Page 46 - IJOCTA-15-1

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

h ∗
∗
∗
|ϑ − ϑ | ≤ |(ϕ ℓ + ℏ) (ϑ − ϑ ) + ℏS −1 v δ (1 − α 1 ) |ϑ − ϑ | ≤ 0,
s ∗
ii (1 − α 2 ) |ω − ω | ≤ 0.
h ∗ 2 2 ∗
×S ϑ ∂ϑ − ϑ ∗ ∂ϑ + b ∂ ϑ 2 − ∂ ϑ ,

∂µ ∂µ ∂µ ∂µ 2
h ∗
∗
∗
|ω − ω | ≤ |(ϕ ℓ + ℏ) (ω − ω ) + ℏS −1 v δ Since, 0 < α 1 , α 2 < 1, therefore |ϑ − ϑ | = 0 and
s ∗ ∗ ∗
h ∗ |ω − ω | = 0, which gives ϑ = ϑ and ω = ω .
∗ ∂ϑ
×S ϑ ∂ω − ∂ω + (ω − ω )
∂µ ∂µ ∂µ This completes the required proof. □
ii
2 2 ∗
−b ∂ ω 2 − ∂ ω . Theorem 2. (Convergence Theorem) Let J 1 :

∂µ ∂µ 2
(40) B 1 → B 1 and J 2 : B 2 → B 2 be mapping (nonlin-
ear) with Banach spaces B 1 and B 2 . Presume that
∗
∗
∥J 1 (ϑ) − J 1 (ϑ )∥ ≤ α 1 ∥ϑ − ϑ ∥ , ∀ϑ, ϑ ∗ ∈ J 1 ,
For Shehu transform, we have, by the help of the ∗ ∗ ∗
and ∥J 2 (ω) − J 2 (ω )∥ ≤ α 2 ∥ω − ω ∥ , ∀ω, ω ∈
convolution theorem,
J 2 , then according to Banach’s fixed point theo-
rem, each mapping J 1 and J 2 has a fixed point.
τ ∗
∗ ∗ R ∂ϑ ∗ ∂ϑ The sequence corresponding to the solution ob-
|ϑ − ϑ | ≤ (ϕ ℓ + ℏ) |ϑ − ϑ | + ℏ ϑ − ϑ
∂µ ∂µ
0 tained by the HASTM with ϑ 0 ∈ J 1 and ω 0 ∈ J 2
δ
2 ∗
2
∂ ϑ ∂ ϑ (τ−ζ) chosen arbitrarily will converge to fixed point of
∂µ ∂µ 2 Γ(δ+1)
+b 2 − dζ,
J 1 and J 2 , respectively and
τ
∗ ∗ R ∂ω
|ω − ω | ≤ (ϕ ℓ + ℏ) |ω − ω | + ℏ |ϑ
∂µ
0 2 2 ∗ ∥ϑ m − ϑ r ∥ ≤ α r 1 ∥ϑ 1 − ϑ 0 ∥ ,
− ∂ω ∗ + (ω − ω ) ∂ ω 2 − ∂ ω 1−α 1

∗ ∂ϑ
∂µ ∂µ + b ∂µ ∂µ 2 ∥ω m − ω r ∥ ≤ α r 2 ∥ω 1 − ω 0 ∥ .
× (τ−ζ) δ dζ, 1−α 2
Γ(δ+1)
Proof. Let us consider (J 1 [η 1 , ∥.∥]) and
or, (J 2 [η 2 , ∥.∥]) of all continuous functions of η 1 and
η 2 with the norm ∥g 1 (τ)∥ = max |g 1 (τ)| and
τ∈η 1
∗
∗
|ϑ − ϑ | ≤ (ϕ ℓ + ℏ) |ϑ − ϑ | ∥g 2 (τ)∥ = max |g 2 (τ)|, respectively. Now, we will

τ ∂ ϑ −ϑ ∗ 2 τ∈η 2
2
∗
2
R ∂ (ϑ−ϑ ) (τ−ζ) δ
1 show that ϑ r and ω r are the Cauchy sequences in
+ℏ dζ,
2 ∂µ +b ∂µ 2 Γ(δ+1)
0 the aforesaid Banach spaces. For that, let
τ ∗
∗ ∗ R ∂(ω−ω )
|ω − ω | ≤ (ϕ ℓ + ℏ) |ω − ω | + ℏ |ϑ
∂µ
0 ∥ϑ m − ϑ r ∥ = max |ϑ m − ϑ r | ,

2 ∗ (τ−ζ) δ τ∈η 1
+ (ω − ω ) ∂ (ω−ω ) dζ,

∗ ∂ϑ
∂µ +b ∂µ 2 Γ(δ+1) ∥ω m − ω r ∥ = max |ω m − ω r | ,
τ∈η 2
or,
or,
∗
∗
|ϑ − ϑ | ≤ (ϕ ℓ + ℏ) |ϑ − ϑ | ∥ϑ m − ϑ r ∥ = max |(ϕ ℓ + ℏ) (ϑ m−1 − ϑ r−1 )
1 ∗ (τ−ζ) δ h τ∈η 1 h
+ℏ (c + d) λ 1 + bλ 2 |ϑ − ϑ | dζ, −1 v δ

2 Γ(δ+1) +ℏS S ϑ m−1 ∂ϑ m−1 − ϑ r−1 ∂ϑ r−1
∗
∗
|ω − ω | ≤ (ϕ ℓ + ℏ) |ω − ω | 2 s 2 ii ∂µ ∂µ
δ +b ∂ ϑ m−1 − ∂ ϑ r−1 ,

(τ−ζ)
∗
+ℏ ((P 1 λ 1 + C 1 + bλ 2 ) |ω − ω |) dζ, ∂µ 2 ∂µ 2
Γ(δ+1)
∥ω m − ω r ∥ = max |(ϕ ℓ + ℏ) (ω m−1 − ω r−1 )
h τ∈η 2 h
2

where, λ 1 = ∂ , λ 2 = ∂ ∂ϑ ≤ C 1 . Using +ℏS −1 v δ S ϑ ∂ω m−1 − ∂ω r−1
∂µ ∂µ 2 and ∂µ s ∂µ ∂µ
integral mean value, we have ∂ϑ ∂ ω m−1 ∂ ω r−1 ii
2
2
+ (ω m−1 − ω r−1 ) − b − ,
∂µ ∂µ 2 ∂µ 2
or,
∗
∗
|ϑ − ϑ | ≤ (ϕ ℓ + ℏ) |ϑ − ϑ |
1 ∗
+ℏ 2 (c + d) λ 1 + bλ 2 |ϑ − ϑ | T 1 ,
∗
∗
|ω − ω | ≤ (ϕ ℓ + ℏ) |ω − ω | ∥ϑ m − ϑ r ∥ ≤ max {(ϕ ℓ + ℏ) |ϑ m−1 − ϑ r−1 |
∗
τ∈η 1 h
+ℏ ((P 1 λ 1 + C 1 + bλ 2 ) |ω − ω |) T 2 , −1 h v δ ∂ϑ m−1 ∂ϑ r−1

+ℏS S ϑ m−1 − ϑ r−1
s ∂µ ∂µ
iio
or, ∂ ϑ m−1 − ∂ ϑ r−1 ,
2
2
∂µ ∂µ
+b 2 2
∥ω m − ω r ∥ ≤ max {(ϕ ℓ + ℏ) |ω m−1 − ω r−1 |
∗
∗
|ϑ − ϑ | ≤ α 1 |ϑ − ϑ | , −1 h v δ τ∈η 2 h

∗
∗

|ω − ω | ≤ α 2 |ω − ω | , +ℏS s S ϑ ∂ω m−1 − ∂ω r−1
∂µ
∂µ
iio
2 ∂ ω r−1
2
∂ ω m−1

or, + (ω m−1 − ω r−1 ) ∂ϑ + b ∂µ 2 − ∂µ 2 .
∂µ
40