Page 167 - IJOCTA-15-3

A numerical method for solving distributed-order multi-term time-fractional telegraph equations involving
µ p
ν p
By putting C D u(x, t), C D u(x, t), and
t
t
α
−(−∆) 2 u(x, t) into Equation (1), and showing 1 X (∆t) −µ p (∆t) −ν p
L
k
k
by U ≈ u(x q , t k ), we acquire Equation (14): T 2 (U ) = L b 1 (µ p ) Γ(2 − µ p ) + b 2 (ν p ) Γ(3 − ν p ) U q k
q
q
p=0
L
  1 X (∆t) −ν p
k
L k−1 + b 2 (ν p )( (U − U k−1 )
1 X (∆t) −µ p X k−j+1 L Γ(3 − ν p ) q q
)
b 1 (µ p )  (U q − U q k−j  p=0
L Γ(2 − µ p )
p=0 j=0 1 X (∆t) −µ p X
k−1
L
− b 1 (µ p )( (U k−j+1 − U k−j ))((j + 1) 1−µ p
L q q
1 X (∆t) −ν p L p=0 Γ(2 − µ p ) j=1
× ((j + 1) 1−µ p − j 1−µ p )) + b 2 (ν p )
L Γ(3 − ν p ) L k−1
p=0 1 X (∆t) −ν p X
− j 1−µ p )) − b 2 (ν p )( (U k−j+1 − 2U k−j
k−1 L Γ(3 − ν p ) q q
X p=0 j=1
× (U q k−j+1 − 2U q k−j + U q k−j−1 )((j + 1) 2−ν p − j 2−ν p )) −α
j=0 + U q k−j−1 )((j + 1) 2−ν p − j 2−ν p )) − κ(∆x) πα
κ(∆x) −α (1 − α)(2 − α)U 0 k 2Γ(α)Γ(3 − α)cos( 2 )
= − πα (1 − α)(2 − α)U k (2 − α)
k
2Γ(α)Γ(3 − α)cos( ) q α × 0 + (U − U )
k
2 α α−1 1 0
q q
q−1
(2 − α) k k X k k k
+ (U − U ) + (U q−i+1 − 2U q−i + (1 − α)(2 − α)U M − (2 − α) (U k − U k )
1
0
q α−1 (M − q) α (M − q) α−1 M M−1
i=0
(1 − α)(2 − α)U k (18)
k 2−α 2−α M
+ U )[(i + 1) − i ] + 0
q−i−1 α Suppose the initial data has an error e as in
q
(M − q)
Equation (19)
M−q−1
(2 − α) k k X k k
− (U M − U M−1 ) + (U q+i−1 − 2U q+i
(M − q) α−1 e 0 0
i=0 ψ = ψ(x q ) + e , q = 1, . . . , M (19)
q
q
o
k
k
e k
k
+U q+i+1 )[(i + 1) 2−α − i 2−α ] + f(x q , t k , U ) Let U and U are the solutions of Equations
q
q
q
(14) (14) and (15). Therefore, it can be depicted as in
under the conditions as in Equation (15): Equation (20)
e k
k
k
0
U = ψ(q∆x), q = 0, 1, . . . , M, T 1 (e k+1 ) = T 2 (e ) + f(x q , t k , U ) − f(x q , t k , U )
q
q
q
q
q
k
k
U = φ 0 (k∆t), U M = φ L (k∆t) k = 0, 1, . . . , N k k e k k (20)
0
(15) in which e q = U − U q . Also, let E =
q
k
k
[e , e k+1 , . . . , e ], k = 0, 1, 2, . . . , N and ∥
1 2 M
k
E k ∥ ∞ = max 1≤m≤M |e |. In proving the main
4. Convergence and stability m
theorems of this manuscript, we consider the fol-
The stability analysis, along with convergence, lowing symbols in Equation (21):
is obtained by the proposed numerical method,
which is presented in Section 3. For simplicity, L
1 X b 1 (µ p )
we can rewrite Equation (14) as follows in Equa- ξ 1 (L, ∆t) = (∆t) −µ p ,
tion (16): L p=0 Γ(2 − µ p )
(21)
L
1 X b 2 (ν p ) −ν p
k
k
T 1 (U q k+1 ) = T 2 (U ) + f(x q , t k , U ) (16) ξ 2 (L, ∆t) = L b 1 (µ p ) Γ(3 − ν p ) (∆t)
q
q
p=0
where it can be expanded as in Equation (17):
4.1. Theorem
1 X b 1 (µ p )(∆t)
L −µ p Let f(x, t, u) be a Lipschitz map. Then, the nu-
T 1 U (k+1) =
q merical method given by Equation (14) with ini-
L Γ(2 − µ p )
p=0 tial boundary conditions Equation (15) is uncon-
b 2 (ν p )(∆t) −ν p (k+1) κ(∆x) −α
+ U q + πα ditionally stable, that is
Γ(3 − ν p ) 2Γ(α)Γ(3 − α) cos
2
(

q−1
X (k) (k) (k) n 0
× U − 2U + U (i + 1) 2−α − i 2−α ∥ E ∥ ∞ ≤ ϱ ∥ E ∥ ∞ , n = 0, 1, . . . , N (22)
q−i+1 q−i q−i−1
i=0
) where ϱ > 0 and independent of the step sizes.

M−q−1
(k) (k) (k) 2−α 2−α According to the conditions Equation (15),
X
+ U − 2U + U (i + 1) − i
q+i−1 q+i q+i+1
i=0 the inequality Equation (22) is satisfied for n = 0.
(17) Suppose Equation (22) satisfies n = 1, . . . , k, as-
and Equation (18): suming that z ∈ {1, 2, . . . , M} such that |e k+1 | =
z
539