Page 34 - IJOCTA-15-2

Analyzing the Black-Scholes equation with fractional coordinate derivatives using . . .
2
Now suppose that g(t) is a derivative func- Lemma 2. Let g(t) ∈ C [0, t n ] and 0 < β < 1,
tion. According to the definition of the Caputo then
′
derivative in 17 and by substituting t = t n , we 1 Z t n g (s) 1 β
have Γ(1 − β) 0 (t n − s) β ds − λ b g(t n )
0
′
β
∂ g(t) 1 Z t g (s)
= ds n−1
X
∂t β Γ(1 − β) 0 (t − s) β − b β − b β g(t l ) − b β g(t 0 )
n−l
n−l−1
n−1
′
Z
1 t n g (s) l=1
= ds
Γ(1 − β) 0 (t n − s) β ≤ 1 1 − β + 4 − β − (1 + 2 −β )
′ Γ(2 − β) 12 2 − β
n Z
1 X t k g (s)
= ds. × max |g (t)|∆t 2−β .
′′
Γ(1 − β) (t n − s) β
k=1 t k−1 0≤t≤t n
′ g(t n ) − g(t n−1 )
Considering g = + O(∆t),
∆t Proof. See. 35
in which O represents the big-O notation, we have
β
∂ g(t n ) 1 X
n
= (g(t n ) − g(t n−1 )) By inserting the n-th time surface in Eq. (14),
∂t β Γ(1 − β)∆t
k=1 we have
Z
t k
+ ∆tO(∆t) (t n − s) −β ds.
2
β
t k−1 ∂ u(x, t n ) ∂ u(x, t n ) ∂u(x, t n )
(18) = A + G
∂t β ∂x 2 ∂x (24)
On the other hand,
− Hu(x, t n ) + f(x, t n ).
Z 1−β
t k ∆t
(t n − s) −β ds = (n − k + 1) 1−β
β
1 − β ∂ u(x, t n )
By placing from Eq. (23) in Eq.
t k−1
∂t
β
− (n − k) 1−β + O(∆t 2−β ). (24), we obtain
(19)
Thus, from Eq. (18) 1 β n−1 β β
X
b u(x, t n ) − b n−l−1 − b n−l u(x, t l )
0
β
n
∂ g(t n ) 1 X λ
= g(t k ) − g(t k−1 ) l=1
∂t β Γ(2 − β)∆t β 2
k=1 β ∂ u(x, t n ) ∂u(x, t n )
− b n−1 u(x, t 0 ) = A ∂x 2 + G ∂x
(n − k + 1) 1−β − (n − k) 1−β + O(∆t 2−β ).
− Hu(x, t n ) + f(x, t n ).
(20) (25)
Consider the following definitions: On the other hand, the BICs are

l := n − k, (
 u(x L , t n ) = λ(t n ),
β
λ := Γ(2 − β)∆t , (21) (26)
 β 1−β 1−β u(x R , t n ) = ζ(t n ),

b := (l + 1) − (l) .
l and
Therefore, we will have u(x, 0) = υ(x). (27)
β
∂ g(t n ) 1 n−1 β
X
= b (g(t n−l ) − g(t n−l−1 ))
∂t β λ l (22) 3.2. Space discretization
l=0
+ O(∆t 2−β ).
On the space domain (x L , x R ), suppose that M x is
P
By rewriting the , we will have the number of node points for space discretization
d
and h is the space step length defined as h =
M x
β

∂ g(t n ) 1 β n−1 β β where d = x R − x L . In this case, we consider
X
= b g(t n ) − b − b
0
∂t β λ n−l−1 n−l the node points as X L = x 0 < x 1 < x 2 < · · · <
l=1
= x R where for i = 0, 1, 2, · · · , M x , we have

x M x
β 2−β
× g(t l ) − b n−1 g(t 0 ) + O(∆t ). x i = x L + ih. In addition, for the space deriva-
(23) tives, we consider the following relations:
229