Page 168 - IJOCTA-15-3

Pakchin et al. / IJOCTA, Vol.15, No.3, pp.535-548 (2025)
max 0≤m≤M |e k+1 | =∥ E k+1 ∥ ∞ , then, we have
m
z
Equation (23):
(1 − α)(2 − α) 2(2 − α) (1 − α)(2 − α)

+ +
q α q α−1 (M − q) α
(ξ 1 (L, ∆t) + ξ 2 (L, ∆t)) ∥ E k+1 ∥ ∞
2(2 − α)
κ(∆x) −α − , b = (ξ 1 k 1−µ p + 2ξ 2 k 2−ν p + ξ 2 + L f ),
= (ξ 1 (L, ∆t) + ξ 2 (L, ∆t))|e k+1 | + πα (M − q) α−1
z
2Γ(α)Γ(3 − α)cos( )
2
q−1
( c = ξ 2 k 2−ν p
X
× (2|e k z−i | − 2|e k z−i |)[(i + 1) 2−α − i 2−α ] (26)
i=0
and for j = 2, . . . , k − 1, we have Equation (27):
M−q−1 )
X
+ (2|e k z+i | − 2|e k z+i |)[(i + 1) 2−α − i 2−α ]
i=0
κ(∆x) −α a = ξ 1 k 1−µ p + ξ 2 k 2−ν p ,
≤ (ξ 1 (L, ∆t) + ξ 2 (L, ∆t))|e k+1 | + πα
z
2Γ(α)Γ(3 − α)cos( )
2 b = ξ 1 k 1−µ p + 2ξ 2 k 2−ν p , (27)
q−1
X
× { (2|e k z−i | − |e k z−i−1 − |e k z−i+1 |[(i + 1) 2−α − i 2−α ]) c = ξ 2 k 2−ν p
i=0
M−q−1
X
+ (2|e k z+i | − |e k z+i−1 | − |e k z+i+1 |) × [(i + 1) 2−α − i 2−α ]} Equation (25) can also be rewritten as follows
i=0 Equation (28)
κ(∆x) −α
≤ |(ξ 1 (L, ∆t) + ξ 2 (L, ∆t))|e k+1 | − πα
z
2Γ(α)Γ(3 − α)cos( )
2
q−1
X k−1
× { (|e k z−i+1 | − 2|e k z−i | + |e k z−i−1 |)[(i + 1) 2−α − i 2−α ] k+1 1 X k−j+1
i=0 ∥ E ∥ ∞ ≤ ξ 1 (L, ∆t) + ξ 2 (L, ∆t) a ∥ E ∥ ∞
M−q−1 j=1
X
+ (|e k z+i+1 | − 2|e k z+i | + |e k z+i−1 |)[(i + 1) 2−α − i 2−α ]}| + b ∥ E k−j ∥ ∞ +c ∥ E k−j−1 ∥ ∞
i=0
κ(∆x) −α ≤ 1 0
≤ |(ξ 1 (L, ∆t) + ξ 2 (L, ∆t))|e k+1 | + πα ξ 1 (L, ∆t) + ξ 2 (L, ∆t) (akδ 1 + bkδ 2 + ckδ 3 ) ∥ E ∥ ∞
z
2Γ(α)Γ(3 − α)cos( )
2
q−1
( (28)
X
× (|e k z−i+1 | − 2|e k z−i | + |e k z−i−1 |) × [(i + 1) 2−α − i 2−α ] From Equation (28), we get Equation (29):
i=0
M−q−1 )

X
+ (|e k z+i+1 | − 2|e k z+i | + |e k z+i−1 |) × [(i + 1) 2−α − i 2−α ]
∥ E k+1 0 (29)
i=0 ∥ ∞ ≤ ϱ ∥ E ∥ ∞
k
k
e k
= |T 1 (e k+1 )| = |T 2 (e ) + f(x z , t k , U ) − f(x z , t k , U )|
z
z
z
z
1
e k
k
k
≤ |T 2 (e )| + |f(x z , t k , U ) − f(x z , t k , U )| where ϱ = max{ ξ 1 (L,∆t)+ξ 2 (L,∆t) (akδ 1 + bkδ 2 +
z
z
z
(23) ckδ 3 )}. Thus, for each arbitrary initial rounding
0
Then, we have Equation (24) error E , there exists ϱ > 0, independent of L,
∆t, and ∆x, such that in Equation (30):
e k
k
k
(ξ 1 (L, ∆t) + ξ 2 (L, ∆t)) ∥ E k+1 ∥ ∞ ≤ |T 2 (e )| + L f |U − U )|
z
z
z
n
0
(24) ∥ E ∥ ∞ ≤ ϱ ∥ E ∥ ∞ (30)
where L f ∈ (0, 1). Substituting Equation (18)
into Equation (24) yields Equation (25) Therefore, our numerical method is uncondi-
tionally stable.
Suppose that the exact and approximate solu-
k−1
e k
k
k+1 X k−j+1 tions of Equation (1) are U and U , respectively.
(ξ 1 (L, ∆t) + ξ 2 (L, ∆t)) ∥ E ∥ ∞ ≤ a ∥ E ∥ ∞ q q
Then, we consider the error function at (x q , t k by
j=1
+ b ∥ E k−j ∥ ∞ +c ∥ E k−j−1 ∥ ∞ the following formula in Equation (31):
(25)
where for j = 1, we have Equation (26):
k
e k
k
e = U − U , k = 1, . . . , N, q = 1, . . . , M (31)
q
q
q
a = (ξ 1 k 1−µ p + ξ 2 k 2−ν p + ξ 1 + 2ξ 2 + W(M, α, z)),
k
k
k
k
in which e = [e , e , . . . , e ]. Due to Equations
−α 1 2 M
κ(∆x) 0
(2) and (15), we have e = [0, 0, . . . , 0]. Then,
W(M, α, z) =
2Γ(α)Γ(3 − α)cos( ) from Equation (14), we have Equation (32):
πα
2
540