Page 44 - IJOCTA-15-1

L.K. Yadav et.al. / IJOCTA, Vol.15, No.1, pp.35-49 (2025)

and
Now, we define a non-linear operator as

0, ℓ ≤ 1,
ϕ ℓ = (18)
1, ℓ > 1.
1
N [Υ 1 (µ, ξ; q) , Υ 2 (µ, ξ; q)] = S [Υ 1 (µ, ξ; q)]
v

Applying the inverse ST to both sides of (16), we − f (µ) + v δ S [Υ 1 (µ, ξ; q) ∂Υ 1 (µ,ξ;q)
s s ∂µ
2
obtain: ∂Υ 2 (µ,ξ;q) ∂ Υ 1 (µ,ξ;q) i
+ + b ,
∂µ ∂µ 2
ϑ ℓ (µ, ξ) N [Υ 1 (µ, ξ; q) , Υ 2 (µ, ξ; q)] = S [Υ 2 (µ, ξ; q)]
2
h − → i v v δ ∂Υ 1 (µ,ξ;q)

= ϕ ℓ ϑ ℓ−1 (µ, ξ) + ℏS −1 R ℓ ϑ ℓ−1 (µ, ξ) , − g (µ) + s S [Υ 2 (µ, ξ; q) ∂µ +
s
i
3
2
∂Υ 2 (µ,ξ;q)
∂ Υ 2 (µ,ξ;q)
∂ Υ 1 (µ,ξ;q)
(19) Υ 1 (µ, ξ; q) ∂µ + a ∂µ 3 − b ∂µ 2 .
(25)
Therefore, the Mth-order approximate solution of
equation (6) is provided as follows: 1
Here q ∈ 0, is an embedding parameter.
n
Liao 32,33 constructed zeroth-order deformation
M
X equation such as
ϑ(µ, ξ) = ϑ ℓ (µ, ξ) . (20)
ℓ=0
(1 − q) S [Υ 1 (µ, ξ; q) − ϑ (µ, 0)]
Moreover, for M → ∞, we get 1
= ℏH (µ, ξ) N [Υ 1 (µ, ξ; q) , Υ 2 (µ, ξ; q)] ,
∞ (26)
X
ϑ(µ, ξ) = ϑ ℓ (µ, ξ) . (21) (1 − q) S [Υ 2 (µ, ξ; q) − ω (µ, 0)]
2
ℓ=0 = ℏH (µ, ξ) N [Υ 1 (µ, ξ; q) , Υ 2 (µ, ξ; q)] .
4. HASTM solution of fractional WBK
Here, S represent the Shehu transform, ℏ is
equation nonzero auxiliary parameter, H (µ, ξ) ̸= 0 denotes
an auxiliary function, ϑ (µ, 0) and ω (µ, 0) indicate
The fractional Whitham-Broer-Kaup equations
initial guesses of ϑ (µ, ξ) and ω (µ, ξ), respectively.
(1) with Caputo fractional derivative can be
Let q = 0 and q = 1 in equation (26), we get
rewritten as
C δ ∂ϑ(µ,ξ) ∂ω(µ,ξ) Υ 1 (µ, ξ; q) = ϑ 0 (µ, ξ) , Υ 1 (µ, ξ; 1) = ϑ (µ, ξ) ,
D ϑ (µ, ξ) + ϑ (µ, ξ) +
ξ ∂µ ∂µ Υ 2 (µ, ξ; q) = ω 0 (µ, ξ) , Υ 2 (µ, ξ; 1) = ω (µ, ξ) .
2
∂ ϑ(µ,ξ)
+b = 0, (27)
∂µ 2
C D ω (µ, ξ) + ω (µ, ξ) ∂ϑ(µ,ξ) + ϑ (µ, ξ) ∂ω(µ,ξ)
δ
ξ
∂µ
∂µ
3
2
+a ∂ ϑ(µ,ξ) − b ∂ ω(µ,ξ) = 0, Thus, if q increases from 0 to 1, then Υ 1 (µ, ξ; q)
∂µ 3 ∂µ 2 varies from initial guess ϑ 0 (µ, ξ) to the solu-
(22)
tion ϑ(µ, ξ), and Υ 2 (µ, ξ; q) varies from initial
guess ω 0 (µ, ξ) to the solution ω(µ, ξ), respectively.
subject to initial conditions
Upon expanding Υ 1 (µ, ξ; q) and Υ 2 (µ, ξ; q) ac-
cording to Taylor’s series near q, we have
ϑ (µ, 0) = f (µ) , ω (µ, 0) = g (µ) , (23)
∞
P ℓ
Υ 1 (µ, ξ; q) = ϑ 0 (µ, ξ) + ϑ ℓ (µ, ξ) q ,
δ
where 0 < δ ≤ 1 and C D ξ present the Caputo ℓ=1
∞ (28)
fractional derivative of order δ. Υ 2 (µ, ξ; q) = ω 0 (µ, ξ) + P ω ℓ (µ, ξ) q ,
ℓ
Applying ST to both side of (22) and on simpli- ℓ=1
fication, we get where

nh ℓ
v

S [ϑ (µ, ξ)] − f (µ) + v δ ϑ (µ, ξ) ∂ϑ(µ,ξ) ϑ ℓ (µ, ξ) = 1 ∂ Υ 1 (µ,ξ;q) ,
s s ∂µ ℓ! ∂q ℓ q=0
2
∂ω(µ,ξ) ∂ ϑ(µ,ξ) io ℓ (29)
+ + b = 0, ω ℓ (µ, ξ) = 1 ∂ Υ 2 (µ,ξ;q) .
∂µ ∂µ 2 ℓ! ∂q ℓ
nh ∂ϑ(µ,ξ) q=0
v

S [ω (µ, ξ)] − g (µ) + v δ ω (µ, ξ)
s s ∂µ
io By proper choosing of ϑ 0 (µ, ξ), ω 0 (µ, ξ), ℏ, and
3
2
+ϑ (µ, ξ) ∂ω(µ,ξ) + a ∂ ϑ(µ,ξ) − b ∂ ω(µ,ξ) = 0 .
∂µ ∂µ 3 ∂µ 2 H (µ, ξ), the series in equation (28) converges at
(24) q = 1 , we will get
38