A linearized difference scheme for a class of fractional partial differential equations with delay Текст научной статьи по специальности «Математика»

Известия Института математики и информатики УдГУ

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 class of non linear fractional partial differential equations with initial and Dirichlet boundary conditions is under consideration. We seek to obtain numerical solutions for this considered class of equations based on finite difference method. The convergence order will be 2 — a in time and four in space. A numerical example is given to support the theoretical results.

Keywords: fractional partial differential equation, linear difference scheme, delay, discrete energy method, convergence analysis.

Introduction

A great significance is devoted to study delay differential equations. They are widely used in many fields of science such as economics, physics, ecology, medicine, transportation scheduling, engineering control, computer aided design, nuclear engineering. They play a very important role in describing a variety of phenomena in the natural and social sciences. Also Fractional order differential equations, as generalizations of classical integer order differential equations, are increasingly used to model problems in fluid flow, finance and other areas of application. In [4,5], numerical approximations for some different classes of fractional differential equations were discussed. There are many contributions in literature which deals with obtaining numerical solutions of space-time fractional partial differential equations such as [6]. This paper presents a practical linear difference scheme for solving space-time fractional partial differential equation with time delay. This linear difference scheme is applied previously for for a class of nonlinear delay partial differential equations [1,7]. In this approach, we extend this idea to time and space fractional partial differential equation with nonlinear delay.

dau d^ u

-q^ ~ d-Q^ = f{x,t,u{x,t),u{x,t - s)), a < x <b, te[0,T], (0.1)

u(a, t) = ua(t), u(b, t) = ub(t), t € [0,T], (0.2)

u(x,t) = p(x,t), x € [a, b], t € [—s, 0), (0.3)

where 0 < a < 11 < ,0 ^ 2, d > 0 is the diffusion coeflici ent and s > 0 is the delay parameter. Throughout this work, we suppose that the function f (x,t,^, v) and the solution u(x,t) are sufficiently smooth and assume that f (x,t, v) has the first order continuous derivative with respect to the first and second components in the e0 neighborhood of the solution such that e0 is a positive constant. Let c0 = max | u(x,t) c1 = max | f„(u(x,t) + ei, u(x,t — s) + e2,x,t)

a<x<b a<x<b, 0<t<T

0<t<T |ei|<eo,|e2|<eo

c2 = max | fv(u(x,t) + e1, u(x,t — s) + e2,x,t)

a<x<b, 0<t<T |ei|^eo,|e2|<eo

§ 1. Derivation of the linearized difference scheme

Take two positive integers M and n, and let h = ^jf, r = ^ such that Xi = a + ih, tk = kr and tk+1 = (k + \)t = \{tk + ifc+i). Cover the domain by Q/jT = il^x Qr, where Q/j = {xi\0 ^ i ^ M},

QT = {tk| -n < k < N}, N = Let W = {v\v = vf, 0 < i < M, -n < k < N} be a grid function space on Define = +

1This work was supported by Act 211 Government of the Russian Federation program 02. A03.21.0006 on 27.08.2013 and by RFBR Grant 13-01-00089.

Kartarv and his group [3] obtained the following approximation for the time Caputo fractional derivative at tk+1:

dau(tk+i,Xi

dta such that

k-1 (ufc+1 - vk) 1

+ +0(r2-«), (1.1)

m=1

^ = ° = 0<a<l. (1.2)

Also, Sun and his group [2] presented the following averaging operator

Avx = cfv(x - h) + (1 - 2cf)v(x) + cfv(x + h), 1 <0 < 2. (1.3)

It is easy to verify that

Av (x) = (1 + cf h2^) v (x). (1.4)

Also,

= + h) — 2v(x) + z/(x —

1

^0*0 = 7-8 E - -^+

k=0

where wf = Aigf, wf = Axgf + Acgf, wf = Aigf + Aogf_i + k > 2,

02 + 3,0 + 2 4 - 02 02 - 30 + 2 f -02 + 0 + 2

Ai —-, An —-, A_i —-, Co —-.

12 ' 6 12 ' 2 24

f 1 f C1 0 + ^ f

= !> 5fc = (1--

They proved some properties concerned with the averaging operator A

(Am, v) = (u, Av), |v ||A = (Av,v), < l|vIIA < l|v||2, (¿f v,v) < 0.

|v II2

3

Remark 1. Riemann-Liouville and Caputo operators have the following property

RD?u{x,t)= cD?u{x,t) - + 1-yM(fc)(x'°)' m= 1,2,3,.... (1.6)

According to (1.1) and the property (1.6), we can write Kartarv approximation 0 < a < 1 at the points tk+1 as follows

r k~l / \ (vk+l -vk)l

m=1

(1.7)

Consider Eq.(O.l) at the points (xi,tk+1), gives 9au(a;i,ifc+i) d^u(xi,tk+1)

-^—-~d-dxf3 2 = -s)), (1.8)

0 ^ i ^ M, 0 ^ k ^ N - 1.

Remark 2. Taylor expansion yields

dßu(xi,tk+i) (faß

(1.9)

dßu(xj,tk+1

¡9^

1 / d2 d2 \

-(-p-ßu(xutk) + —I^-ßu(xi,tk+1) J + 0(r2),

2 V dx2

1 /dßu(xj,tk) i

2 V dx/3

+ ■

+ O(T2 ).

Remark 3. Taylor expansion yields

'U(, t

fc+i •

fc-n+i __ 1 2*

u{xt,tk+h -s) = u;-"^ = rA+l-n + rA-n + o(r2).

After substitution with (1.7), (1.11), and (1.12), (1.13) into (1.8), we obtain

k-l

UJiUi + - Wfc_m)u™ + f _2 - Wfcj

m=1 (i

- Wk ' + CT

2l-c

d fdßu(xj,tk) dßu(xj,tk+1)\

3 k

^ ! 2 Ul

2 V dxß dxß )

1 i._i 1 1

k—n

t 2—a) + 0(t 2),

such that

0 ^ i ^ M, 0 ^ k ^ N — 1. By Operating with the averaging operator A on both sides of (1.14), we have

(1.10) (1.11)

(1.12) (1.13)

(1.14)

k— 1

A

Wl'k + (Wk—m+1 - Wk—+ (

t

m=1

r(1 - a)

fc+é \ o , K - Ui)

2 -wfc + (7- 1

21-

_ d fdßu(xj,tk) ^(a^ifc+ih 2 V 9a;/3 9a;/3 J

k

11 2Ui oUi

1 fc-1 fc+l-ri , }_„.k—n

1 2 2

t 2—a) + 0(t 2).

(1.15)

Recall the properties of the averaging operator A (1.3)—(1.5), then (1.15) can be written as follows

A

k-1

Wiuf + (Wfc-m+l - Uk-m)UT + X

m=1

- Wk ' + CT

Kfc+1 - «?)

2l-a

= 2 + A/ (a*, ifc+1, - + -utn) + 0(r2"«) + 0(r2) + 0(/>4). (1.16)

Then, we can write

k—1

A

W1 Uk + (Wk—m+1 - Wk—m)Um + (

j.—a .

-ujk)Ui + CT-

m=1

r(1 - a)

21—a

-uk 2 1

+ -Uk

+ Rk

(1.17)

such that

1 < i < M - 1, 0 < k < N - 1,

—a

and

K cs(T2-a + T2 + h4).

Noting that the initial and boundary conditions after partition will be:

Uo = u„(tfc), UkM = u6(ifc), 1 < k < N, Uk = p(xi, tk), 0 < i < M, -n < k < 0.

(1.18)

(1.19)

Omit the small term Rk in (1.18) and replace Uk with uk, the constructed linear difference scheme will have the following form

A

k-1

Wl"Uk + E (wfc-m+l - u)k-m)u™ + ( J* - Wfc)

m=1

= + A/(x,,ifc+i, ?^uk - i

^ n

21-a

1 fc+l-n. I l„fc-ra

2 + 2

(1.20)

such that

1 < i < M - 1, 0 < k < N - 1,

u0k = ua(tk), uM = ub(ik), 1 < k < N, (1.21)

uk = p(xi, tk), 0 < i < M, -n < k < 0. (1.22)

Remark 4. When 0 = 2, (1.20) coincides with the the linear difference scheme for the time fractional

partial differential equation with delay

dau d 2u

-^■-d-^ = f(x,t,u(x,t),u(x,t-s)), a < x < b, t£[0,T],

u(a,t)= ua(t), u(b, t)= ub(t), t G [0,T], u(x,t) = p(x,t), x G [a, b], t G [-s, 0),

where 0 < a < 1, d is the diffusion coefficient and s > 0 is the delay parameter. And the resulted difference scheme will have the form

(1.23)

(1.24)

(1.25)

A

fc-i f-^ Wl"Uk + E (Wfc-m+l - Wfc_m)u™ + f _2 - Wfc J

m=1 (1

l0, (<+ - ) n -wfc vï + a^-^

21-c

= dölu°+* + A/(^,ifc+i, -u* - -u\

1 k— 1 ^ fc+1—ri I ri

The averaging operator A will have the following form

1

(1.26)

Az/(x) = (1 + c22h262x)v{x) = (1 + — h262x)v{x) = c22v{x - h) + ( 1 - 2c|)i/(a;) + î|ï/(x + fc) =

12

v (x — h) + 10v (x) + v (x + h)

Remark 5. If we replace the averaging operator A by the unit ope rator I, then we obtain the following 2 - a order in time and second order in space difference scheme

such that

a

k-1 t wlui + E (UJk-m+1 - +

1

r(1 - a)

n (U" ' " - UÏ

21-c

1

jxß k+i I + ^ k 1 k-1 1 fc+l-n I 1 fc-n

+f(Xi,tk+i, -Ui - -Ut +2Mi

1 < i < M - 1, 0 < k < N - 1,

Ua (tk ),

'M

U5(tk), 1 < k < N,

uk = p(xj,tk), 0 < i < M, -n < k < 0.

(1.27)

(1.28) (1.29)

u

u

2. Convergence and stability of the proposed scheme

Denote ek = Uk-uk, 0 < i < M, -n < k < N and subtract (1.20)^(1.22) from (1.17)^(1.19),

ek = Uk - uk we obtain the error difference scheme

A

k~l / ^fc+i \ (ek+1 - ek)

+ " wfc-m)e™ + (r J - wfcJe° + * ^ i!

m=1 ^ '

.1 -\ut\\uk+l~n+ \utn

= c16:^ +*f[(xi,t^,±u? - -2Ut\+ ^

- f(xi,tk+h, |uk - + \vk-n)] + ^ (2-1)

3 k 1 k_1 1 k+i_« ■ 1 IlA — ti ' a — ti '

t 2 » 2 2

1 < i < M - 1, 0 < k < N - 1,

kk

e0 = 0, eM = 0, 1 < k < N, (2.2)

ek = 0, 0 < i < M, -n < k < 0. (2.3)

If the spatial domain [a, b] is covered bv = (x^ | 0 ^ i ^ M,} and let

Vh = (v | v = (vo,..., vm), vo = vm = 0}

be a grid function space on Qh.

For any u, v € Vh, define the discrete inner products and corresponding norms as

M_1 M

(u,u) = UiVi, (5xu,5xv) = /»^(¿xV^ft^-i)-i=1 i=1 2 2

and ___

|| u ||= v (u,u), | u |i= v (¿xu, ¿xu), || u ||™= max | u |.

The following inequalities are achieved

.. .. \/b — a......b — a . . A.

II u lloo^—^— I u I1' II u 11^ I u li- (2-4)

Lemma 2.1. For any u € Vh ^ holds that

(a [WlU* + E (Wfc-nH-l " + " Wfc) + (T^""^] , ) >

m=1 ^ '

> f II 7/k+1 II2 - II 7/k II2 ^ 22-a ^ H I 'St II U% Hsi )■

Lemma

2.2 ([7]). Suppose that (Fk | k ^ 0} 6e a non negative consequence and satisfies Fk+1 < A + Bt Ek=1 FL, k = 0,1,..., then Fk+1 < Aexp(Bkr), k = 0,1,..., sucA iAai A B are non negative constants.

For the difference scheme (1.20)—(1.22) and by using the previous lemmas, we can deduce the following convergence result.

Theorem 2.1. Let u(x, t), x € [a, b], -s ^ t ^ T be the solution of (0.1)-(0.3) and (uk | 0 ^ i ^ M, -n ^ k ^ N} 6e the solution of the considered difference scheme (1.20)-(1.22) denote etk = Uk - uk, 0 < i < M, -n < k < N and

„ MVb^L M2(10e2 + 5c? + Co). 1

C =-c3 exp(-—5-), e =

6e2 " 3Tr(2 - a)22-a '

then if

we have

ek c(t2-a + h4), 0 < k < N. (2.5)

To discuss the stability of the difference scheme (1.20)—(1.22), we use the discrete energy method

in the same way like the discussion of the convergence. Let | 0 ^ i ^ M, 0 ^ k ^ N} be the solution of

A

k~1 , t, , i \ (i/k+l iM - fc+2 ,, V.,0 I .^i ~Vi)

--Wfc H/j + a-

wlvi + Y 1 ~ + ( ~ Wfc)

m=1 ^ '

21-c

3 u 1 1 1 kl1_n 1

= dô^t + Af(Xi,tk+i, ¡uk - -uk~\+ -vtn} (2.6)

2 2 2 2 2

such that

1 ^ i ^ M - 1, 0 ^ k ^ N - 1,

V? = ua(tk), vM = ub(tk), 1 < k < N, (2.7)

vk = p(xi, tk) + 0k, 0 ^ i < M, — n < k < 0, (2.8)

where 0? is the perturbation of p(x^,tk).

Theorem 2.2. Let n? = v? — uk, 0 ^ i ^ M, — n ^ k ^ N. TAen iAere exist constants c7,c8,h0,t0 such that

0 M-1

II n? C7T £ || 0? ||, 0 < k < N, || 0? ||= , h ^ (0?)2,

k=-n \ i=1

only if

h ^ h0, t ^ t0

and

max | 0? c8.

§ 3. Test example

Consider the following time-space fractional partial differential equation with delay

d?u

-2—,=f(x,t,u(x,t),u(x,t-0.1)), 1 < x <2, i€(0,l], (3.1)

dta dx?

—31, ,3

u(l,t) = —— (r-2t-l), u(2,t)=0, ¿e(0,1], (3.2)

32

u(a:,i) = (—x6 -x)(t3 -2t-l), x e (1,2), te [-0.1,0), (3.3)

32

where 0 < a < 1, 1 < ^ ^ 2,

f(x, t, u(x, t),u{x, t - 0.1)) = u{x, t - 0.1)2 - 2£i + 6 - ~ %?{{t ~ 0.1)3 - 2{t - 0.1) - l)2,

32

such that

32 r(7) — ¡3 r(2 — (3) )[t 1 h

6 - ( r(4) t3- - 2F(2) - x)

~ r(4 — a) T(2-ay 32 Xh

The exact solution is

u{x,t) = (—x6-x)(t3 - 2t - 1). 32

Let uk | 0 ^ i ^ M, 0 ^ k ^ N is the solution of the constructed difference scheme (1.20)—(1.22), define the maximum norm error

t) = max | u(xj,tfc) — uk

o^MM o^k^N

In the following table, we present the maximum errors for different numerical solutions obtained with different step sizes when (a = 0.1 fi = 1.9).

h Т Eoo(h,T) lnr Eco{h,T)

10ё2 Есю(к/2,т/4)

i 10 i 20 1 40 1 80 1 100 1 400 1 1600 1 6400 3.25 х 10"5 2.08 x 10"6 1.310 x 10"7 8.215 x 10"9 3.96578 3.98894 3.99516 *

§ 4. Conclusion

This work is related to a class of fractional partial differential equations with non linear delay. A linearized difference scheme was constructed to solve this sort of equations. Un conditional convergence and stability for the numerical difference scheme were proved. A numerical example supported our theoretical results. Our difference scheme can be easily applied for two dimensional delay problems with fractional orders.

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Received 01.10.2015

Hendy Akhmed Said, Department of Computational Mathematics, Ural Federal University, pr. Lenina, 51, Yekaterinburg, 620083, Russia. E-mail: [email protected]

А. С. Хенди

Линеаризованная разностная схема для класса дифференциальных уравнений с частными производными дробного порядка с запаздыванием

Ключевые слова: дифференциальные уравнения с частными производными дробного порядка, линейная разностная схема, запаздывание, анализ сходимости.

УДК 517.958, 530.145.6

Хенди Ахмед Сайд, Уральский федеральный университет, 620083, Россия, г. Екатеринбург, пр. Ленина, 51. E-mail: [email protected]