THE SOLUTION OF COUPLED MODIFIED KDV SYSTEM BY THE ...

M. GHOREISHI, A.I.B.MD. ISMAIL, A. RASHID: COUPLED MODIFIED KDV SYSTEM ... 125
are defined. Differentiating equations (3)-(5) m-times with respect to the parameter p and then
dividing by m! and setting p = 0, gives the linear equations [10]
with the initial conditions
where
and we have
L u [u m (x, t) − χ m u m−1 (x, t)] = R 1m (⃗u m−1 ), (15)
L v [v m (x, t) − χ m v m−1 (x, t)] = R 2m (⃗v m−1 ), (16)
L z [z m (x, t) − χ m z m−1 (x, t)] = R 3m (⃗z m−1 ), (17)
R 1m (u m−1 ) =
R 2m (v m−1 ) =
R 3m (z m−1 ) =
u m (x, 0) = 0,
v m (x, 0) = 0,
z m (x, 0) = 0,
1
(m − 1)! × ∂m−1 N 1 (φ(x, t; p))
∂p m−1 , (18)
1
(m − 1)! × ∂m−1 N 2 (ϕ(x, t; p))
∂p m−1 , (19)
1
(m − 1)! × ∂m−1 N 3 (ψ(x, t; p))
∂p m−1 , (20)
χ m =
{
0, m ≤ 1,
1, m > 1.
Now, the solution of the m-order deformation equations (15)-(17) for m ≥ 1 becomes
u m (x, t) = χ m u m−1 (x, t) +
v m (x, t) = χ m v m−1 (x, t) +
z m (x, t) = χ m z m−1 (x, t) +
∫ t
0
∫ t
0
∫ t
0
H 1 (x, t)R 1m (⃗u m−1 )dt, (21)
H 2 (x, t)R 2m (⃗v m−1 )dt, (22)
H 3 (x, t)R 3m (⃗z m−1 )dt. (23)
The detailed analysis of the convergence of the HAM is discussed by Liao in [11]. We note
that the HAM only utilities the initial and makes no use of the boundary conditions.
3. Numerical solutions
In this section, the HAM will be demonstrated on examples of new coupled MKdV. For our
numerical computation, let the expression
ψ m (x, t) =
m−1
∑
k=0
u k (x, t), (24)
denote the m-term HAM approximation to u(x, t). We compare the approximate analytical
solution is obtained using HAM for our new coupled MKdV with the exact solution. We define
E m (x, t) to be the absolute error between the exact solution and m-term approximate HAM
solution ψ m (x, t) as follows
E m (x, t) = |u(x, t) − ψ m (x, t)|. (25)