The Ginzburg-Landau equation

Computers Math. Applic. Vol. 29, No. 3, pp. 3-4, 1995

Pergamon

Copyright©1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221(94)00222.3 0898-1221/95 $9.50 + O.00

The

Ginzburg-Landau

Equation

G. ADOMIAN General Analytics Corporation, 155 Clyde Road Athens, G A 30605, U.S.A.

R. E. MEYERS U.S. Army Research Laboratories W h i t e Sands, NM, U.S.A.

(Received July 199~; acceptedAugust 199~) Abstract--The

decomposition method is applied to the Ginzburg-Landau equation.

Keywords--Decomposition

method, Real G-L equation, Complex G-L equation, An polynomials.

The general equation is ut=(v+ia)uxx_(k+i/3)lul2u+~u

'

u(x,O)= f(x), O _ < x < co,

u(O,t)=g(t), 0 < t < (x~,

where a, ~, 7, v, k are real, v > O, k > O. Let's consider first the real G-L equation where a=B=O. Then ut + vuxx - klul2u + 7u = O. Let Lt = ~,° L-it = fol(.) dt, u = ~-~n=o°°un, f ( u ) = lul2u = ~n°°=oAn ,

u = =(0) - n?l~ ~ un - L; ln=0

un + k rt~0

An, n~---0

uo = u(t = o) = / ( z ) , Ul

=

-L-tlTuo - L71v

u2

=

-L~'l"/ul

~

( )

L~lv ~

-

uo + kAo,

ul + kA1,

When the An polynomials [1, 2] are evaluated, we can determine u as closely as necessary by m--1 computing an m-term approximant ~om = ~ n = o un which converges to u. For the general case, proceeding in the same manner, •

02

oo

oo

o o

u = u(t = O) + L71(v + za)-~x2 ~ un - L~l(k + i/~) ~ An + L717 ~ un. n-~O

n~O

n-~O

This work was supported by the Atmospheric Sciences Laboratory under the auspices of the U.S. Army Research Office Scientific Services Program.

Typeset by ~4A48-TEX

4

G . ADOMIAN AND R . E. MEYERS

Hence, the decomposition components are: [1] uo = ~(t = o), •

~2

ul = L t l ( v + ~)-ff~x2 uo - L-ZI(k + ij3)Ao + L t l T U o , 02 u2 = L71(v + ia)-~x 2 ul - L t l ( k + il3)A1 + L'~lTul,

•

02

Un = L t l ( v + z a ) ~ x 2 un-1 - L t l ( k + i~)An-1 + L t l T u n - 1 . To evaluate the f ( u ) in terms of the An polynomials, we write f(~) = ~J~l 2, ~](u) = H(u) - H ( - u ) ,

[u[ = uT/(u);

where H is the Heaviside (step) function of the first kind and ~] is the Heaviside (step) function of the second kind H ( u ) = +1, for u > 0 and 0 for u < 0, 7/(u) -- +1,

for u > 0 and - 1 for u < 0.

Thus, I~l 2 = u 2 v 2 ( u ) ,

-- u2[H2(u) - 2 H ( u ) H ( - u ) + H 2 ( - u ) ] , y(u) : (u)(u%2~) = ~%2(u). Hence, OG

f(u) =~-~2(u)An(u3), n=0

A0 -- u03, A1 z 3U2Ul,

A2 = 3u2u2 + 3u~u0, A3 = u 3 + 3u~u3 + 6UoUlu2,

A4 -- 3u~u4 + 3u~u2 + 3~guo + 6~0u1~3, A5 = 3u2u5 + 3u~u3 + 3u22u1 + 6uoulu4 + 6UoU2U3,

The f ( u ) is singular at the origin since the function is piecewise-differentiable there which is due oo to the modelling. We can use ~-~n=OAn for f ( u ) as long as we avoid the origin. Thus, one can m-1 now write the m - t e r m approximant ~om En=O Un using the above An. We can also retain this f ( u ) but replace it with a smooth approximation as in [3] which avoids any problem at the origin. -~

REFERENCES 1. G. Adomian, Solving Frontier Problems of Physics--The Decomposition Method, Kluwer Acad. Publ., (1994). 2. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, (1986). 3. G. Adomian and R. Rach, Smooth polynomial expansions of piecewise-differentiable functions, Appl. Math. Lett. 2 (4), 377-379 (1989)•