Page 37 - IJOCTA-15-2

M. M. Parsa, K. Sayevand, H. Jafari, I. Masti / IJOCTA, Vol.15, No.2, pp.225-244 (2025)
n
n
0

In summary, let U = U , U , · · · , U n T
1 2 M x−1
T
β
and V = [υ 1 , υ 2 , · · · , υ M x−1 ] . Furthermore, ∂ u(x, t n ) 1 β n n−1 β β
X
= b u − b − b u l
∂t β λ 0 n−l−1 n−l
l=1
(
n
n
U = λ n , U M x = ζ n , n = 0, 1, 2, · · · , N t , − b β u 0 + r ,
n
0
n−1
∆t
0
U = υ i , i = 0, 1, 2, · · · , M x , (53)
i
(48)
and also, according to Lemma 2
then, (35) can be expressed in the following ma-
trix form: n 2−β
r ≤ C u ∆t . (54)
 ∆t
0
U = V, On the other hand, by using Eq. (28), we have



 1 0 1 1
 ΛU = (−Λ + 2I) U + 2λF + M ,
n
n
2
 ΛU = −ΛU n−1 + 2λF + M + 2b β U 0 1 n−1
X
 n−1 β n β β l β 0
 b U −
b − b U − b U
n−1 β β 0 i i n−1 i
 P
 l l−1 n−l−1 n−l
 + b − b U + U , λ
l=1 n−l−1 n−l l=1
(49) U n − 2U − U n
n
= A i+1 i i−1
where Λ = (γ ij ) (M x−1)×(M x−1) and h 2

 U n − U n
n
µ 2 = 1 + 2Ω + Φ, i = j, + G i+1 i−1 − HU + f + r ,
n
n

 i i i
 2h

µ 1 = Ω + Ψ, i = j + 1,

γ ij = (50) (55)
−µ 1 , i = j − 1, and also



0, O.W,


n−1
and also 1 b U n−1 − X b β − b β U l−1 − b β U 0
β
 0 i n−l−1 n−l i n−1 i
T λ
 n n− 1 n− 1 n− 1 l=1
2
F = f 1 2 , f 2 2 , · · · , f M x−1 ,


U − 2U − U
 n−1 n−1 n−1
= A i+1 i i−1
h 2



M = [−µ 1 (λ n + λ n−1 ), 0, · · · , 0, µ 1 (ζ n + ζ n−1 )] , U i+1 − U i−1 n−1 n−1 n−1
 n T
 n−1 n−1
(51) + G 2h − HU i + f i + r i .
n
n− 1 f + f n−1 (56)
i
where f 2 = i .
i
2 Hence, we obtain
n−1
1 β n− 1 X β β l− 1 β
b U 2 − b − b U 2 − b U 0
4. Truncation error, stability and λ 0 i n−l−1 n−l i n−1 i
l=1
convergence analysis n− 1 n− 1 1
U i+1 − 2U i − U i−1
2
2 2 n−
= A
In the follow-up truncation error, stability, and h 2
convergence analysis are discussed for the pro- n− 1 1
2
2 n− 1 1 1
posed model of the Black-Scholes equation with + G U i+1 − U i−1 − HU n− 2 + f n− 2 + R n− 2 ,
fractional coordinate derivatives. 2h i i i
(57)
n
Theorem 2. For i = 1, 2, · · · , M x − 1 and n = n− 1 2 r + r n−1
i
i
n− 1 where R i := .
1, 2, · · · , N t , the truncation error R 2 of the 2
i
Crank-Nicolson scheme presented is as follows: From Eq. (54) and Eq. (28), we have
n− 1 n 2−β 2
n− 1 2−β 2 |R 2 | ≤ C i ∆t + h , (58)
R 2 ≤ C(∆t + h ), (52) i
i n
where C i are constsnts. Let us suppose that
where C is constant. n
C = max C , then
i
1≤i≤M x,1≤n≤N t
n− 1 2−β 2
|R 2 | ≤ C(∆t + h ). (59)
Proof. According to the defined fractional deriv- i
ative, we have
232