415
Internat. J. Math. & Math. Sci.
VOL. 19 NO. 2 (1996) 415
THE FIFTH-ORDER KORTEWEG-DE VRIES EQUATION
G. ADOMIAN
General Analytics Cooration
155 Clyde Rd.
Athens, GA 30605, USA
(Received December 13, 1994)
ABSTRACT. Decomposition is applied to the 5th-order KdV equation.
KEY WORDS AND PHRASES. Decomposition method, Adomian Polynomials.
1991 AMS SUBJECT CLASSIFICATION CODES.
is given by:
The 5th-order KdV equation
ut
Writing Lt
u
where
0
fd (o)dt, we have
O/Ot and L-f
Ltu
Lt
L-I u
+ 6uux + ux + uzz
6uuz uxzz uzzzzz
6L-[luuz L-luzzz
u(O) 6L-[1A,{uu.} L-[I(o3/Ox3)u n-(OS/cgxS)u
A,.,{uu} represents the Adomian Polynomials [2] for uu:. Letting
u,, decomposition
u
n--0
yields
o
(0)
u2
-6L-[Ao L-[ (O3/cgx3)uo
-6L-[1A1- Ll (O3/Ox3)ul L[ ( /i:gxS)u
U.+l
-6L-[1A, L-[I (o3/Ox3)u, L-[I (c /OxS)u,.,
Ul
-
Using primes to indicate the differentiation with respect to x [2]
Ao
A
Uo Uo
uoU
uuo
A,
,o + ,- +
A1
"alU0
+ uu +
+ uC,_ +
Now all components of u are determinable and we can write the n-term approximant
n-1
m=0
which approaches u as rn
ofn.
co.
It has been shown that high accuracy can be achieved for small values
REFERENCES
nonlocal
for capillary-gravity waves: 5th-degree KdV equation,
solutions
BOYD,
Weakly
J.P.,
[1]
Physica D, 48 (1991), 129-146.
[2] ADOMIAN, G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer
Acad. Publ., 1994.
[3] ABBAOUI, K. and CHERRUAULT, Y., Convergence of Adomian’s method applied to differential
equations, Comp. Math. Applic., 28, no. 5 (1994), 103-109.