Page 169 - IJOCTA-15-3

A numerical method for solving distributed-order multi-term time-fractional telegraph equations involving

L
1 X
2−α 2−α µ p
× [(i + 1) − i ]} = − b 1 (µ p )b (∆t) 2−µ p
1
L k−1 L
1 X (∆t) −µ p X k−j+1 k−j p=0
b 1 (µ p )( (U q − U q ))
L
L Γ(2 − µ p ) 2 2 X 4−ν p
′′
p=0 j=0 × ∂ u(x q , η k ) + h F 1 (µ) − 1 b 2 (ν p )b ν p (∆t)
2
2
2
∂ U(x q , η k ) ∂t 24 L p=0 24
µ p
× ((j + 1) 1−µ p − j 1−µ p ) + b (∆t) 2−µ p )
1 2 4 ′ 2 4−α 4
∂t × ∂ U(x q , η ) h ′′ α (∆t) ∂ U
k
L
k−1
h 2 ′′ 1 X (∆t) −ν p X k−j+1 ∂t 4 + 24 F 2 (ν) + b 24 ∂x 4 (θ q , η k )
− F 1 (µ) + b 2 (ν p )( (U q ∂f
k
24 L Γ(3 − ν p ) + f(x q , t k , U ) + ∆t[ (x q , ζ k+1 , U(x q , ζ k+1 ))
p=0 j=0 q ∂t
− 2U q k−j + U q k−j−1 )((j + 1) 2−ν p − j 2−ν p ) + ∂f (x q , ζ k+1 , U(x q , ζ k+1 )) ∂U (x q , ζ k+1 )]
′
4
(∆t) 4−ν p ∂ U(x q , η ) h 2 ∂U ∂t
ν p k ′′ (33)
+ b ) − F 2 (ν))
2 4
24 ∂t 24 Applying the definition of T 1 and T 2 , we can
κ(∆x) −α (1 − α)(2 − α)U k rewrite Equation (33) as in Equation (34):
= − πα { 0
2Γ(α)Γ(3 − α)cos( ) q α
2
k
k
q−1 T 1 (U k+1 ) = T 2 (U ) + f(x q , t k , U )
(2 − α) k k X k k q q q
+ (U − U ) + (U q−i+1 − 2U q−i
0
1
q α−1 L ∂ u(x q , η k )
2
i=0 − 1 X b 1 (µ p )b (∆t) 2−µ p
µ p
(1 − α)(2 − α)U k L 1 ∂t 2
k
+ U q−i−1 )[(i + 1) 2−α − i 2−α ] + M p=0
(M − q) α L
h 2 1 X (∆t) 4−ν p
M−q−1 ′′ ν p
(2 − α) k k X k + F 1 (µ) − b 2 (ν p )b 2
− (U M − U M−1 ) + (U q+i−1 24 L p=0 24
(M − q) α−1
i=0 4 ′ 2
∂ U(x q , η ) h ′′ (34)
k
k
k
− 2U q+i + U q+i+1 )[(i + 1) 2−α − i 2−α ]} × + F 2 (ν)
∂t 4 24
4
(∆t) 4−α ∂ U (∆x) 4−α ∂ U
4
α
+ b (θ q , η k ) + b α (θ q , η k )
24 ∂x 4 24 ∂x 4
∂f
k
+ f(x q , t k , U ) + ∆t[ (x q , ζ k+1 , U(x q , ζ k+1 )) ∂f
q
∂t + ∆t[ ∂t (x q , ζ k+1 , U(x q , ζ k+1 ))
∂f ∂U
+ (x q , ζ k+1 , U(x q , ζ k+1 )) (x q , ζ k+1 )] ∂f ∂U
∂U ∂t + (x q , ζ k+1 , U(x q , ζ k+1 )) (x q , ζ k+1 )]
(32) ∂U ∂t
in which η k ∈ (0, t k ), η ′ ∈ (0, t k ), θ q ∈ (x q−1 , x q ) Thus, the error e k q for q = 1, . . . , M and
k
and ζ k ∈ (t k , t k+1 ). We can rewrite Equation (14) k = 1, . . . , N satisfies Equation (35)
as follows in Equation (33):
( 0
e = 0,

q
(k+1) k (k+1) k ˜ k
T 1 U q = T 2 U q + R q + f(x q , t k , U ) − f(x q , t k , U )
q
q
(35)
L k−1 in which it can be defined as in Equation (36)
1 X (∆t) −µ p X k−j+1 k−j
b 1 (µ p )( (U − U ))
L Γ(2 − µ p ) q q
p=0 j=0 L
2
1 X ∂ u(x q , η k )
L k+1 µ p 2−µ p
1 X (∆t) −ν p R q = − b 1 (µ p )b (∆t)
1
× ((j + 1) 1−µ p − j 1−µ p ) + b 2 (ν p )( L ∂t 2
L Γ(3 − ν p ) p=0
p=0
L
′
4
k−1 h 2 1 X (∆t) 4−ν p ∂ U(x q , η )
X ′′ ν p k
× (U k−j+1 − 2U k−j + U k−j−1 )((j + 1) 2−ν p − j 2−ν p ) + F 1 (µ) − b 2 (ν p )b
q q q 24 L 2 24 ∂t 4
j=0 p=0
4
κ(∆x) −α (1 − α)(2 − α)U 0 k (2 − α) h 2 (∆x) 4−α ∂ U
′′
+ πα { + + F 2 (ν) + b α (θ q , η k )
2Γ(α)Γ(3 − α)cos( ) q α q α−1 4
2 24 24 ∂x
q−1 ∂f
k
k
k
k
k
× (U − U ) + X (U q−i+1 − 2U q−i + U q−i−1 ) + ∆t[ (x q , ζ k+1 , U(x q , ζ k+1 ))
0
1
∂t
i=0
∂f ∂U
k
(1 − α)(2 − α)U M (2 − α) + (x q , ζ k+1 , U(x q , ζ k+1 )) (x q , ζ k+1 )]
2−α
2−α
× [(i + 1) − i ] + − ∂U ∂t
(M − q) α (M − q) α−1
(36)
M−q−1 k+1 k+1 k+1 k+1
k
k
k
k
k
× (U M − U M−1 ) + X (U q+i−1 − 2U q+i + U q+i+1 ) Consider R = [R 1 , R 2 , . . . , R M ], k =
i=0 0, 1, . . .. From Equation (37), we get
541