Page 43 - IJOCTA-15-1

Approximate analytical solutions of fractional coupled Whitham-Broer-Kaup equations . . .

Definition 4. The Shehu transform of the Ca- and q. Liao, 32,33 constructed the zeroth-order de-
puto derivative is given as [31] formation equation as:

s
h i δ
δ
S C D ϑ (µ, ξ) = V (µ, s, v) ℏq N [Y (µ, ξ; q)]
ξ
v
= (1 − q) S [[Y (µ, ξ; q)] − ϑ 0 (µ, ξ)] . (10)
m−1
X s δ−r−1
− ϑ (r) (µ, 0) .
v
r=0 Evidently, for q = 0 and q = 1 in equation (10),
(5) the following results holds
In Table 1 show different special functions of
Shehu transformation. Y (µ, ξ; 0) = ϑ 0 (µ, ξ), Y (µ, ξ; 1) = ϑ(µ, ξ),
(11)
Table 1. The Shehu transform of
some special functions respectively. Further, if q varies from 0 to 1, the
series solution Y (µ, ξ; q) converges from ϑ 0 (µ, ξ)
Functional Form Shehu Transform Form to the solution ϑ(µ, ξ). Upon expanding Y (µ, ξ; q)
1 v
s 2 with the help of Taylor’s series near to q, one can
ξ v get
s 2
e ξ v
s−av
sin (ξ) v 2 ∞
2
s +v 2 X ℓ
cos (ξ) sv Y (µ, ξ; q) = ϑ 0 (µ, ξ) + ϑ ℓ (µ, ξ)q , (12)
2
s +v 2
ξ n v n+1 ℓ=0

n! , n = 0, 1, 2, .... s
where
ℓ
1 ∂ Y (µ, ξ; q)
3. Basic idea of proposed technique ϑ ℓ (µ, ξ) = . (13)
Γ (ℓ + 1) ∂q ℓ
q=0
Consider a nonlinear fractional differential equa-
tion with the Caputo fractional derivative If the auxiliary linear operator, the initial guess,
the auxiliary parameter ℏ, and the auxiliary func-
tion are appropriately selected, the series given by
C δ
D {ϑ(µ, ξ)} − ℜ [ϑ(µ, ξ)] − ℵ [ϑ(µ, ξ)]
ξ (12) converges at q = 1, and
= h(µ, ξ), 0 < δ ≤ 1, (6)
∞
X
ℓ
ϑ(µ, ξ) = ϑ 0 (µ, ξ) + ϑ ℓ (µ, ξ) q , (14)
with the initial condition
ℓ=0
ϑ(µ, 0) = Φ(µ). (7) is the solution of the original problem described
by equation (6). Define the vector as
Applying the ST to both sides of equation (6) and − →
after simplification, we get: ϑ ℓ (µ, ξ) = {ϑ 0 (µ, ξ), ϑ 1 (µ, ξ), . . . , ϑ ℓ (µ, ξ)} ,
(15)
υ δ
υ
S [ϑ(µ, ξ)] − [Φ(µ)] + S[ℜ [ϑ(µ, ξ)]
s s Initially, by differentiating Equation (10) ℓ times
+ ℵ [ϑ(µ, ξ)] − h(µ, ξ)] = 0 . (8) with respect to q, followed by evaluation at q = 0,
and ultimately dividing by Γ (ℓ + 1), we derive the
th
ℓ -order deformation equation.
Now, we define nonlinear operator as
− →
h i
υ S [ϑ ℓ (µ, ξ) − ϕ ℓ ϑ ℓ−1 (µ, ξ)] = ℏR ℓ ϑ ℓ−1 (µ, ξ) ,
N [Y (µ, ξ; q)] = S [Y (µ, ξ; q)] − Φ(µ)
s (16)
υ where
δ
+ S[ℜ [Y (µ, ξ; q)]
s
+ ℵ [Y (µ, ξ; q)] − h(µ, ξ)]. (9) h − → i 1 ∂ ℓ−1
R ℓ ϑ ℓ−1 (µ, ξ) = Y (µ, ξ; q) ,
Here, q ∈ [0, 1] represents an artificial embedding (ℓ − 1)! ∂q ℓ−1 q=0
parameter, and Y (µ, ξ; q) is a real function of µ, ξ, (17)
37