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A linearized difference scheme for a class of fractional partial differential equations with delay.

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Известия Института математики и инорматики УдУ
2015. Вып. 2 (46)
MSC: 35R11, 65M06, 26A33
A. S. Hendy
A LINEARIZED DIFFERENCE SCHEME FOR A CLASS OF FRACTIONAL
PARTIAL DIFFERENTIAL EQUATIONS WITH DELAY1
A lass of non linear frational partial dierential equations with initial and Dirihlet boundary onditions is under
onsideration. We seek to obtain numerial solutions for this onsidered lass of equations based on nite dierene
method. The onvergene order will be 2 ? ? in time and four in spae. A numerial example is given to support the
theoretial results.
Keywords : frational partial dierential equation, linear dierene sheme, delay, disrete energy method, onvergene
analysis.
Introdution
A great signiane is devoted to study delay dierential equations. They are widely used in
many elds of siene suh as eonomis, physis, eology, mediine, transportation sheduling, engineering ontrol, omputer aided design, nulear engineering. They play a very important role in
desribing a variety of phenomena in the natural and soial sienes. Also Frational order dierential equations, as generalizations of lassial integer order dierential equations, are inreasingly
used to model problems in uid ow, nane and other areas of appliation. In [4, 5?, numerial
approximations for some dierent lasses of frational dierential equations were disussed. There
are many ontributions in literature whih deals with obtaining numerial solutions of spaetime
frational partial dierential equations suh as [6?. This paper presents a pratial linear dierene
sheme for solving spaetime frational partial dierential equation with time delay. This linear
dierene sheme is applied previously for for a lass of nonlinear delay partial dierential equations [1, 7?. In this approah, we extend this idea to time and spae frational partial dierential
equation with nonlinear delay.
?? u
? ?u
?
d
= f (x, t, u(x, t), u(x, t ? s)),
?t?
?x?
u(a, t) = ua (t),
u(x, t) = ?(x, t),
a < x < b,
u(b, t) = ub (t),
x ? [a, b],
t ? [0, T ],
t ? [?s, 0),
t ? [0, T ],
(0.1)
(0.2)
(0.3)
where 0 < ? < 1, 1 < ? 6 2, d > 0 is the diusion oeient and s > 0 is the delay parameter.
Throughout this work, we suppose that the funtion f (x, t, µ, ?) and the solution u(x, t) are suiently smooth and assume that f (x, t, µ, ?) has the rst order ontinuous derivative with respet
to the rst and seond omponents in the ?0 neighborhood of the solution suh that ?0 is a positive
onstant. Let c0 = max | u(x, t) |, c1 =
max
| fµ (u(x, t) + ?1 , u(x, t ? s) + ?2 , x, t) |,
a<x<b
0<t<T
c2 =
max
a<x<b, 0<t<T
|?1 |6?0 ,|?2 |6?0
a<x<b, 0<t<T
|?1 |6?0 ,|?2 |6?0
| f? (u(x, t) + ?1 , u(x, t ? s) + ?2 , x, t) |.
џ 1. Derivation of the linearized dierene sheme
s
Take two positive integers M and n, and let h = b?a
M , ? = n suh that xi = a + ih, tk = k? and
1
1
tk+ 1 = (k + 2 )? = 2 (tk + tk+1 ). Cover the domain by ?h? = ?h Ч ?? , where ?h = {xi |0 6 i 6 M },
2
?? = {tk | ? n 6 k 6 N },
N = [ T? ]. Let W = {?|? = ?ik , 0 6 i 6 M, ?n 6 k 6 N } be a grid
k+ 21
funtion spae on ?h? . Dene ?i
= 21 (?ik + ?ik+1 ).
1
This work was supported by At 211 Government of the Russian Federation program 02.A03.21.0006 on 27.08.2013
and by RFBR Grant 13-01-00089.
236
Kartary and his group [3? obtained the following approximation for the time Caputo frational
derivative at tk+ 1 :
2
? ? u(tk+ 1 , xi )
2
?t?
suh that
k?1
h
X
(uk+1 ? uk ) i
(?k?m+1 ? ?k?m)um ? ?k u0 + ? i 1?? i + O(? 2?? ),
= ? 1 uk +
2
m=1
1
1
?i = ?((i + )1?? ? (i ? )1?? ),
2
2
?=
1
1
,
?(2 ? ?) ? ?
0 < ? < 1.
(1.1)
(1.2)
Also, Sun and his group [2? presented the following averaging operator
A?x = c?2 ?(x ? h) + (1 ? 2c?2 )?(x) + c?2 ?(x + h),
It is easy to verify that
Also,
1 < ? 6 2.
(1.3)
A?(x) = (1 + c?2 h2 ?x2 )?(x).
(1.4)
??
?(x)
= ?x? ?(x) + O(h4 ),
A
?x?
(1.5)
1
?(x
+
h)
?
2?(x)
+
?(x
?
h)
,
h2
+?
1 X ?
wk ?(x ? (k ? 1)h) + O(h2 ),
?x? ?(x) = ?
h
?x2 ?(x) =
k=0
?
?
where w0? = ?1 g0? , w1? = ?1 g1? + ?0 g0? , wk? = ?1 gk? + ?0 gk?1
+ ??1 gk?2
, k > 2,
?1 =
? 2 + 3? + 2
,
12
?0 =
4 ? ?2
,
6
??1 =
? 2 ? 3? + 2
,
12
c?2 =
?? 2 + ? + 2
,
24
?+1 ?
)gk?1 .
k
They proved some properties onerned with the averaging operator A
g0? = 1,
gk? = (1 ?
hAu, ?i = hu, A?i,
|?k2A = hA?, ?i,
k?k2
6 k?k2A 6 k?k2 ,
3
h?x? ?, ?i 6 0.
R e m a r k 1. RiemannLiouville and Caputo operators have the following property
?
R Dt u(x, t) =
?
c Dt u(x, t) ?
m?1
X
k=0
tk??
u(k) (x, 0),
?(k + 1 ? ?)
m ? 1 < ? 6 m,
m = 1, 2, 3, . . . .
(1.6)
Aording to (1.1) and the property (1.6), we an write Kartary approximation 0 < ? < 1 at the
points tk+ 1 as follows
2
?
R Dt u(xi , tk+ 1 )
2
h
k
= ?1 u +
k?1
X
m
(?k?m+1 ??k?m )u +
m=1
Consider Eq.(0.1) at the points (xi , tk+ 1 ), gives
t??
k+ 1
(uk+1 ? uk ) i
??k u0 +? i 1?? i +O(? 2?? ).
?(1 ? ?)
2
(1.7)
2
2
? ? u(xi , tk+ 1 )
2
?t?
?d
? ? u(xi , tk+ 1 )
2
?x?
= f (xi , tk+ 1 , u(xi , tk+ 1 ), u(xi , tk+ 1 ? s)),
0 6 i 6 M,
2
0 6 k 6 N ? 1.
237
2
2
(1.8)
R e m a r k 2. Taylor expansion yields
? ? u(xi , tk+ 21 )
?x?
? ? u(xi , tk+ 12 )
?x?
=
? 2 2??
1) ,
I
u(x
,
t
i
k+
2
?x2
1 ? 2 2??
? 2 2??
I
u(x
,
t
)
+
I
u(x
,
t
)
+ O(? 2 ),
i
k
i
k+1
2 ?x2
?x2
1 ? ? u(xi , tk ) ? ? u(xi , tk+1 ) + O(? 2 ).
+
2
?x?
?x?
=
=
(1.9)
(1.10)
(1.11)
R e m a r k 3. Taylor expansion yields
k+ 21
u(xi , tk+ 12 ) = ui
1
3
?
= uki ? uik?1 + O(? 2 ),
2
2
(1.12)
1
1
?
+ uik?n + O(? 2 ).
= uk+1?n
2 i
2
(1.13)
k?n+ 21
u(xi , tk+ 21 ? s) = ui
After substitution with (1.7), (1.11), and (1.12), (1.13) into (1.8), we obtain
h
?1 uki
+
k?1
X
(?k?m+1 ?
?k?m)um
i
+
m=1
t??
k+ 1
(uk+1 ? uk ) i
? ?k u0i + ? i 1?? i ?
?(1 ? ?)
2
2
d ? ? u(xi , tk ) ? ? u(xi , tk+1 ) +
=
2
?x?
?x?
3
1
1
1
= f xi , tk+ 1 , uki ? uik?1 , uik+1?n + uik?n + O(? 2?? ) + O(? 2 ),
2 2
2
2
2
?
suh that
0 6 i 6 M,
(1.14)
0 6 k 6 N ? 1.
By Operating with the averaging operator A on both sides of (1.14), we have
k?1
t??
h
X
(uk+1 ? uk ) i
k+ 21
m
k
(?k?m+1 ? ?k?m )ui +
A ? 1 ui +
? ?k u0i + ? i 1?? i ?
?(1 ? ?)
2
m=1
d ? ? u(xi , tk ) ? ? u(xi , tk+1 ) ? A
+
=
2
?x?
?x?
1
1
1
3
= Af xi , tk+ 1 , uki ? uik?1 , uik+1?n + uik?n + O(? 2?? ) + O(? 2 ).
(1.15)
2 2
2
2
2
Reall the properties of the averaging operator A (1.3)(1.5), then (1.15) an be written as follows
k?1
t??
h
X
(uk+1 ? uk ) i
k+ 21
m
k
? ?k u0i + ? i 1?? i =
(?k?m+1 ? ?k?m )ui +
A ? 1 ui +
?(1 ? ?)
2
m=1
1
1
1
3
+ Af xi , tk+ 1 , uki ? uik?1 , uik+1?n + uik?n + O(? 2?? ) + O(? 2 ) + O(h4 ). (1.16)
2 2
2
2
2
Then, we an write
k+ 21
= d?x? ui
k?1
t??
h
X
(U k+1 ? U k ) i
k+ 21
m
k
(?k?m+1 ? ?k?m )Ui +
A ? 1 Ui +
? ?k Ui0 + ? i 1?? i
=
?(1 ? ?)
2
m=1
k+ 21
= d?x? Ui
suh that
3
1
1
1
+ Af xi , tk+ 1 , Uik ? Uik?1 , Uik+1?n + Uik?n + Rik ,
2 2
2
2
2
1 6 i 6 M ? 1,
0 6 k 6 N ? 1,
238
(1.17)
and
| Rik |6 c3 ? 2?? + ? 2 + h4 .
Noting that the initial and boundary onditions after partition will be:
U0k = ua (tk ),
k
UM
= ub (tk ),
1 6 k 6 N,
(1.18)
Uik = ?(xi , tk ),
0 6 i 6 M,
?n 6 k 6 0.
(1.19)
Omit the small term
in (1.18) and replae
will have the following form
Rik
Uik
k?1
h
X
(?k?m+1 ? ?k?m)um
+
A ?1 uki +
i
m=1
k+ 21
= d?x? ui
suh that
with
uki ,
the onstruted linear dierene sheme
t??
k+ 1
(uk+1 ? uk ) i
? ?k u0i + ? i 1?? i =
?(1 ? ?)
2
2
1
1
1
3
+ Af xi , tk+ 1 , uki ? uik?1 , uik+1?n + uik?n ,
2 2
2
2
2
1 6 i 6 M ? 1,
(1.20)
0 6 k 6 N ? 1,
uk0 = ua (tk ),
ukM = ub (tk ),
1 6 k 6 N,
(1.21)
uki = ?(xi , tk ),
0 6 i 6 M,
?n 6 k 6 0.
(1.22)
R e m a r k 4. When ? = 2, (1.20) oinides with the the linear dierene sheme for the time frational
partial dierential equation with delay
?2u
??u
?
d
= f (x, t, u(x, t), u(x, t ? s)),
?t?
?x2
u(a, t) = ua (t),
a < x < b,
u(b, t) = ub (t),
u(x, t) = ?(x, t),
x ? [a, b],
t ? [0, T ],
t ? [0, T ],
t ? [?s, 0),
(1.23)
(1.24)
(1.25)
where 0 < ? < 1, d is the diusion oeient and s > 0 is the delay parameter.
And the resulted dierene sheme will have the form
k?1
t??
h
X
(uk+1 ? uk ) i
k+ 21
m
k
? ?k u0i + ? i 1?? i =
(?k?m+1 ? ?k?m )ui +
A ? 1 ui +
?(1 ? ?)
2
m=1
1
1
1
3
+ Af xi , tk+ 1 , uki ? uik?1 , uik+1?n + uik?n .
2 2
2
2
2
The averaging operator A will have the following form
k+ 21
= d?x2 ui
(1.26)
1 2 2
h ?x )?(x) = c22 ?(x ? h) + (1 ? 2c22 )?(x) + c22 ?(x + h) =
12
1
=
?(x ? h) + 10?(x) + ?(x + h) .
12
A?(x) = (1 + c22 h2 ?x2 )?(x) = (1 +
R e m a r k 5. If we replae the averaging operator A by the unit operator I , then we obtain the following
2 ? ? order in time and seond order in spae dierene sheme
k?1
h
t??
X
(uk+1 ? uk ) i
k+ 12
k
m
? ?k u0i + ? i 1?? i =
?1 u i +
(?k?m+1 ? ?k?m )ui +
?(1 ? ?)
2
m=1
k+ 21
= d?x? ui
suh that
3
1
1
1
+ f xi , tk+ 12 , uki ? uik?1 , uk+1?n
+ uik?n ,
i
2
2
2
2
1 6 i 6 M ? 1,
uk0 = ua (tk ),
uki = ?(xi , tk ),
(1.27)
0 6 k 6 N ? 1,
= ub (tk ),
1 6 k 6 N,
(1.28)
0 6 i 6 M,
?n 6 k 6 0.
(1.29)
ukM
239
џ 2. Convergene and stability of the proposed sheme
Denote eki = Uik ?uki , 0 6 i 6 M,
we obtain the error dierene sheme
?n 6 k 6 N and subtrat (1.20)(1.22) from (1.17)(1.19),
k?1
t??
h
X
(ek+1
? eki ) i
k+ 21
0
m
k
i
? ?k ei + ?
=
(?k?m+1 ? ?k?m)ei +
A ?1 ei +
?(1 ? ?)
21??
m=1
h
1
1
1
3
xi , tk+ 1 , Uik ? Uik?1 , Uik+1?n + Uik?n ?
2 2
2
2
2
i
3
1
1
1
? f xi , tk+ 1 , uki ? uik?1 , uik+1?n + uik?n + Rik ,
2 2
2
2
2
1 6 i 6 M ? 1, 0 6 k 6 N ? 1,
k+ 21
= d?x? ei
+ Af
ek0 = 0,
eki
= 0,
ekM = 0,
(2.2)
1 6 k 6 N,
0 6 i 6 M,
(2.1)
(2.3)
?n 6 k 6 0.
If the spatial domain [a, b] is overed by ?h = {xi | 0 6 i 6 M, } and let
Vh = {? | ? = (?0 , . . . , ?M ),
?0 = ?M = 0}
be a grid funtion spae on ?h .
For any u, ? ? Vh , dene the disrete inner produts and orresponding norms as
hu, ?i = h
M
?1
X
ui ? i ,
h?x u, ?x ?i = h
k u k=
p
hu, ui,
(?x ui? 1 )(?x ?i? 1 ),
2
2
i=1
i=1
and
M
X
| u |1 =
p
h?x u, ?x ui,
The following inequalities are ahieved
?
b?a
k u k? 6
| u |1 ,
2
k u k? = max | u | .
06i6M
b?a
k u k6 ? | u |1 .
6
(2.4)
L e m m a 2.1. For any u ? Vh , it holds that
k?1
t??
D h
X
(uk+1 ? uk ) i k+ 1 E
k+ 21
m
k
(?k?m+1 ? ?k?m )ui +
A ?1 ui +
? ?k u0i + ? i 1?? i , ui 2 >
?(1 ? ?)
2
m=1
? k 2
2
.
> 2?? k uk+1
?
k
u
k
k
i
A
A
i
2
L e m m a 2.2P( [7?). Suppose that {F k | k > 0} be a non negative onsequene and satises
k+1
F
6 A + B? kl=1 F L , k = 0, 1, . . . , then F k+1 6 A exp(Bk? ), k = 0, 1, . . . , suh that A, B
are non negative onstants.
For the dierene sheme (1.20)(1.22) and by using the previous lemmas, we an dedue the
following onvergene result.
T h e o r e m 2.1. Let u(x, t), x ? [a, b], ?s 6 t 6 T be the solution of (0.1)(0.3) and {uki |
0 6 i 6 M, ?n 6 k 6 N } be the solution of the onsidered dierene sheme (1.20)(1.22), denote
eki = Uik ? uki , 0 6 i 6 M, ?n 6 k 6 N and
?
M 2 (10?2 + 5c21 + c22 )
1
M b?a
c3 exp(
), ? =
,
C=
2
?
6?
3T ?(2 ? ?)22??
then if
? 1
? 1
0 4
0 2??
, h6
,
4C
4C
k ek k? 6 C ? 2?? + h4 , 0 6 k 6 N.
?6
we have
240
(2.5)
To disuss the stability of the dierene sheme (1.20)(1.22), we use the disrete energy method
in the same way like the disussion of the onvergene. Let {?ik | 0 6 i 6 M, 0 6 k 6 N } be the
solution of
suh that
k?1
t??
h
X
(? k+1 ? ? k ) i
k+ 21
m
k
(?k?m+1 ? ?k?m)?i +
A ? 1 ?i +
? ?k ?i0 + ? i 1?? i
=
?(1 ? ?)
2
m=1
1
1
1
3
= d?x? ?ik + Af xi , tk+ 1 , ?ik ? ?ik?1 , ?ik+1?n + ?ik?n ,
2 2
2
2
2
1 6 i 6 M ? 1,
?0k
?ik
= ua (tk ),
= ?(xi , tk ) +
k
?M
?ki ,
(2.6)
0 6 k 6 N ? 1,
= ub (tk ),
(2.7)
1 6 k 6 N,
0 6 i 6 M,
?n 6 k 6 0,
(2.8)
where ?ki is the perturbation of ?(xi , tk ).
T h e o r e m 2.2. Let ?ik = ?ik ? uki ,
c7 , c8 , h0 , ?0 suh that
k ? k k? 6 c7 ?
0
X
0 6 i 6 M,
k ?k k,
?n 6 k 6 N. Then there exist onstants
v
u M ?1
u X
k
(?ki )2 ,
k ? k= th
0 6 k 6 N,
i=1
k=?n
only if
h 6 h0 ,
and
? 6 ?0
max | ?ki |6 c8 .
?n6k60
06i6M
џ 3. Test example
Consider the following time-spae frational partial dierential equation with delay
??u
?? u
?
2
= f (x, t, u(x, t), u(x, t ? 0.1)), 1 < x < 2, t ? (0, 1],
?t?
?x?
?31 3
u(1, t) =
(t ? 2t ? 1), u(2, t) = 0, t ? (0, 1],
32
1
u(x, t) = ( x6 ? x)(t3 ? 2t ? 1), x ? (1, 2), t ? [?0.1, 0),
32
where 0 < ? < 1, 1 < ? 6 2,
f (x, t, u(x, t), u(x, t ? 0.1)) = u(x, t ? 0.1)2 ? 2?1 + ?2 ? (
suh that
(3.1)
(3.2)
(3.3)
1 6
x ? x)2 ((t ? 0.1)3 ? 2(t ? 0.1) ? 1)2 ,
32
1 ?(7)
?(2)
x6?? ?
x1?? )(t3 ? 2t ? 1),
32 ?(7) ? ?
?(2 ? ?)
2?(2) 1?? 1 6
?(4) 3??
t
?
t
)( x ? x).
?2 = (
?(4 ? ?)
?(2 ? ?)
32
?1 = (
The exat solution is
1 6
x ? x)(t3 ? 2t ? 1).
32
Let uki | 0 6 i 6 M , 0 6 k 6 N is the solution of the onstruted dierene sheme (1.20)(1.22),
dene the maximum norm error
u(x, t) = (
E? (h, ? ) = max | u(xi , tk ) ? uki |.
06i6M
06k6N
In the following table, we present the maximum errors for dierent numerial solutions obtained
with dierent step sizes when (? = 0.1, ? = 1.9).
241
h
?
1
10
1
20
1
40
1
80
1
100
1
400
1
1600
1
6400
E? (h, ? )
3.25 Ч 10?5
2.08 Ч 10?6
1.310 Ч 10?7
8.215 Ч 10?9
log2
E? (h,? )
E? (h/2,? /4)
3.96578
3.98894
3.99516
?
џ 4. Conlusion
This work is related to a lass of frational partial dierential equations with non linear delay.
A linearized dierene sheme was onstruted to solve this sort of equations. Un onditional
onvergene and stability for the numerial dierene sheme were proved. A numerial example
supported our theoretial results. Our dierene sheme an be easily applied for two dimensional
delay problems with frational orders.
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Reeived 01.10.2015
Hendy Akhmed Said, Department of Computational Mathematis, Ural Federal University, pr. Lenina, 51,
Yekaterinburg, 620083, Russia.
E-mail: ahmed.hendyfs.bu.edu.eg
А. С. Хенди
Линеаризованная разностная схема для класса диеренциальных уравнений с частными
производными дробного порядка с запаздыванием
диеренциальные уравнения с частными производными дробного порядка, линейная
разностная схема, запаздывание, анализ сходимости.
Ключевые слова:
УДК 517.958, 530.145.6
Хенди Ахмед Саид, Уральский едеральный университет, 620083, оссия, г. Екатеринбург, пр. Ленина,
51. E-mail: ahmed.hendyfs.bu.edu.eg
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class, equations, scheme, differential, different, partial, linearized, dela, fractional
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