Page 170 - IJOCTA-15-3

Pakchin et al. / IJOCTA, Vol.15, No.3, pp.535-548 (2025)
following formula Equation (41) was used to cal-
L culate the convergence order:
1 X
k+1 µ p 2
|R q | ≤ P 1 b 1 (µ p )b (∆t) 2−µ p + P 2 h
1
L e(h, ∆t)
p=0 Rate h = log ( ),
2
h
L e( , ∆t)
2
1 X ν p 2 (41)
+ P 3 b 2 (ν p )b (∆t) 4−ν p + P 4 h e(h, ∆t)
2
L Rate ∆t = log ( ).
p=0 2 e(h, ∆t )
2
α
+ P 5 b (∆t) 4−α + P 6 ∆t
5.1. Example 1
≤ b 1 P 1 max {b 1 (µ)}(∆t) 2−µ L + P 2 h 2
µ∈(0,1) Consider Equation (42):
+ b 2 P 3 max {b 2 (ν)}(∆t) 4−ν L + P 4 h 2
ν∈(1,2) Z 1
α 4−α C µ
+ P 5 b (∆t) + P 6 ∆t Γ(5 − µ) D u(x, t)dµ
t
0
(37) Z 2
in which it can be defined as in Equation (38): + Γ(5 − ν) D u(x, t)dν = (42)
C
ν
t
1
α 2
2
∂ u(x q , t) − (−∆) 2 u(x, t) + u (x, t) + g(x, t)
′′
P 1 = max | |, P 2 = max |F 1 |,
t∈[0,T] ∂t 2 µ∈(0,1) where in Equation (43):
4
1 ∂ U(x q , t)
′′
P 3 = max | |, P 4 = max |F 2 |, 4 2
t∈[0,T] 24 ∂t 4 ν∈(1,2) g(x, t) = 24(x(1 − x)) 3 (t − t ) − t (x(1 − x)) 6
8
4
1 ∂ U ln t
απ
P 5 = max | (x, t k )|, + g 1 (x, t)g 1 (x, t) = (2cos( )) −1 4
t
24 x∈[0,L] ∂x 4 2
∂f Γ(4)
P 6 = max |{ (x q , ζ k+1 , U(x q , ζ k+1 )) × (x 3−α + (1 − x) 3−α )
ζ k+1 ∈[t k ,t k+1 ] ∂t Γ(4 − α)
∂f ∂U 3Γ(5)
+ (x q , ζ k+1 , U(x q , ζ k+1 )) (x q , ζ k+1 )}|, − (x 4−α + (1 − x) 4−α )
∂U ∂t Γ(5 − α)
ν p
µ p
b 1 = max{b }, b 2 = max{b } 3Γ(6)
p 1 p 2 + (x 5−α + (1 − x) 5−α )
(38) Γ(6 − α)
α
and b α = max α {b }. Then, we get Equation (39): Γ(7)
− (x 6−α + (1 − x) 6−α )
Γ(7 − α)
k+1 2−µ L 2 4−ν L (43)
∥ R ∥ ∞ ≤ C((∆t) + h + (∆t)
(39) with conditions in Equation (44):
+ (∆t) 4−α + (∆t))
α
where C = max{b 1 P 1 , P 2 , b 2 P 3 , P 4 , b P 5 , P 6 , }. u(x, 0) = 0, x ∈ (0, 1),
(44)
u(0, t) = 0, u(1, t) = 0, t ∈ (0, 1]
5. Illustrative examples The exact solution is u(x, t) = t (x(1 −
4
3
Some numerical examples are demonstrated in x)) . We solved this problem using the numer-
this part to show the effectiveness of the studied ical method developed in this study. The method
numerical approach. The absolute errors for these was applied with L = 20, M = N = 30, and
illustrative examples at different points (x, t) are various values of α. Figure 1 presents the ap-
computed as follows in Equation (40): proximate solutions obtained by our method for
these parameter values and different choices of α.
k
|e(x q , t k )| = |u(x q , t k ) − u |, (40) This figure provides a visual representation of how
q
k = 1, . . . , N, q = 1, . . . , M, the solution evolves across the spatial domain and
time for different choices of α, offering insight into
k
in which u(x q , t k ) and u show the exact and ap- the influence of the fractional order on the over-
q
proximate solutions, respectively. Additionally, all behavior of the solution. Figure 2 compares
all numerical simulations presented in this study the exact solution with the approximate solution
were conducted using Mathematica software (ver- at L = 20, M = N = 30, and various choices of
sion 11). The computations were performed on a α in u(x, 0.5). Figure 3 illustrates the absolute
laptop equipped with an Intel Core i5 processor error functions for L = 20, M = N = 30, and
(2.40 GHz) and 16.00 GB of RAM. Moreover, the various choices of α. These graphs illustrate the
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