Page 166 - IJOCTA-15-3

Pakchin et al. / IJOCTA, Vol.15, No.3, pp.535-548 (2025)
evaluate these integrals. This choice was moti-
k−1 vated by the fact that quadrature methods pro-
(∆t) −ν X ν
= b (u(x, t k−j+1 ) vide a flexible and accurate approach to numeri-
j
Γ(3 − ν) (7)
j=0 cal integration, especially when dealing with func-
− 2u(x, t k−j ) + u(x, t k−j−1 )) tions that are difficult to handle analytically. Fur-
thermore, quadrature methods are widely used for
in which k 1 = (1 − j)∆t, k 2 = −j∆t, k 3 = handling fractional integrals due to their ability to
ν
(−j − 1)∆t and b = ((j + 1) 2−ν − j 2−ν ).
j approximate the integrals over complex domains
Also, by applying a similar technique, we get
with high precision. Therefore, we have Equation
Equation (8):
(10):
1 Z t L
µ ′ −µ Z b
D u(x, t) = u (x, τ)(t − τ) dτ γ X
t C C γ p
Γ(1 − µ) a b(γ) D u(x, t)dγ = h b(γ p ) D u(x, t)
t
t
Z t−a a p=0
1 ′ −µ
= u (x, t − τ)τ dτ h 2
Γ(1 − µ) 0 − F (γ), γ ∈ (a, b)
′′
24
1 k−1 Z (j+1)∆t ′ (10)
X
= u (x, t − τ)τ −µ dτ
Γ(1 − µ) j∆t where γ p = η p−1 +η p , p = 1, . . . , L, γ p ∈ [η p−1 , η p ],
j=0 2
and the interval [η p−1 , η p ] is a partition of the L
k−1
1 X u(x, t + k 1 ) − u(x, t + k 2 ) subintervals of [a, b] with equal amplitude h =
= b−a C γ
Γ(1 − µ) ∆t . Here F is F = b(γ) D u(x, t).
t
j=0 L
Then, using Equation (10), we have Equation
Z (j+1)∆t 1
× µ dτ (11):
j∆t τ
k−1 Z 1 L
(∆t) −µ X µ C µ 1 X C µ p
= b (u(x, t k−j+1 ) − u(x, t k−j )) b 1 (µ) D u(x, t)dµ = b 1 (µ p ) D u(x, t)
t
t
j
Γ(2 − µ) 0 L
j=0 p=0
(8) h 2
′′
µ 2−µ 2−µ − F 1 (µ), µ ∈ (0, 1)
in which b = ((j +1) −j ). For 1 < α < 2,
j 24
the Riesz fractional operator can be estimated as (11)
45
in Equation (9) : where µ p = η p−1 +η p for p = 1, . . . , L. We con-
2
sider the interval [η p−1 , η p ] with equal amplitude
(∆x) −α h = b−a such that [η p−1 , η p ] ⊆ [0, 1], F 1 =
L
α/2
− (−∆) u(x, t) = − πα b 1 (µ) D u(x, t). Similar to the procedures in
µ
C
2Γ(α)Γ(3 − α) cos t
2
Equation (11), we have Equation (12):
(1 − α)(2 − α)u 0 (2 − α)
× + (u 1 − u 0 )
q α q α−1
Z 2 L
q−1 1 X
X C ν C ν p
2−α 2−α b 2 (ν) D u(x, t)dν = b 2 (ν p ) D u(x, t)
+ (u q−i+1 − 2u q−i + u q−i−1 ) (i + 1) − i t t
1 L
i=0 p=0
(1 − α)(2 − α)u M (2 − α) h 2
′′
+ − (u M − u M−1 ) − F 2 (ν), ν ∈ (1, 2)
(M − q) α (M − q) α−1 24
M−q−1 ) (12)
X
2−α 2−α ′ ′
+ (u q+i−1 − 2u q+i + u q+i+1 ) (i + 1) − i where ν p = η p−1 +η p for p = 1, . . . , L and con-
2
i=0 sidering the interval [η ′ , η ] with equal ampli-
′
(9) b−a p−1 ′ p ′
in which ∆x = b−a , x = x q = a + q∆x, u 0 = tude h = L such that [η p−1 , η ] ⊆ [1, 2] and
p
M C ν
u(a, t) = u(x−q∆x, t), u 1 = u(a+∆x, t) = u(x− F 2 = b 2 (ν) D u(x, t)
t
2
(q − 1)∆x, t), . . . , u q−i = u(x − i∆x, t), . . . , u q = Neglecting ◦(h ) in Equations (11) and (12),
u(x, t) = u(a + q∆x, t) Equation (1) can be approximated as Equation
(13):
3. Description of the numerical method L L
1 X C µ p 1 X C ν p
In this section, we present the discretization b 1 (µ p ) D u(x, t) + b 2 (ν p ) D u(x, t) =
t
t
L L
method used for approximating the integral terms p=0 p=0
α
in the distributed-order fractional operator. The − κ(−∆) 2 u(x, t) + f(x, t, u(x, t))
proposed method utilized a quadrature rule to (13)
538