Page 48 - IJOCTA-15-1

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

and so on.
2
3
ω 2 (µ, ξ) = 2 ℏ (1 + ℏ) θ κ 1 csch (κ 1 (µ + ι)) If we set δ = 1 and ℏ = −1, then the solutions of
2 2
× coth (κ 1 (µ + ι)) ξ δ − 2 ℏ θ κ 1 4 (45) can be written as
Γ(δ+1)
M
2
×csch (κ 1 (µ + ι)) ϑ (µ, ξ) = P ϑ ℓ (µ, ξ) ,
2
× 2 + 3csch (κ 1 (µ + ι)) ξ 2δ , (M) ℓ=0
Γ(2δ+1) (47)
M
P
and so on. ω (M) (µ, ξ) = ω ℓ (µ, ξ) .
ℓ=0
If we set δ = 1 and ℏ = −1, then the solutions of
When M → ∞, then solutions (47) converges to
(41) can be written as
exact solution of (45)
M
P ϑ(µ, ξ) = θ − 2κ 1 coth (κ 1 ((µ + ι) − θξ)),
ϑ (M) (µ, ξ) = ϑ ℓ (µ, ξ), 2 2
ℓ=0 (43) ω(µ, ξ) = − 2κ 1 csc h (κ 1 ((µ + ι) − θξ)).
M (48)
P
ω (M) (µ, ξ) = ω ℓ (µ, ξ) . Table 2. The absolute error of
ℓ=0
ϑ HAST M in comparison with
4
34
When M → ∞, then solutions (43) converges to ADM ,VIM, 35 and OHAM at
exact solution of (41) ℏ = −1, ι = 10, θ = 0.005, and
κ 1 = 0.10 for example 1
ϑ(µ, ξ) = θ − κ 1 coth (κ 1 ((µ + ι) − θξ)), (µ, ξ) ADM 34 VIM 30 OHAM 4 ϑ HASTM ϑ HASTM
2
2
ω(µ, ξ) = − κ 1 csc h (κ 1 ((µ + ι) − θξ)). (0.1, 0.1) 7.10000 × 10 −9 2.77556 × 10 −17 4.8863 × 10 −6 (δ = 0.75) (δ = 1)
3.2976 × 10 −6
2.77556 × 10 −17
(44) (0.1, 0.3) 6.50000 × 10 −9 2.77556 × 10 −17 8.4857 × 10 −6 4.9784 × 10 −6 2.77556 × 10 −17
5.1896 × 10 −6
3.33067 × 10 −16
3.33067 × 10 −16
(0.1, 0.5) 5.90000 × 10 −9
1.0335 × 10 −6
(0.2, 0.1) 2.82000 × 10 −8 2.77556 × 10 −17 4.7120 × 10 −6 3.2131 × 10 −6 2.77556 × 10 −17
(0.2, 0.3) 2.59000 × 10 −8 4.16334 × 10 −17 8.2675 × 10 −6 4.8503 × 10 −6 4.16334 × 10 −17
(0.2, 0.5) 2.41000 × 10 −8 3.60882 × 10 −17 1.0069 × 10 −6 5.0562 × 10 −6 3.60882 × 10 −17
Example 2 If a = 1 and b = 0, then fractional (0.3, 0.1) 6.33670 × 10 −8 1.38778 × 10 −17 4.5915 × 10 −6 3.1310 × 10 −6 1.38778 × 10 −17
WBK equation (22) can be rewritten as (0.3, 0.3) 5.85000 × 10 −8 2.77556 × 10 −17 8.0560 × 10 −6 4.7263 × 10 −6 2.77556 × 10 −17
3.19189 × 10 −16
4.9268 × 10 −6
9.8118 × 10 −6
3.19189 × 10 −16
(0.3, 0.5) 5.40000 × 10 −8
(0.4, 0.1) 1.12400 × 10 −7 1.38778 × 10 −17 4.4746 × 10 −6 3.0513 × 10 −6 1.38778 × 10 −17
(0.4, 0.3) 1.03900 × 10 −7 3.19189 × 10 −16 7.8510 × 10 −6 4.6060 × 10 −6 3.19189 × 10 −16
(0.4, 0.5) 9.61000 × 10 −8 3.19189 × 10 −16 9.5621 × 10 −6 4.8014 × 10 −6 3.19189 × 10 −16
C D ϑ (µ, ξ) + ϑ (µ, ξ) ∂ϑ(µ,ξ) + ∂ω(µ,ξ) = 0, (0.5, 0.1) 1.75500 × 10 −7 0 4.3613 × 10 −6 2.9740 × 10 −6 0
δ
ξ ∂µ ∂µ (0.5, 0.3) 1.62200 × 10 −7 5.55112 × 10 −17 7.6522 × 10 −6 4.4893 × 10 −6 5.55112 × 10 −17
C D ω (µ, ξ) + ω (µ, ξ) ∂ϑ(µ,ξ) + ϑ (µ, ξ) ∂ω(µ,ξ) (0.5, 0.5) 1.50100 × 10 −7 3.19189 × 10 −16 9.3199 × 10 −6 4.6798 × 10 −6 3.19189 × 10 −16
δ
ξ
∂µ
∂µ
3
+ ∂ ω(µ,ξ) = 0, Table 3. The absolute error of
∂µ 3 ω HAST M in comparison with
4
(45) ADM, 34 VIM, 35 and OHAM at
subject to initial conditions ℏ = −1, ι = 10, θ = 0.005, and κ 1 =
0.10 for example 1.
ϑ (µ, 0) = θ − 2κ 1 coth [κ 1 (µ + ι)] , (µ, ξ) ADM 34 V IM 4 OHAM 30 ω HASTM ω HASTM
(δ = 1)
(δ = 0.75)
2
2
ω (µ, 0) = −2κ 1 csch [κ 1 (µ + ι)] . (0.1, 0.1) 9.5512 × 10 −10 1.73472 × 10 −18 1.2632 × 10 −6 8.6140 × 10 −7 1.7347 × 10 −18
(0.1, 0.3) 8.0600 × 10 −10 2.60209 × 10 −17 2.2165 × 10 −6 1.3004 × 10 −6 2.60209 × 10 −17
(46) (0.1, 0.5) 6.7700 × 10 −10 1.80411 × 10 −16 2.6998 × 10 −6 1.3558 × 10 −6 1.80411 × 10 −16
(0.2, 0.1) 3.8210 × 10 −9 3.46945 × 10 −18 1.2242 × 10 −6 8.3477 × 10 −7 3.46945 × 10 −18
(0.2, 0.3) 3.2240 × 10 −9 2.34188 × 10 −17 2.1480 × 10 −6 1.2602 × 10 −6 2.34188 × 10 −17
(0.2, 0.5) 2.7060 × 10 −9 1.73472 × 10 −16 2.6163 × 10 −6 1.3139 × 10 −6 1.73472 × 10 −16
(0.3, 0.1) 8.5970 × 10 −9 3.46945 × 10 −18 1.1866 × 10 −6 8.0917 × 10 −7 3.46945 × 10 −18
According to Eqs. (33)-(35), we get following re- (0.3, 0.3) 7.2520 × 10 −9 1.99493 × 10 −17 2.0821 × 10 −6 1.2216 × 10 −6 1.99493 × 10 −17
(0.3, 0.5) 6.0910 × 10 −9 1.61329 × 10 −16 2.5361 × 10 −6 1.2736 × 10 −6 1.61329 × 10 −16
sults (0.4, 0.1) 1.5284 × 10 −8 2.60209 × 10 −18 1.1505 × 10 −6 7.8454 × 10 −7 2.60209 × 10 −18
(0.4, 0.3) 1.2893 × 10 −8 1.73472 × 10 −17 2.0188 × 10 −6 1.1844 × 10 −6 1.73472 × 10 −17
(0.4, 0.5) 1.0827 × 10 −8 1.52656 × 10 −16 2.4589 × 10 −6 1.2348 × 10 −6 1.52656 × 10 −16
(0.5, 0.1) 2.3880 × 10 −8 8.67362 × 10 −19 1.1157 × 10 −6 7.6084 × 10 −7 8.67362 × 10 −19
ϑ 0 (µ, ξ) = θ − 2κ 1 coth (κ 1 (µ + ι)) , (0.5, 0.3) 2.0144 × 10 −8 2.08167 × 10 −17 1.9578 × 10 −6 1.1486 × 10 −6 2.08167 × 10 −17
(0.5, 0.5) 1.6916 × 10 −8
2.3816 × 10 −6
1.43982 × 10 −16
1.43982 × 10 −4
1.1975 × 10 −6
2
2
ω 0 (µ, ξ) = −2 κ 1 csch (κ 1 (µ + ι)) , Table 4. The absolute error of
2
2
ω 1 (µ, ξ) = 2 ℏ θ κ 1 csch (κ 1 (µ + ι)) ξ δ , ϑ HAST M in comparison with
Γ(δ+1) ADM, 34 VIM, 35 and OHAM 4 at
3
2
ω 1 (µ, ξ) = 4 ℏ κ 1 θcsch (κ 1 (µ + ι))
ξ δ ℏ = −1, iota = 10, θ =
× coth (κ 1 (µ + ι)) , 0.005, and κ 1 = 0.10 for example 2.
Γ(δ+1)
2
2
ϑ 2 (µ, ξ) = 2 ℏ (1 + ℏ) θ κ 1 csch (κ 1 (µ + ι))
3
2 2
2
× ξ δ − 4 ℏ θ κ 1 csch (κ 1 (µ + ι)) (µ, ξ) ADM 34 V IM 30 OHAM 4 ϑ HASTM ϑ HASTM
(δ = 0.75)
(δ = 1)
Γ(δ+1) (0.1, 0.1) 1.04892 × 10 −4 1.23033 × 10 −4 1.07078 × 10 −4 3.6854 × 10 −5 1.1102 × 10 −16
ξ 2δ (0.1, 0.3) 9.64474 × 10 −5 3.69597 × 10 −4 3.04565 × 10 −4 5.2751 × 10 −5 2.2204 × 10 −16
× coth (κ 1 (µ + ι)) , (0.1, 0.5) 8.88312 × 10 −5 6.16873 × 10 −4 4.81303 × 10 −4 5.6645 × 10 −5 1.3323 × 10 −15
Γ(2δ+1) (0.2, 0.1) 4.25408 × 10 −4 1.19869 × 10 −4 1.04388 × 10 −4 3.5906 × 10 −5 0
2
3
ω 2 (µ, ξ) = 4 ℏ (1 + ℏ) θ κ 1 csch (κ 1 (µ + ι)) (0.2, 0.3) 3.91098 × 10 −4 3.60098 × 10 −4 2.97260 × 10 −4 5.1394 × 10 −5 2.7756 × 10 −16
(0.2, 0.5) 3.60161 × 10 −4 6.01006 × 10 −4 4.70136 × 10 −4 5.5188 × 10 −5 1.2878 × 10 −15
2 2
× coth (κ 1 (µ + ι)) ξ δ − 4 ℏ θ κ 1 4 (0.3, 0.1) 9.71922 × 10 −4 1.16789 × 10 −4 1.01776 × 10 −4 3.4988 × 10 −5 1.1102 × 10 −16
(0.3, 0.3) 8.93309 × 10 −4
5.0880 × 10 −5
2.90150 × 10 −4
3.50866 × 10 −4
5.5511 × 10 −17
Γ(δ+1) (0.3, 0.5) 8.22452 × 10 −4 5.85610 × 10 −4 4.59590 × 10 −4 5.3776 × 10 −5 1.1112 × 10 −15
2
2
×csch (κ 1 (µ + ι)) 2 + 3csch (κ 1 (µ + ι)) (0.4, 0.1) 1.75596 × 10 −3 1.13829 × 10 −4 9.92418 × 10 −5 3.4097 × 10 −5 5.5511 × 10 −17
(0.4, 0.3) 1.61430 × 10 −3 3.41948 × 10 −4 2.83229 × 10 −4 4.8805 × 10 −5 1.1102 × 10 −16
2δ
× ξ , (0.4, 0.5) 1.48576 × 10 −3 5.70710 × 10 −4 4.49116 × 10 −4 5.2407 × 10 −5 1.3323 × 10 −15
1.10936 × 10 −4
3.3234 × 10 −5
9.67808 × 10 −4
5.5511 × 10 −17
(0.5, 0.1) 2.79519 × 10 −3
Γ(2δ+1)
(0.5, 0.3) 2.56714 × 10 −3 3.33274 × 10 −4 2.78492 × 10 −4 4.7569 × 10 −5 0
(0.5, 0.5) 2.63184 × 10 −3 5.56235 × 10 −4 4.38895 × 10 −4 5.1080 × 10 −5 9.9920 × 10 −16
42