Page 35 - IJOCTA-15-2

M. M. Parsa, K. Sayevand, H. Jafari, I. Masti / IJOCTA, Vol.15, No.2, pp.225-244 (2025)
We obtain
 " #
 dg(x i ) g(x i+1 ) − g(x i−1 ) 1 β U + U n−1 n−1 U + U l−1
l
n
2
X
U
 = + O(h ), b i i − b β − b β i i − b β n−1 i 0
 dx 2h λ 0 2 n−l−1 n−l 2


l=1
n−1 n−1 n−1 !
n
A U n − 2U − U n U − 2U − U
 2 = i+1 i i−1 + i+1 i i−1

 d g(x i ) g(x i+1 ) − 2g(x i ) − g(x i−1 ) 2 h 2 h 2
= + O(h ).
 2

dx 2 h 2 U n − U n U n−1 − U n−1 !
(28) + G i+1 i−1 + i+1 i−1
2 2h 2h
n n−1 n n−1
U + U f + f
− H i i + i i .
By inserting the i-th space surface in Eq. (25), 2 2
we have (33)
Consider the following definitions:
 n n−1
n−1 n− 1 U + U
1 β X β β  U 2 := i i ,
b u(x i , t n ) − b − b u(x i , t l ) i 2

λ 0 n−l−1 n−l 1 n n−1 (34)
l=1  n− f + f i
i
 f 2 := .
2 i
∂ u(x i , t n ) ∂u(x i , t n ) 2
β
− b n−1 u(x i , t 0 ) = A + G
∂x 2 ∂x Therefore
− Hu(x i , t n ) + f(x i , t n ),
(29) " #
1 β n− 1 n−1 β β l− 1 β 0
X
U
b U i 2 − b − b U i 2 − b n−1 i
0
λ n−l−1 n−l
where U i n = u(x i , t n ) and f i n = f(x i , t n ) for  n− 1 l=1 n− 1 n− 1 
i = 0, 1, 2, · · · , M x , n = 0, 1, 2, · · · , N t . U i+1 2 − 2U i 2 − U i−1 2
= A  
h 2
Based on the definitions and by placing Eq.
(28) in Eq. (29), we have  n− 1 n− 1 
U 2 − U 2 n− 1 n− 1
+ G  i+1 i−1  − HU 2 + f 2 .
2h i i
" #
1 β n n−1 β β l β 0 (35)
X
b U − b − b U − b U
λ 0 i n−l−1 n−l i n−1 i
l=1 The BICs will be discretized as follows:
n n n n n
U i+1 − 2U − U i−1 U i+1 − U i−1
i
= A + G
h 2 2h 
n
U = λ n := λ(t n ),

n
n
− HU + f .  0
i
i
(30) n = 0, 1, 2, · · · , N t , (36)

 n
U = ζ n := ζ(t n ),

Again, we derive the Crank-Nicolson scheme. M x
We have
0
" # U = υ i := υ(x i ), i = 0, 1, 2, · · · , M x . (37)
1 β n n−1 β β l β 0 i
X
b U − b − b U − b U
i
i
λ 0 n−l−1 n−l n−1 i
l=1 To better understand the method, its compu-
n n n n n
U − 2U − U U − U tational molecule is presented in Figure 1
= A i+1 i i−1 + G i+1 i−1
h 2 2h
n
n
− HU + f ,
i
i
(31)
and also
1 β n−1 n−1 β β l−1
X
b U − b − b U
λ 0 i n−l−1 n−l i
l=1
n−1 n−1 n−1
U − 2U − U
β 0 i+1 i i−1 (32)
U
− b n−1 i = A
h 2
n−1 n−1
U − U
+ G i+1 i−1 − HU i n−1 + f i n−1 . Figure 1. The computational molecule of the pro-
2h posed method
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