=Paper=
{{Paper
|id=Vol-1894/num2
|storemode=property
|title=Compact difference scheme for semi-linear delayed diffusion wave equation with fractional order in time
|pdfUrl=https://ceur-ws.org/Vol-1894/num2.pdf
|volume=Vol-1894
|authors=Ahmed S. Hendy,Vladimir G. Pimenov
}}
==Compact difference scheme for semi-linear delayed diffusion wave equation with fractional order in time==
https://ceur-ws.org/Vol-1894/num2.pdf
Compact difference scheme for semi-linear delayed diffusion
wave equation with fractional order in time
Ahmed S. Hendy1,2 Vladimir G. Pimenov1,3
ahmed.hendy@fsc.bu.edu.eg v.g.pimenov@urfu.ru
1 – Institute of Natural Sciences and Mathematics, Ural Federal University (Yekaterinburg, Russia)
2 – Department of Mathematics, Benha University (Benha, Egypt)
3 – Krasovskii Institute of Mathematics and Mechanics (Yekaterinburg, Russia)
Abstract
A compact difference scheme for a class of non-linear fractional
diffusion-wave equations with fixed time delay is considered. Analysis
of the constructed difference scheme is done in L∞ -norm by means of
the discrete energy method. A numerical test example is introduced to
illustrate the accuracy and efficiency of the proposed method.
1 Introduction
An important class of fractional differential equations which has been studied widely in recent years is the
time fractional diffusion-wave equation (FDWE). The time FDWE is obtained from the classical diffusion-wave
equation by replacing the second-order time derivative term by a fractional derivative of order 1 < α ≤ 2.
The fractional diffusion equation was introduced in physics by Nigmatullin [16] to describe diffusion in media
with fractal geometry, which is a special type of porous media. Gorenflo et al. [10] presented the scale-invariant
solutions for the time-fractional diffusion-wave equation in terms of the generalized Wright function. Agrawal [2]
extended this formulation to a diffusion-wave equation that contains a fourth-order space derivative term in a
bounded space domain. Recently, simulations of the approximation solutions of time-fractional wave, forced wave
(shear wave) and damped wave equations are given in [1]. A novel fractional diffusion-wave equation with non-
local regularization for noise removal was presented in [24]. The existence and uniqueness of solutions for Dirichlet
initial-boundary value problem associated to the semi linear fractional wave equation was recently studied in [14].
As a numerical approach to solve FDWE, Sun and his co-authors proposed a high order difference methods for the
fractional diffusion-wave equation [8]. Also, in [22], the efforts of the authors were devoted to the application of
fractional multi-step method to obtain a numerical solution of time fractional diffusion-wave equation. Compact
finite difference schemes for the modified anomalous fractional sub-diffusion equation and fractional diffusion-
wave equation were studied in [21]. A numerical solution for a general class of diffusion problem was considered
in [3], where the standard time derivative is replaced by a fractional one. An efficient numerical method was
constructed in [26] to solve this moving boundary problem.
Time delay occurs in many realistic applications which are modeled mathematically, e.g. [9, 6, 13, 4, 15, 5].
Reaction-diffusion equations with time delay effect have been proposed as models for the population ecology, the
cell biology and the control theory in recent years [19]. The existence of mild solutions for initial value problem
for nonlinear time fractional non-autonomous evolution equations with delay in Banach space E was studied
in [7]. Numerically, a linearized quasi-compact difference scheme was proposed for semi-linear space-fractional
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A.A. Makhnev, S.F. Pravdin (eds.): Proceedings of the International Youth School-conference «SoProMat-2017», Yekaterinburg,
Russia, 06-Feb-2017, published at http://ceur-ws.org
325
diffusion equations with a fixed time delay [11]. A linearized compact finite difference scheme was presented for
the semi-linear fractional delay convection-reaction–diffusion equation in [23]. The authors of the manuscript at
hand, recently proposed a difference scheme for a class of non-linear delay distributed order fractional diffusion
equations in [17]. As an extension to this contribution and depending on Sun ’s work [20], we seek to derive a
compact linear difference scheme to solve numerically FDWE effected with a non-linear delayed source function,
more specific we consider
∂ α u(x, t) ∂ 2 u(x, t)
α
=K + f (x, t, u(x, t), u(x, t − s)), t > 0, 0 ≤ x ≤ L, (1a)
∂t ∂x2
with the following initial and boundary conditions
∂u(x, 0) ∂ r̃(x, t)
u(x, t) = r̃(x, t), 0 ≤ x ≤ L, t ∈ [−s, 0], = ψ̃(x) = lim , (1b)
∂t t→−0 ∂t
u(0, t) = φ0 (t), u(L, t) = φL (t), t > 0, (1c)
where s > 0 is the delay parameter, K is a positive constant. The fractional derivative of order 1 < α ≤ 2 is
defined in Caputo sense.
In order to transform (1) to a system with zero Dirichlet boundary conditions, we define h(x, t) := φ0 (t) +
x
L (φL (t) − φ0 (t)) and introduce the new function v(x, t) = u(x, t) − h(x, t). Hence, we have
∂ α v(x, t) ∂ 2 v(x, t)
= K + f (x, t, v(x, t), v(x, t − s)), t > 0, 0 ≤ x ≤ L, (2a)
∂tα ∂x2
with the following initial and boundary conditions
∂v(x, 0) ∂r(x, t)
v(x, t) = r(x, t), 0 ≤ x ≤ L, t ∈ [−s, 0], = ψ(x) = lim , (2b)
∂t t→−0 ∂t
v(0, t) = v(L, t) = 0, t > 0. (2c)
We need to overcome two degrees of complexity; how to employ a suitable approximation of the time fractional
derivative on the one hand, and how to approximate the non linear delay source function linearly on the other,
in order to obtain a numerical solution for (2). Throughout this work and by the aid of [23], we suppose that the
function f (x, t, µ, v) and the solution u(x, t) of (2) are sufficiently smooth in the following sense:
• Let m be an integer
Smsatisfying ms ≤ T < (m + 1)s, define Ir = (rs, (r + 1)s), for r = −1, 0, . . . , m − 1,
Im = (ms, T ), I = q=−1 Iq and assume that u(x, t) ∈ C (6,3) ([0, L] × [0, T ]),
• The partial derivatives fµ (x, t, µ, v) and fv (x, t, µ, v) are continuous in the 0 -neighborhood of the solution.
Define
c1 = sup |fµ (x, t, u(x, t) + 1 , u(x, t − s) + 2 )| , (3a)
0 |aij |;
j6=i
10 bα0 K 1 bα0 K
aii = + 2, ai+1,i = − 2 = ai−1,i .
12 µ̄τ h 12 µ̄τ 2h
Therefore, the coefficient matrix is nonsingular and the theorem is readily proved by strong induction.
3 Convergence and stability for the difference scheme
Now, we introduce the uniqueness, stability and convergence theorems in L∞ -norm using the discrete energy
method for the proposed difference scheme.
The spatial domain [0, L] is covered by Ωh = {xi | 0 ≤ i ≤ M, } and let Vh = {v | v = (v0 , . . . , vM ), v0 = vM =
0} be a grid function space on Ωh . For any u, v ∈ Vh , define the discrete inner products and corresponding norms
as
M
X −1 M
X
hu, vi = h ui vi , hδx u, δx vi = h (δx ui−1/2 )(δx vi−1/2 ),
i=1 i=1
1
hδx2 u, vi = −hδx u, δx vi, δx ui = (ui − ui−1 ),
h
p p
kuk = hu, ui, |u|1 = hδx u, δx ui, kuk∞ = max |u|,
0≤i≤M
and denote v v
u M u M −1
u X u X
2
|v|1 = th (δx vi )2 , kδx vk∞ = th (δx2 vi )2 .
i=1 i=1
According to [25], the following inequalities are fulfilled
√
L L
kuk∞ ≤ |u|1 , kuk ≤ √ |u|1 . (11)
2 6
For the analysis of the difference scheme, we will use the following lemmas:
Lemma 3. Let v ∈ Vh , we have kδx2 vk ≤ h2 |v|1 .
√
Lemma 4. [18] Let v ∈ Vh and v0 = vM = 0. Then, we have kvk∞ ≤ 2L |v|1 .
Lemma 5. [20] For any G = {G1 , G2 , G3 , . . .} and ψ, we obtain
m
" k−1
# m
1−αl 2−αl
X
α
X α αl αl tm X tm
b0 Gk − (bk−j−1 − bk−j )Gj − bk−1 ψ Gk ≥
l
τ G2k − ψ2 .
j=1
2(2 − α l ) 2(2 − α l )
k=1 k=1
329
Lemma 6. Gronwall inequality [25]. Suppose that {F k | k ≥ 0} is a nonnegative sequence and satisfies
k+1
Pk
F ≤ A + Bτ l=1 F l , k ≥ 0, for some nonnegative constants A and B. Then F k+1 ≤ A exp(Bkτ ).
Theorem 2. (Convergence). Let v(x, t) ∈ [0, L]×[−s, T ], be the solution of (2) such that v(xi , tk ) = Vik and
vi (0 ≤ i ≤ M, −n ≤ k ≤ N ) be the solution of the difference scheme (9), denote eki = Vik − vik , for 0 ≤ i ≤ M ,
k
−n ≤ k ≤ N , r
3L 3T 1
C= exp (5c21 + c22 ) , θ = T α−1 Γ(3 − α),
8θK 8θK 2
then if
1
3−α 41
0 0
τ ≤ τ0 = , h ≤ h0 = , (12)
4C 4C
one has that
k ek k∞ ≤ C τ 3−α + h4 , 0 ≤ k ≤ N,
(13)
where c1 , c2 and 0 are those from (3).
The proof uses the previous formulated lemmas in the sense of our results in [12].
To discuss the stability of the difference scheme (9), we also use the discrete energy method in the same way
like the discussion of the convergence. Let {zik | 0 ≤ i ≤ M, 0 ≤ k ≤ N } be the solution of
k−1
1 k−1/2
X j−1/2
A bα 0 δt zi − (bα α
k−j−1 − bk−j )δt zi − bα
k−1 ψ(xi )
µ̄ j=1
k−1/2 1 1 1 1
= Kδx2 zi + Af xi , tk−1/2 , zik + zik−1 , zik−n−1 + zik−n , (14a)
2 2 2 2
such that 1 ≤ i ≤ M − 1, 1 ≤ k ≤ N, and supplied by appropriate initial and boundary conditions
z0k = φ0 (tk ), k
zM = φL (tk ), 1 ≤ k ≤ N, (14b)
zik = r(xi , tk ) + ρki , 0 ≤ i ≤ M, −n ≤ k ≤ 0, (14c)
where ρki is the perturbation of ψ(xi , tk ).
Following the same steps as in the proof of the convergence theorem, the stability of the scheme is obtained.
Theorem 3. (Stability). Let θik = zik − vik , for 0 ≤ i ≤ M, −n ≤ k ≤ N. Then there exist some arbitrary
positive constants c4 , c5 , h0 , τ0 which fulfill
v
0 u M −1
k
√ X
k k
u X
kθ k∞ ≤ c4 τ kρ k, 0 ≤ k ≤ N, kρ k = th (ρki )2 ,
k=−n i=1
conditioned by
h ≤ h0 , τ ≤ τ0 , max |ρki | ≤ c5 .
−n≤k≤0
0≤i≤M
4 Numerical Verification
Let νik be the solution of the constructed difference scheme (9) with the step sizes τ and h. Define the maximum
norm error by E(τ, h) = max kVik − νik k∞ . Also, define the following error rates
0≤i≤M
0≤k≤N
E(2τ, h) E(τ, 2h)
rate1 = log2 , rate2 = log2 .
E(τ, h) E(τ, h)
Consider the following numerical test example
∂ α u(x, t) ∂ 2 u(x, t)
= + f (x, t, u(x, t), u(x, t − s)), t ∈ (0, 1), 0 < x < π, (15a)
∂tα ∂x2
330
Γ(4)
f (x, t, u(x, t), u(x, t − s)) = sin(x) (t3 + 2t + 4) + Γ(4−α) t3−α − u(x, t − s) + sin(x)((t − s)3 + 2(t − s) + 4),
with the following initial and boundary conditions
u(x, t) = (t3 + 2t + 4) sin(x), 0 ≤ x ≤ π, t ∈ [−s, 0), s > 0, (15b)
u(0, t) = u(π, t) = 0, t ∈ [0, 1]. (15c)
The exact solution of this problem is
u(x, t) = (t3 + 2t + 4) sin(x). (16)
Results are presented in Tables (1) - (2) (time) and Table 3 (space). From these numerical results, we can see a
good agreement between theoretical and numerical results.
Table 1: Errors and convergence order of the difference scheme (9) for (15) in time variable with α = 1.25,
h = π/3000 and with time delay s = 1.
τ E(τ, h) rate1
1
10 0.00015
1
20 0.00004 1.732
1
40 0.00001 1.738
1
80 4.049 × 10−6 1.741
1
160 1.209 × 10−6 1.744
1
320 3.597 × 10−7 1.749
Table 2: Errors and convergence order of the difference scheme (9) for (15) in time variable with α = 1.75,
h = π/3000 and with time delay s = 41 .
τ E(τ, h) rate1
1
40 0.00003
1
80 0.00001 1.232
1
160 5.415 × 10−6 1.238
1
320 2.288 × 10−6 1.243
1
640 9.625 × 10−7 1.249
Table 3: Errors and convergence order of the difference scheme (9) for (15) in space variable with α = 1.5,
τ = 1/10000 and with time delay s = 12 .
h E(τ, h) rate2
π
4 0.00027
π
8 0.000018 3.872
π −6
16 1.2525 × 10 3.880
π
32 8.24587 × 10−8 3.925
π
64 5.28023 × 10−9 3.965
π
128 3.33926 × 10−10 3.983
Acknowledgements
This work was supported by Government of the Russian Federation Resolution N 211 of March 16, 2013.
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