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 L1-norm by means of the discrete energy method. A numerical test example is introduced to illustrate the accuracy and efficiency of the proposed method.
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 = K + f (x; t; u(x; t); u(x; t s)); t > 0; 0 x
L; with the following initial and boundary conditions u(x; t) = r~(x; t); 0 x
L; t 2 [ s; 0]; u(0; t) = 0(t); u(L; t) = L(t); t > 0; = ~(x) = lim t! 0 ; where s > 0 is the delay parameter, K is a positive constant. The fractional derivative of order 1 < 2 is defined in Caputo sense.
In order to transform (1) to a system with zero Dirichlet boundary conditions, we define h(x; t) := 0(t) + Lx ( L(t) 0(t)) and introduce the new function v(x; t) = u(x; t) h(x; t). Hence, we have = K + f (x; t; v(x; t); v(x; t s)); t > 0; 0 x
L; with the following initial and boundary conditions v(x; t) = r(x; t); 0 x
L; t 2 [ s; 0]; v(0; t) = v(L; t) = 0; t > 0: = (x) = lim t! 0 ;
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; ; v) and the solution u(x; t) of (2) are sufficiently smooth in the following sense: Let m be an integer satisfying ms T < (m + 1)s, define Ir = (rs; (r + 1)s), for r = Im = (ms; T ), I = Sqm= 1 Iq and assume that u(x; t) 2 C(6;3) ([0; L] [0; T ]), 1; 0; : : : ; m 1, The partial derivatives f (x; t; ; v) and fv(x; t; ; v) are continuous in the 0-neighborhood of the solution. Define c1 = c2 =
sup 0<x<L; 0<t T j 1j 0;j 2j 0
max 0<x<L; 0<t T j 1j 0;j 2j 0 jf (x; t; u(x; t) + 1; u(x; t s) + 2)j ; jfv(x; t; u(x; t) + 1; u(x; t s) + 2)j : The structure of this paper is arranged as: a derivation of the linear difference scheme is done in the following section. Next, in the third section, the solvability, convergence and stability for the difference scheme are carried out. In the fourth section, numerical examples are given to illustrate the accuracy of the presented scheme and to support our theoretical results. 2
Construction of the difference scheme A numerical solution based on the Crank-Nicholson method is derived. Before we continue, some further notations are fixed. Take two positive integers M and n, let h = ML ; = ns and denote xi = i h for i = 0; : : : ; M ; tk = k and tk 1=2 = k 12 = 12 (tk +tk 1), for k = n; : : : ; N , where N = T . Using the points xi in space and tk in time we cover the space-time domain by h = h , where h = fxi j 0 i M g and = ftk j n k N g. (1a) (1b) (1c) (2a) (2b) (2c) (3a) (3b) Let W = w : h ! R j w(xi; tk) = Wik; i = 0; 1; : : : ; M ; k = h . For w 2 W, we define Wik 1=2 := 12 Wik + Wik 1 .
Lemma 1. Let q(x) 2 C6([xi 1; xi+1]), then n; n + 1; : : : ; N be a grid function space on where !i 2 (xi 1; xi+1):[25]
In [20], an approximation for the time Caputo fractional derivative at tk 1=2 with 1 <
h12 (q(xi 1) 2q(xi) + q(xi+1)) = 2h440 q(
1 = 0 bk j) tVi < 2 was given: 1 where (x) is defined in (2b), jrkj
bk = 2 (3 1 ) 2 2 (k + 1)2 k2 ; = (2
); and for any function v : [0; L] [ s; +1) ! R one denotes v(xi; tj) = Vij for i 2 N, j 2 Z and defines xVik 1=2 = h1 Vik Vik 1 ; x2Vik = h12 Vik+1 2Vik + Vik 1 : (4e) We are now in a position to apply and combine the above, that is (4), to (2a) at the points (xi; tk 1=2); and arrive at 2 0
1 4 bk j 1 bk j tVi such that i = 0; : : : ; M; k = 1; : : : ; N:
Lemma 2. For g = (g0; g1; : : : ; gM ); let the linear operator A be defined as
1 Agi = 12 (gi 1 + 10gi + gi+1); 1 i
M
1:
s)); (
1 A 4 where k 1 X j=1
13
= K x2Vik 1=2 + Af xi; tk 1=2; 32 Vik 1 + 1 V k 2; 1 V k n 1 + 1 V k n
2 i 2 i 2 i
+ Rik 1=2; (
C 3 + h4 ; 1 i
M 1; 1 k
N:
(
Proof. By using the following Taylor expansions
in (
1 4
2 ;
s) = 12 Vik n + 21 Vik n 1 + O
so applying A to (
1 + where we used the continuity of the derivatives of f in its third and fourth component when letting According to Lemma 1, we have ! 0.
h4 @6v = x2Vik + 240 @x6 ik; tk ; k i 2 (xi 1; xi+1); 1
3 = K2 x2 Vik + Vik 1 + Af xi; tk 1=2; 23 Vik 1 1 V k 2; 1 V k n + 1 V k n 1 ; 2 i 2 i 2 i as v(x; t) 2 C(6;3)([0; L] [0; T ]). Define Rik 1=2 = Ark + O h4 + O achieved and the proof is complete.
The final form of our difference scheme is obtained by neglecting Rik 1=2 and replacing Vik with vik in (
1
N; and supplying appropriate initial and boundary conditions = K x2vik 1=2 + Af xi; tk 1=2; 32 vik 1
21 vik 2; 21 vik n 1 + 12 vik n ; (9a) v0k = 0(tk); vMk = L(tk); 1 vik = r(xi; tk); 0 i
M;
k n k
N; 0: (9b) (9c) Recall that v(xi; tk) = Vik and vik is the solution of the difference scheme, hoping to have v(xi; tk) vik, we discuss in the next section ik := jVik vikj.
We now prove that our difference scheme admits a unique solution. Next, we show that the obtained solution solves (2).
Theorem 1. (Solvability). The difference scheme (
Avk = 'k(vk 1; vk 1; : : : ; v n): Now, we introduce the uniqueness, stability and convergence theorems in L1-norm using the discrete energy method for the proposed difference scheme.
The spatial domain [0; L] is covered by h = fxi j 0 i M; g and let Vh = fv j v = (v0; : : : ; vM ); v0 = vM = 0g be a grid function space on h. For any u; v 2 Vh; define the discrete inner products and corresponding norms as and denote According to [25], the following inequalities are fulfilled
M 1 M hu; vi = h X uivi; h xu; xvi = h X( xui 1=2)( xvi 1=2);
i=1 i=1 h x2u; vi = h xu; xvi; xui = (ui
ui 1); kuk = phu; ui; juj1 = ph xu; xui; kuk1 = max juj;
0 i M 1 h uv M jvj1 = tuh X( xvi)2; i=1
uv M 1 k xvk1 = tuh X ( x2vi)2: 2
i=1 kuk1 pL 2 juj1; kuk
L p6 juj1: For the analysis of the difference scheme, we will use the following lemmas:
Lemma 3. Let v 2 Vh; we have k x2vk h2 jvj1.
Lemma 4. [18] Let v 2 Vh and v0 = vM = 0: Then, we have kvk1 Lemma 5. [20] For any G = fG1; G2; G3; : : :g and , we obtain p 2L jvj1.
(
Gk
t1m l 2(2 l) m X G2k k=1
t2m l 2(2 l)
2: then if one has that
Lemma 6. Gronwall inequality [25]. Suppose that fF k j k 0g is a nonnegative sequence and satisfies F k+1 A + B Plk=1 F l, k 0, for some nonnegative constants A and B. Then F k+1 A exp(Bk ).
Theorem 2. (Convergence). Let v(x; t) 2 [0; L] [ s; T ]; be the solution of (2) such that v(xi; tk) = Vik and
vik (0 i M; n k N ) be the solution of the difference scheme (
C = r 3L 8 K exp = + f (x; t; u(x; t); u(x; t s)); t 2 (0; 1); 0 < x < ; (15a)
N; and supplied by appropriate initial and boundary conditions z0k = 0(tk); zMk = L(tk); 1 zik = r(xi; tk) + ik; 0 i
M; k
N; n k 0; where ik is the perturbation of (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 ik = zik vik, for 0 i M; n k N: Then there exist some arbitrary positive constants c4; c5; h0; 0 which fulfill
k k k1 c4p
k ; 0 k k k
N;
uv M 1 k kk = tuh X ( ik)2; i=1 conditioned by f (x; t; u(x; t); u(x; t s)) = sin(x) (t3 + 2t + 4) + with the following initial and boundary conditions (4) t3 ) u(x; t s) + sin(x)((t u(x; t) = (t3 + 2t + 4) sin(x); 0 x ; t 2 [ s; 0);
s > 0; u(0; t) = u( ; t) = 0; t 2 [0; 1]:
u(x; t) = (t3 + 2t + 4) sin(x): The exact solution of this problem is 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.
This work was supported by Government of the Russian Federation Resolution N 211 of March 16, 2013.
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