- ,

52 = 2

'n. -

(v_~.

(2.11)

(3,,"-2,~--1"~ ( ~ o ~ ) '/~ \ 5] '

c~ : ( 1 8 0 , ¢ - 2 4 0 , , ~ + 8 2 , , + : : ~ 2 )

''~'

15.

P=3 al

,, 2 / 3

=

aa 0

~/3

a3

,

,~ 1/3

a2

,Ja o

=

2/3

a3

,

t)'2 = 9~ ( 3 n -- 2 ) a 0l/3 a 2/3 3 ,

t)o

b3 = 2 ( ' , ' ~ , -

2/3 1/3 a0 a3

ct = 12 ( 1 5 n 'e - i 0 n - 1) -

')(3~ -

~

1)a3,

--

1) (I(}~/:~(l;~1.':;.

1 2 , ' - ,) (~o,

co

20;

(2.12)

c9 = 12 ~ (15n 3 - 20n 2 + 2n. + 2) (z() a:( 5H

'

c3

th

,

1/3

-

5

2'nao.

~

12 (Sn, 3

t0n. 2 + 3~ + 2) <~ bit

1)=4 al

bo = 2 n a o ,

tq = bs = ( - 8

Co

=

a2

oa 0

:

(8

)1/,13/4

+ ~n) a o

a4

-~+8n

128

04

a4 %

C1

12n2ao,

c2 = \ - ~ ca = (8-~ + -25() 15~)

,, J/2 1/2

3/4 1/4 qa 0 a 4 ,

=

,

=

_128 _

+ 15~t

,

--

_

96t~, 4_ 487~2) ao1/4(143,"4

+ 2'tt

"

,) ~ l.n) %

be = ( - 8 +

b4

~

1/4 3/1

.

(1'2, 4a0 (11

i/2

I/ O l

.

a4,

(2. ta) --

32n + 48'n 2

(I 0:~/~ (I ~/'4 4

\ 1,,'2, 9 6 n + 72n '-)) % 1//2(~4'

,

C'I

/ (224 \ 15

128

o/

nL - -- 3 2 n + 127)- (~1. 15n ,

4

P.L.

In fact, if we assume

fo(r)

SACHDEV AND CH. SRINIVASA RAO

to have the form k

al/p\p fo(r)=r 2n a~/p+ ~ - )

(2.14)

for all p = 2, 3, 4 , . . . , (this is obviously true, for p = 2, 3, and 4) then the solution of (2.7) gives

i ) p t-3n+p) f 2 ( r , : - 2 r 2~ a l l p + ~1/p\ -) p-1 (_3al/Pn + al/P

,

(2.15)

7-

4 rUn [ ,/p

f4 (r) = _15 n

a 1/p~'-2

~a °

x [ - 45a2/,n 3

+g/, (-45~

+ 2~_-)

+.aol/p-1/p6tp( - 9 0 n 3

+ 30n2Pr + 3pn)

(2.16)

+ ao~2v - 5~p~ + 6p~ - 2v ~) T2

in decreasing powers of r for all p = 2, 3, 4 , . . . . Other fi(r), i > 4 can hence be found from (2.6), recursively. It is clear from the above, t h a t each representation has two arbitrary constants, a0 = c - n , where c is the old age constant, and ap. It is also evident, t h a t the solutions t h a t we have constructed have the linear solution as their asymptote in the limit t --+ oc. One may construct higher-order solutions for p > 4 in the same manner, although fi now become more complicated. T h e Reynolds number of the N-wave solution of (1.1) is defined as 1

R (t) = a I

~ (x,

t) dx.

(2.17)

Integrating (1.1) with respect to x from - o o to x0 and using vanishing conditions at x = - o c , we have 1 R' (t) = ~ : ~ (x0, t). (2.18) Equations (2.3) and (2.14) give

uz(xo,t)-

1

ta/2 { c l + ~ j

c2

~-v/n

,

(2.19)

an/p, c2 = alp/p, and

a0 = c - n ; c is the old age constant. From (2.18) and (2.19) we have R(t) for different values of n and p.

where

c1 =

R (t) : R (to) + ± log

(c, + c2/t'/2)

,

?Z _ - - p

(2.20) 1

n

= R (to) + - - c2 n - p where

(h (t) - h (to)),

52 ~ (n-p)/n

h (~) = (~, + ~-G/~; Here, ci =

a~Ip and

c2 =

alIp

n ¢p,

(2.21)

modifi(~d Burgers Equation

5

Now we show t h a t as r -~ oo,

f2i(r) ~

,~z~(2i)! (t()T 2 rt , i!

(2.22)

f2'i+1(7) = O(T),

(2.23)

for i > 0. Equations (2.22),(2.23) imply t h a t the sohltion ~t in (2.3) tends to the ~,hl ag(~ solution (2.2) as r -~ oc. Since we know f0, .f~, .h, and .~)3 front (2.8),(2.9) and (2.1-1) (2.16), we m a y c o m p u t e f.1, f.~,..., etc. fronl (2.6) as fbllows: 'tz(i -F 2)

i!

fofi+l

= --21?, E

i

,-1

J'h: f z + l - ~ '

k=l i -

1

i -

fk+l

+(n4-1) E

k=O

ft.'

fi+l

h,

A:=I I

fi-A"

f/,:

.fi-l-~'

k! (i- l-k)! + 2t'2E I,:! (i7i7)c)!

(2.24)

t,'=0

i-1 f'k

fi-l-k

-- 21~ ~

~'=0

ft,"

fi-l-L'

~:=0

4~z2(2"a)O'-l)/2r (i

f,i-,~ 4,Jz(2d)( .... 1 ) / 2 ,t~)! -

(i

fi--r, tz 1)!"

Now assume t h a t i is odd, i = 2j + 1, j >_ 1 and asymptotic relations (2.22),(2.23) ~r(' true fi)r all fk, k = 0, 1 . . . . , i - 1. It m a y be checked t h a t

.i~/o/,+,

,---,a~,n3+%-4''

_F(12?zJ+lT4r ,

l=l

( 2 j + 2 -- 2 / ) ( 4 /

~d--TTT)~ 23)

7![Y--TTTg

Since E I =j -I1 ((2J @ 2 -- 2/)(4l - 2 j ) ) / ( l ! ( j

}

7 (2.25)

,

as

- l + 1)!) = 0 is an identity for each positive integer .j,

7z(i + 2 ) f o f i + , ~ (4j + 6)a2tza+2r4,~"

i!

+

as r --, oc.

(2.26)

d!

Using the a s y m p t o t i c relation (2.22) for Jo, we have

+ fi 4-1 ~

( j + 1)!

2'! ~ a0r2~.

~'tS T

OC,

(2.27)

as required in (2.22). The proof of (2.23) is similar. Tile approach of the a s y m p t o t i c sohltion (2.3) to the old age fbrm (2.2) for large t is thus rigorously established. We studied equation (1.1) subject to IC (1.2) for ~tz = 2 and ~, - 4 numeric~dl~ ~ ilsiug ~t scheme due to [8]. We observed t h a t when 6 is very small, tile analysis of Lee-Bapty ~md Crighton [3] and Harris [4] shows an excellent agreement. We briefly discuss the numerical results for (I.1),(1.2) for J~ = 4, 6 = 0.005 (see [9] for nlore details). T h e Reynolds numbers (2.20) obtained by the nonlinearisation m e t h o d shows an error less t h a n 4% for t >_ 3(10 when p = 2, 3; this error further, decreases as p is increased. Our analysis begins to apply at t ~ 300, and s m o o t h l y leads to the linear solution. \¥e have c o m p a r e d the numerical solution of (1.1) with the IC (1.2) with the a s y m p t o t i c analytic solul;ion (2.3) at t = 400 and ?t - 4; here the old age constant c ~md tile other constant c2 = @/P are found to be - ~ _ 2 and 0.0007. respectively. The agre<;ment of the analytical solution (2.3) with the nnmerical one is relnark&bly good over most part of t.h(~ N-wave.

6

P. L. SACHDEVAND CH. SI:IINIVASARAO

REFERENCES 1. G.A. Nariboli and W.C. Lin, A new type of Burgers' equation, Z. Angew. Math. Mech. 53, 505-510, (1973). 2. M. Teymur and E. Suhubi, Wave propagation in dissipative or dispersive nonlinear media, J. Inst. Math. Applics. 21, 25-40, (1978). 3. I.P. Lee-Bapty and D.G. Crighton, Nonlinear wave motion governed by the modified Burgers' equation, Phil. Tran. R. Soc. Lond. A 323, 173-209, (1987). 4. S.E. Harris, Sonic shocks governed by the n.odified Burgers' equation, Euro. J. Appl. Math. 7, 201-222, (1996). 5. P.L. Sachdev and K.T. Joseph, Exact'representations of N-wave solutions of generalized Burgers equations, In Nonlinear Diffusion Phenomenon, (Edited by P.L. Sachdev and R.E. Grundy), pp. 197-219, Narosa Publishing House, New Delhi, (1994). 6. P.L. Sachdev, K.T. Joseph and K.R.C. Nair, Exact N-wave solutions for the nonplanar Burgers equation, Proc. Roy. Soc. Lond. A 445, 501-517, (1994). 7. P.L. Sachdev, K.T. Joseph and B. Mayil Vaganan, Exact N-wave solutions of generalized Burgers equations, Stud. Appd. Math. 9'7, 349-367, (1996). 8. C.N. Dawson, Godunov-mixed methods for advective flow problems in one space dimension, Siam J. Numer. Anal. 28, 1282-1309, (1991). 9. Ch. Srinivasa Rao, Some analytic studies on generalized Burgers equations, Ph.D. Thesis, Indian Institute of Science, India, (1999).