An improved heat conduction model with Riesz fractional Cattaneo–Christov flux

International Journal of Heat and Mass Transfer 103 (2016) 1191–1197

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Review

An improved heat conduction model with Riesz fractional Cattaneo– Christov flux Lin Liu a,b, Liancun Zheng b,⇑, Fawang Liu c, Xinxin Zhang a a

School of Energy and Environmental Engineering, University of Science and Technology Beijing, Beijing 100083, China School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China c School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld 4001, Australia b

a r t i c l e

i n f o

Article history: Received 26 March 2016 Received in revised form 31 May 2016 Accepted 24 July 2016 Available online 30 August 2016 Keywords: Heat conduction Cattaneo–Christov flux Fractional equation Numerical solution

a b s t r a c t An improved constitutive model is proposed in which the time space upper-convected derivative is used to characterize heat conduction phenomena. The space Riesz fractional Cattaneo–Christov model is the generalization of Fourier law which takes the effects of relaxation time, fractional parameter and convection velocity into account. Formulated governing equation possesses the coexisting characteristics of parabolic and hyperbolic. Solutions are obtained numerically by the shifted Grünwald formula for space fractional derivatives and the theoretical analyses are presented for special cases. Three interesting characteristics are found: (a) for spatial evolution, the distribution is symmetrical for u = 0 while asymmetrical for u – 0. (b) for temporal evolution, the distribution is oscillating decreasing for f – 0 but monotone decreasing for f = 0. (c) for fractional parameters evolution, the distribution is approximately linearly descending. Moreover, the influences of the involved parameters on the temperature distribution are also shown graphically and discussed in detail. Ó 2016 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3. 4.

5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical discretization method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solvability, stability and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Heat conduction is a widespread nature phenomenon occurring in many fields [1,2]. The classical heat conduction equation is based on the Fourier law [3]

q ¼ k grad T; ⇑ Corresponding author. E-mail address: [email protected] (L. Zheng). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.07.113 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

ð1Þ

1191 1192 1193 1193 1194 1194 1194 1195 1196 1197 1197 1197

where q and k are heat flux vector and thermal conductivity, respectively. The classical Fourier law has been highly successful in a very wide range of engineering applications. However, one unphysical property is that it issues an infinite velocity of propagation, i.e., any initial disturbance is felt instantly throughout the whole of the medium. In order to overcome this problem, Cattaneo [4] modified the Fourier law by taking the finite velocity of propagation into account:

1192

qþn

L. Liu et al. / International Journal of Heat and Mass Transfer 103 (2016) 1191–1197

@q ¼ k grad T; @t

ð2Þ

where the nonnegative constant n refers particularly to the relaxation time. This extension turns the diffusion equation from a parabolic equation to a hyperbolic one [5]. In recent years, many researchers devoted themselves to the study of heat conduction with Cattaneo model. Christov and Jordan [6] pointed out that the usual form of Maxwell–Cattaneo leads to a paradoxical result if the body is in motion, the governing equation cannot be reduced to a single transport equation in the multidimensional case. Qi and Guo [7] studied the initial-boundary value problem of generalized Cattaneo equation, the exact solution was obtained in series forms in terms of the H-function by utilizing the Laplace transform. Compte and Metzler [8] generalized the Cattaneo equation to study the anomalous transport, three possible generalizations were proposed which can be used to describe continuous time random walks, non-local transport theory, and delayed flux-force relation. More literatures related to Cattaneo models are given in Refs. [9– 12]. Recently, Christov [13] generalized the Cattaneo flux model by using Oldroyds’ upper-convected derivative, the modified Cattaneo–Christov flux model is given by:


 @q þ V  rq  q  rV þ ðr  VÞq ¼ k grad T; qþn @t

(FDE). The employed methodology was useful for analysis of the effects of drugs and cancers. Chen et al. [24] presented an investigation for a variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures, results showed that the proposed variable-order fractional derivative model agree significantly well with experimental data. Moreover, the fractional model was also verified by accurately predicting chloride concentration profiles in a period of 200 days. Different important physical phenomena: cooling of particles and signals, particle and wave traps, Maxwell’s Demon, etc. represented some domains where fractional kinetics proved to be valuable by Zaslavsky [25]. Literatures related to fractional model can be seen in Refs. [26–28]. Motivated by the above mentioned works, we extend the study of heat conduction equation with space fractional Cattaneo– Christov model. For simplicity, we consider the velocity as a constant and analyze the 1-D model. By introducing the space fractional derivative, the constitutive Eq. (3) is modified as:

" !#   @q @q 1 @ b1 T @ b1 T þu ¼ k  qþn  ; @t @x 2 cosðpb=2Þ @xb1 @ðxÞb1

where u is the convection velocity along the x direction. The parameter b (1 < b 6 2) is the order of space fractional derivative and the symbols

ð3Þ

where V is the velocity vector. The Cattaneo–Christov flux model was gotten a large of researcher’s attention. Straughan [14] studied the thermal convection in a horizontal layer of incompressible Newtonian fluid with gravity acting downward, results showed that the thermal relaxation effect is significant for a sufficiently large Cattaneo number and the convection mechanism switches from stationary convection to oscillatory convection with narrower cells. Tibullo and Zampoli [15] investigated the behavior of the Cattaneo– Christov equation when applied to incompressible fluids, the uniqueness of solutions for boundary-initial value problems involving an incompressible fluid was established. Using Cattaneo– Christov constitutive equation, Han et al. [16] studied coupled flow and heat transfer of viscoelastic fluid over a stretching plate by employing the Cattaneo–Christov heat flux model. Haddad [17] considered the thermal instability in a Brinkman porous media incorporating fluid inertia with the Cattaneo–Christov theory. Hayat et al. [18] investigated steady two-dimensional magnetohydrodynamic flow and heat transfer of Oldroyd-B fluid over a stretching surface with homogeneous-heterogeneous reactions based on the Cattaneo–Christov heat flux. Owing to the nonlocal nature of fractional operators [19,20], the factional differential equations have attracted considerable attention in many fields of engineering, science and technology. For example, in the studying of heat conduction processes, the transports take place in a highly non-homogeneous medium, the traditional second-order Fokker–Planck equation is not adequate. The non-homogeneities of the medium may alter the traditional Fourier law. On the basis of this idea, many modified constitutive relations are proposed. Taloni et al. [21] presented a generalized elastic model, the fractional Langevin equation was derived for a particle transport in such a physical system which was published in physical review letters. Their results indicated that the fractional Langevin equation was the only one fulfilling the fluctuation dissipation relation within a new family of fractional Brownian motion equations. Du et al. [22] showed that the fractional model perfectly fitted the test data of memory phenomena in different disciplines, moreover, he found that a physical meaning of the fractional order was an index of memory. In a very recently paper, Namazi et al. [23] worked on modelling and prediction of the effects of chemotherapy on cancer cells using fractional diffusion Equation

ð4Þ

@ b1 @xb1

and

@ b1 @ðxÞb1

are the left and right Riemann–Liouville

fractional derivative [29,30] are defined, respectively, as

@ b1 Tðx; tÞ 1 @ ¼ @xb1 Cð2  bÞ @x @ b1 Tðx; tÞ @ðxÞ

b1

¼

1 @ Cð2  bÞ @x

Z

x

Z

1 ðx  nÞb1

0 1

1 ðn  xÞb1

x

Tðn; tÞdn;

ð5Þ

Tðn; tÞdn;

ð6Þ

The symbol CðÞ refers to the Euler gamma function. For parameters b ¼ 2 and u ¼ 0, the Cattaneo model (2) is recovered while the classical diffusion model of Fourier’s law (1) is recovered for parameters b ¼ 2; n ¼ 0 and u ¼ 0. 2. Mathematical formulation By the combination of constitutive Eq. (4) and the following conservation equation of energy

cq

@T @T þ c qu þ div q ¼ 0; @t @x

ð7Þ

we arrive at the space Riesz fractional Cattaneo–Christov equation

n

2 @2T @ 2 T @T @T @bT 2@ T þ þ u þ nu þ 2nu  a ¼ 0; @x@t @t @x @x2 @t 2 @jxjb

ð8Þ

where c; q and a ¼ k=ðcqÞ is the specific heat capacity, mass density and thermal diffusivity coefficient, respectively. The symbol

@b @jxjb

is

the space Riesz fractional derivative [31]:

@bT b

@jxj

¼ C b

! @bT @bT ; þ @xb @ðxÞb

Cb ¼

1 : 2 cosðpb=2Þ

ð9Þ

The initial conditions and boundary conditions are given as:

Tðx; 0Þ ¼ T 0 sin

px @Tðx; 0Þ l

;

@t

¼0

ð10Þ

and

Tð0; tÞ ¼ Tðl; tÞ ¼ 0;

ð11Þ

respectively. Introducing the non-dimensional quantities as

T ! T0T; t !

b

b

l  l  1b t ; x ! lx ; n ! n ; u ! l au ; a a

ð12Þ

1193

L. Liu et al. / International Journal of Heat and Mass Transfer 103 (2016) 1191–1197

we obtain the dimensionless governing equation and the initial and boundary conditions (here we omit the superscript ⁄ for simplicity), by:

@2T @ 2 T @T @T @2T @bT þ þu þ nu2 2  n 2 þ 2nu ¼ 0; @x@t @t @x @x @t @jxjb Tðx; 0Þ ¼ sin px;

@Tðx; 0Þ ¼ 0; @t

Tð0; tÞ ¼ Tð1; tÞ ¼ 0:

Basing on the definitions 1–5, we can obtain the final discrete scheme of (13) as: Cb

ð13Þ

hb

m X

xbliþ1 T lj þ

l¼iþ2



nu2 h2

 C C j þ hbb xb0 þ hbb xb2 T iþ1

  2C b j b 2nu2 1 u þ sn2 þ 2nu sh þ s þ h  h2 þ hb x1 T i

ð14Þ

i2  2  X Cb Cb Cb j b b 2nu u þ nu xbilþ1 T lj 2  sh  h þ b x0 þ bþ1 x2 T i1 þ b h h h h

ð15Þ

 j1 2nu j1 n j2 2nu 1 ¼ 2n s2 þ sh þ s T i  sh T i1  s2 T i

:

ð25Þ

l¼0

Defining a new matrix G1 with the coefficients of (25): 3. Numerical discretization method Prior to getting the numerical solution of Eq. (13), some useful definitions of the difference scheme are presented. We define xi ¼ ih (i ¼ 0; 1; 2; . . . ; m) and tj ¼ js (j ¼ 0; 1; 2; . . . ; n) where h and s are the grid size in space and time, respectively. Definition 1. The backward difference scheme is used for approximating the first order space derivative:

@Pðxi ; tk Þ Pðxi ; t k Þ  Pðxi1 ; tk Þ  rh Pðxi ; tk Þ ¼ þ OðhÞ: @x h

ð17Þ

ð18Þ

Definition 4. The second order time derivative is approximated:

@Pðxi ; tk Þ Pðxi ; t k Þ  2Pðxi ; t k1 Þ þ Pðxi ; t k2 Þ þ OðsÞ: ¼ @t s2

ð19Þ

@ b Pðxi ; tÞ b

@ðxÞ



xbl Pðxilþ1 ; tÞ þ OðhÞ;

b

h

ð20Þ

l¼0

X 1 miþ1

xbl Pðxiþl1 ; tÞ þ OðhÞ;

ð21Þ

l¼0

where x are the Grünwald weight coefficients and they are defined by:

  bþ1 xb0 ¼ 1; xbl ¼ 1  xbl1 l

ð22Þ

The weight coefficients xbl satisfy [33,34]:

ðiÞ xb1 ¼ b; ;xbl > 0 for 1 X

m X

l¼0

l¼0

xbl ¼ 0;

l ¼ i  1; l 6 i  2;

0 2n B B B B B B¼B B B B @

s2

1

1 þ 2nu sh þ s

 2nu hs

2n

s2

1 þ 2nu sh þ s

 2nu hs

.. ..

. 2nu 1 . 2n s2 þ sh þ s 2nu 2n 2nu 1  hs s2 þ sh þ s

C C C C C C C C C A

;

MM

we can obtain a simplified form of Eq. (25), given by:

n

s2

T j2 :

ð28Þ

The initial conditions and boundary conditions can be discretized as:

Tðx; 0Þ ¼ sin pxi ;

@Tðx; 0Þ ¼ 0; @t

ð29Þ

and

ð30Þ

4. Solvability, stability and convergence For the theoretical analyses, the special form of Eq. (13), i.e. n ¼ 0 is considered for simplicity. Then the simplified forms of the discrete scheme (25) and the matrix G1 are given as follows:

b l

ðiiÞ

ð26Þ

l ¼ i;

Tð0; tÞ ¼ Tð1; tÞ ¼ 0:

Definition 5. The shifted Grünwald formulae [32] are used for approximating the space fractional derivative of order 1 < b 6 2:

@ Pðxi ; tÞ 1  b @xb h

l ¼ i þ 1;

and a new matrix B:

G1 T j ¼ BT j1 

@Pðxi ; tk Þ Pðxi ; t k Þ  Pðxi ; tk1 Þ þ OðsÞ: ¼ @t s

iþ1 X

xilþ1

l P i þ 2;

ð27Þ

Definition 3. The backward difference scheme is used for approximating the first order time derivative:

b

hb

ð16Þ

Definition 2. The central difference scheme for the second order space derivative is approximated as:

@ 2 Pðxi ; t k Þ Pðxiþ1 ; t k Þ  2Pðxi ; t k Þ þ Pðxi1 ; tk Þ 2 ¼ þ Oðh Þ: 2 @x2 h

8 Cb b x > > > hb liþ1 > > 2 Cb Cb b b nu > > > h2 þ hb x0 þ hb x2 > < 2C b b 2nu2 1 u G1il ¼ sn2 þ 2nu sh þ s þ h  h2 þ hb x1 > > > nu2 2nu u C b b Cb b > >  sh  h þ hb x0 þ hb x2 > > h2 > > : Cb b

l ¼ 0; 2; 3; . . . ;

xbl < 0 for any

m P 1:

ð23Þ ð24Þ

1

s

T ij ¼

" # iþ1 X b j u j 1X 1 miþ1 j b j þ ðT i  T i1 Þ þ C b b xl T ilþ1 þ b xl T iþl1 h h l¼0 h l¼0 1

s

T j1 i ;

ð31Þ

and

8 Cb b > x > > hb liþ1 > > Cb Cb b b > > > > hb x0 þ hb x2 < 2C G1il ¼ 1s þ uh þ hbb xb1 > > > >  u þ C b xb þ Cb xb > > 0 2 h hb hb > > > Cb b : hb

xilþ1

l P i þ 2; l ¼ i þ 1; l ¼ i; l ¼ i  1; l 6 i  2;

:

ð32Þ

1194

L. Liu et al. / International Journal of Heat and Mass Transfer 103 (2016) 1191–1197

Using the inequalities (34) and (35), we have:

4.1. Solvability The discrete scheme (31) is uniquely solvable if the matrix G1 is strictly diagonally dominant, i.e. the coefficients G1il satisfy: m1 X

G1il < jG1ii j;

i ¼ 1; 2; :::; m  1;

i0 þ1







m





X X

j j u j

j

j u j

Cb Cb xbþ1 xbþ1

ei0 6 ei0 þ h ei0 þ hbþ1 i0 lþ1 ei0 þ hbþ1 li0 þ1 ei0  h ei0

ð33Þ

l¼1;l–i C

b þ hbþ1

First, we state the following inequalities: m1 X

Cb h

l¼iþ2

Cb h

xbliþ1 < 0;

b

xb0 þ

b

Cb h

b

i2 X

Cb h

xbilþ1 < 0;

b

ð34Þ

l¼0

u h

xb2 < 0;  þ

Cb h

xb0 þ

b

Cb b

h

xb2 < 0:

C

m

X

1

Gil

l¼1;l–i











m i2

Cb X b Cb X b C b b C b b

¼ hb xliþ1 þ hb xilþ1 þ hb x0 þ hb x2





l¼iþ2 l¼0



u Cb b Cb b

þ  h þ hb x0 þ hb x2

C

m X

i2 X

xbliþ1  Chbb

l¼iþ2

l¼0

xbilþ1  Chbb xb0  Chbb xb2

þ uh  hbb xb0  hbb xb2 C

C

¼  hbb

miþ1 X

6 1s þ uh þ ¼

C

l¼0 2C b hb

xbl  Chbb

xb1

jG1ii j

m X







l¼0 l–i0



j u j

xbþ1 li0 þ1 ei0  h ei0

l ¼ i0  1 l–i0

i0 þ1



 

X



j

Cb Cb Cb xbþ1 þ hbþ1 xbþ1 þ uh eij0 þ hbþ1 xbþ1 6 1 þ hbþ1 1 1 i0 lþ1 el

ð35Þ

For a given i, the sum of the coefficients G1il (l ¼ 1; 2; :::; m  1 and l–i) satisfies:

¼  hbb

l¼i0 1

l¼0

i0 þ1



 

X



j

Cb Cb Cb xbþ1 þ hbþ1 xbþ1 þ uh eij0 þ hbþ1 xbþ1 ¼ 1 þ hbþ1 1 1 i0 lþ1 ei0

b þ hbþ1

m X l¼i0 1 l–i0

l¼0 l–i0





j su j

xbþ1 li0 þ1 el  h ei0 1

:



i0 þ1

  X

Cb Cb Cb j 6 1 þ hbþ1 xbþ1 þ hbþ1 xbþ1 þ uh eij0 þ hbþ1 xbþ1 1 1 i0 lþ1 el

l¼0

l–i0



m

X Cb j su j

xbþ1 þ hbþ1 li0 þ1 el  h ei0 1

l¼i0 1

l–i0



i0 þ1 m



X X

Cb Cb j j

bþ1 xbþ1 e þ x e ¼ eij0 þ uh eij0  uh eij0 1 þ hbþ1 bþ1 i0 lþ1 l li0 þ1 l

h

l¼0 l¼i0 1



j1 0

¼ ei0 6 ei0

ð39Þ

iþ1 X l¼0

xbl þ 2Chbb xb1 þ uh

j

0

The inequality kE k1 6 kE k1 holds for j ¼ 1; 2; . . . ; n.

:

ð36Þ

4.3. Convergence

Thus, the proof is completed.

Prior to proving the convergence of the numerical method of Eq. (31), we define Pðxi ; t j Þ (i ¼ 1; 2; . . . ; m  1; j ¼ 0; 1; 2; . . . ; n) as the exact solution at the mesh point ðxi ; tj Þ. By defining

4.2. Stability

qij ¼ Pðxi ; tj Þ  Pij , yields the following equation:

Prior to proving the stability of the numerical method of Eq. (31), we first define T ij (i ¼ 1; 2; . . . ; m  1; j ¼ 0; 1; 2; . . . ; n) as the

approximate solution and e ¼  as the roundoff error at the mesh point ðxi ; t j Þ, then the roundoff error equation is given as follows: j i

1

s

"

u h

j eij þ ðeij  ei1 Þ þ Cb

T ij

T ij

iþ1 1X

h

j xbl eilþ1 þ

b l¼0

X 1 miþ1 b

h

l¼0

#

qij þ

su h

j ðqij  qi1 Þs

h

iþ1 m c X 1c X j j j1 xbþ1 xbþ1 þ Rij ; ilþ1 ql  s bþ1 liþ1 ql ¼ qi

bþ1

l¼0

h

l¼i1

ð40Þ

1

j ¼ ej1 xbl eiþl1 : s i

ð37Þ By defining Ek ¼ ðek1 ; ek2 ; . . . ; ekm1 Þ

T

and kE j k1 ¼ max16i6m1 jeij j ¼jeij0 j,

the numerical scheme (31) is unconditionally stable with respect to the initial date, and there exists the following inequality:

kE j k1 6 kE0 k1 ;

for

j ¼ 1; 2; . . . ; n:

ð38Þ

Table 1 The errors and convergence order of Eq. (43) versus grid size reduction at t ¼ 0:1. h 1/8 1/16 1/32 1/64

E2 ðh; sÞ 0.0016877 0.00044094 0.00011779 0.000036509

Order 1.9364 1.9044 1.6899 1.1741

Fig. 1. Distribution curves versus x for parameters n ¼ 0:1; u ¼ 0 and t ¼ 0:1 with different values of space fractional parameter b.

L. Liu et al. / International Journal of Heat and Mass Transfer 103 (2016) 1191–1197 T

j where q0i ¼ 0; R j ¼ ðR1j ;R2j ; . .. ; Rm1 Þ and Rij0 ¼ maxi¼1;2;...;m1;j¼1;2;...;n jRij j

6 C 1 sðh þ sÞ; C 1 is a constant. By

defining

j Y j ¼ ðq1j ; q2j ; . . . ; qm1 Þ

T

and

kY j k1 ¼

maxi¼1;2;...;m1 jqij j ¼ jqij0 j, the numerical scheme (31) is uncondition-

f ðx;tÞ ¼ 2tx2 ð1xÞ2 þuðt2 þ1Þ2xð1xÞð12xÞ 2 3 Cð3Þ 2a ð4Þ ð5Þ 3a 4a 2 CCð4 þ CCð5 ; Cð3aÞ x aÞ x aÞ x 2 4 5 þðt þ1ÞC b 2a 3a 4a Cð3Þ Cð4Þ Cð5Þ þ Cð3aÞ ð1xÞ 2 Cð4aÞ ð1xÞ þ Cð5aÞ ð1xÞ ð46Þ

ally convergent, and there exists a positive constant C  > 0 such that:

and the exact solution is given:

kY j k1 6 C ðh þ sÞ; for j ¼ 1; 2; . . . ; n:

Tðx; tÞ ¼ ðt2 þ 1Þx2 ð1  xÞ2 :



ð41Þ

Using the inequalities (34) and (35), we obtain:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u m1 u X 2 E2 ðh; sÞ ¼ th ½Pðxi ; tj Þ  Pij  ;

l¼0

þ hbþ1

m X l¼i0 1





j u j

xbþ1 li0 þ1 qi0  h qi0

and

i0 þ1





X



j

Cb Cb Cb ¼ 1 þ hbþ1 xbþ1 þ hbþ1 xbþ1 þ uh qij0 þ hbþ1 xbþ1 1 1 i0 lþ1 qi0

þ hbþ1

m X l¼i0 1 l–i0





j u j

xbþ1 li0 þ1 qi0  h qi0

ð48Þ

i¼1



Cb

ð47Þ

The L2 -norm error [35] and the convergence order are defined as:

i0 þ1







X

j j u j

j

Cb xbþ1

qi0 6 qi0 þ h qi0 þ hbþ1 i0 lþ1 qi0

Cb

1195

log2 ½E2 ðh; sÞ=E2 ðh=2; sÞ

l¼0 l–i0

ð49Þ

respectively.The errors and convergence order of Eq. (43) versus grid size reduction at t ¼ 0:1 are shown in Table 1. It can be seen that the discrete scheme is stable. Moreover, as Table 1 shows, the numerical solution is in good agreement with the exact solution

i0 þ1



 

X



j

Cb Cb Cb xbþ1 þ hbþ1 xbþ1 þ uh qij0 þ hbþ1 xbþ1 6 1 þ hbþ1 1 1 i0 lþ1 ql

C

b þ hbþ1

m X l¼i0 1 l–i0





j u j

xbþ1 q q 



i0 1

li0 þ1 l h

l¼0 l–i0

:



i0 þ1

  X

Cb Cb Cb j 6 1 þ hbþ1 xbþ1 þ hbþ1 xbþ1 þ uh qij0 þ hbþ1 xbþ1 1 1 i0 lþ1 ql

l¼0

l–i0



m X

Cb j u j þ hbþ1 xbþ1 q  q li0 þ1 l h i0 1

l¼i0 1

l–i0



i0 þ1



m X X

j u j u j Cb Cb j j

bþ1 bþ1 ¼ qi0 þ h qi0  h qi0 1 þ hbþ1 xi0 lþ1 ql þ hbþ1 xli0 þ1 ql



l¼0 l¼i 1 0

j



X



j 0

l

¼ qj1

Ri0

i0 þ Ri0 6 qi0 þ

Fig. 2. Distribution curves versus t for parameters n ¼ 0:1; u ¼ 0 and x ¼ 0:5 with different values of space fractional parameter b.

l¼1

6 C 1 J sðh þ sÞ ð42Þ





The inequality qij0 6 C  ðh þ sÞ holds for j ¼ 1; 2; . . . ; n where the constant C  ¼ C 1 Js. 5. Numerical example In order to confirm that our numerical method recovers the correct solution behavior, a source term is introduced to construct an exact solution. Then we have the following equation:

@T @T @ b T þu  ¼ f ðx; tÞ; @t @x @jxjb

ð43Þ

subject to the initial and boundary conditions:

Tðx; 0Þ ¼ x2 ð1  xÞ2 ;

@Tðx; 0Þ ¼0 @t

ð44Þ

and

Tð0; tÞ ¼ Tð1; tÞ ¼ 0; respectively, where

ð45Þ

Fig. 3. Distribution curves versus x for parameters b ¼ 2; u ¼ 0 and t ¼ 0:1 with different values of parameter n.

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L. Liu et al. / International Journal of Heat and Mass Transfer 103 (2016) 1191–1197

Fig. 4. Distribution curves versus t for parameters b ¼ 2; u ¼ 0 and x ¼ 0:5 with different values of parameter n.

Fig. 5. Distribution curves versus b for parameters n ¼ 0:1; t ¼ 0:1 and x ¼ 0:5 with different values of parameter n.

Fig. 6. Distribution curves versus x for parameters n ¼ 0:1; b ¼ 2 and t ¼ 0:1 with different values of velocity u.

Fig. 7. Distribution curves versus t for parameters n ¼ 0:1; b ¼ 2 and x ¼ 0:5 with different values of velocity u.

with the accuracy of Oðh þ sÞ which indicates the correctness of numerical results.

6. Results and discussion Figs. 1 and 2 show the influences of different values of space fractional parameter b on the spatial and temporal evolution of temperature for the fixed parameters n ¼ 0:1 and u ¼ 0, respectively. The parameter b is the nonlocality of heat conduction. Fig. 1 shows that the smaller the parameter b is, the larger the magnitude of temperature will be for a fixed position at parameter t ¼ 0:1. Fig. 2 shows that the temperature distributions oscillate and decay until reaching zero with the increase of time. Moreover, the amplitude of temperature becomes stronger, i.e., the hyperbolicity enhances for a larger parameter b. Figs. 3 and 4 show the influences of different values of relaxation parameter n on the spatial and temporal evolution of temperature, respectively. The parameter n has the effect of delay and the larger parameter n leads a stronger delay effect. Fig. 3 shows that the magnitude of temperature is larger for a larger parameter n at parameter t ¼ 0:1. As is known, Eq. (13) reduces to a fractional heat conduction equation which is deduced by the Fourier’s law

Fig. 8. Distribution curves versus b for parameters n ¼ 0:1; t ¼ 0:1 and x ¼ 0:5 with different values of velocity u.

for parameters n ¼ 0 and u ¼ 0. At this condition, it only has the parabolicity, i.e., the oscillating phenomenon does not occur. For parameter n–0, the temperature distributions oscillate and decay

L. Liu et al. / International Journal of Heat and Mass Transfer 103 (2016) 1191–1197

until reaching zero with the increase of time. Moreover, it can be seen from Fig. 4 the amplitude of the temperature (hyperbolicity) becomes stronger and the temperature decay becomes slower for a larger parameter n. Fig. 5 shows the fractional parameter evolution of the temperature at the influence of different values of relaxation parameter n. Here we only discuss the region of (1.6, 2) for simplicity. Fig. 5 shows that the magnitude of temperature is approximately linearly reduced with the growing of parameter b. For a fixed parameter b, the larger the parameter n is, the larger the magnitude of temperature will be. Figs. 6–8 show the distribution curves of the spatial, temporal and fractional parameter evolution of temperature with different values of parameter u when the relaxation parameter is fixed. Results indicate that the convection velocity can influence the symmetry of temperature distribution. The Cattaneo–Christov flux model reduces to the Cattaneo one for parameter u ¼ 0 and the temperature diffuses along the positive and negative directions of x axis symmetrically while the curves are not symmetrical for convection velocity u–0. In this paper, we only discuss the positive convection velocity, for the negative one, we can deduce by a similar analysis. The larger the parameter u is, the larger the magnitude of temperature distributions will be at parameter t ¼ 0:1. As Fig. 7 shows, the temperature distributions oscillate and decay until reaching zero with the increase of time. Moreover, the parabolicity enhances and the corresponding amplitude of temperature distributions becomes weaker for a larger parameter u. Fig. 8 shows that the value of temperature distribution is approximately linearly reduced with the growing of parameter b. For a fixed parameter b, the larger the parameter u is, the larger value of the solution will be. 7. Conclusions An improved space Riesz fractional Cattaneo–Christov constitutive model is first proposed to characterize heat conduction phenomena. The numerical discretization method is used to obtain the solution where the Riesz fractional derivatives are approximated by shifted Grünwald formulae. The theoretical analyses (solvability, stability and convergence) for the special case of the governing equation are presented. Moreover, the transport behaviors with different dynamic parameters effect are analyzed by graphical illustrations and discussed in detail. Further research about the heat conduction with various modified fractional constitutive relations needs to be studied and analyzed. Acknowledgement The work is supported by the National Natural Science Foundation of China (No. 51276014). References [1] T.H. Ning, X.Y. Jiang, Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation, Acta. Mech. Sin. 27 (2011) 994–1000. [2] X.Y. Jiang, M.Y. Xu, The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems, Phys. A 389 (2010) 3368–3374. [3] M.N. Özisik, Heat Conduction, third ed., John Wiley & Sons, New York, 1993.

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