Implicit finite difference approximation for time Fractional heat conduction under boundary Condition of second kind

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1 Internationa Journa of Appied Mathematica Research, 4 () (205) c Science Pubishing Corporation doi: 49/ijamr.v4i.3783 Research Paper Impicit finite difference approximation for time Fractiona heat conduction under boundary Condition of second kind T.N. Mishra, K.N. Rai 2 DST-CIMS, Facuty of Science, BHU, Varanasi 2 Mathematica Sciences, IIT BHU, Varanasi *Corresponding author E-mai: t.mishra0@gmai.com Copyright c 205 Mishra and Rai. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. Abstract The time fractiona heat conduction in an infinite pate of finite thickness, when both faces are subjected to boundary conditions of second kind, has been studied. The time fractiona heat conduction equation is used, when attempting to describe transport process with ong memory, where the rate of heat conduction is inconsistent with the cassica Brownian motion. The stabiity and convergence of this numerica scheme has been discussed and observed that the soution is unconditionay stabe. The whoe anaysis is presented in dimensioness form. A numerica exampe of particuar interest has been studied and discussed in detais. Keywords: Boundary condition of second kind; Caputo fractiona derivative; Impicit finite difference scheme; Time fractiona heat conduction; Unconditionay stabe.. Introduction The fractiona cacuus is a name for the theory of integra and derivatives of arbitrary orders, which unify and generaize the notions of integer-order differentiation and n-fod integration. In recent years, fractiona cacuus has been taken by Scientist and Engineers and appied in an increasing numbers of fieds, namey in the fied of materia and mechanics, signa processing, anomaous diffusion, bioogica system, finance, hydroogy and many others see [8],[2],[24],[27],[2],[],[]. Fractiona sub-diffusion equation is a cass of anomaous diffusive system, which is obtained from the cassica heat conduction equation by repacing the first orders time derivative to a fractiona derivative of order with 0 < < (in Riemann-Liouvie or Caputo sense). It is a known fact that the anomaous diffusion is characterized by a diffusion constant and the mean square dispacement of diffusing species in the form x 2 (t) t, t where, (0 < < ) is the anomaous diffusion exponent. Numerica approaches to different types of fractiona diffusion modes have been increasingy appearing in iterature. Meerchaert [3], considered the stochastic soution of space-time fractiona diffusion equation. Huang and Liu [6], considered the time fractiona diffusion equation in whoe space and aso in haf space. Yuste and Acedo [26], and Yuste [25], presented an expicit scheme and weighted average finite difference methods for the

2 36 Internationa Journa of Appied Mathematica Research fractiona diffusion equation and anayzed these two schemes stabiity by von Neumann method. Zhuang and Liu [30], approximated the time fractiona diffusion equation by impicit difference method. For the fractiona sub-diffusion equation Chen and Liu [3], constructed the difference scheme based on Grunwad-Letnikov formua and showed the stabiity and convergence of the difference scheme using the Fourier method. Zhuang et a. [3], introduced a new way for soving sub-diffusion equation by integration of the origina equation on the both sides to obtain an impicit numerica method. Stabiity and convergence of the scheme were proved by the energy method. Murio [4] constructed impicit finite difference approximation for time fractiona diffusion equations but in showing stabiity of the method by Fourier method he made some faw. Ding and Zhang [4] showed the stabiity of impicit finite difference approximation for time fractiona diffusion equations. Chen et a. [2] constructed finite difference method for the fractiona reaction-sub-diffusion equation. Zhang [28] considered the unconditionay stabe finite difference method for fractiona partia differentia equation. Singh et a. [23] soved the bioheat equations by finite difference method and homotopy perturbation method. The work mentioned above are deaing with Dirichet boundary conditions, where no boundary discretization errors are invoved. However, for Neumann boundary condition (i.e boundary condition of second kind), the discretization of boundary conditions must deat with carefuy to match goba accuracy. Langands and Henry [9] deveoped an impicit difference scheme with convergence order o(τ + h 2 ) based on L approximation for Riemann- Liouvie fractiona derivative and numericay verified the unconditiona stabiity of difference scheme but without goba convergence anaysis. Recenty, Zhao and Sun [29] proposed a Box-type scheme for soving a cass of fractiona sub-diffusion equation with homogeneous Neumann boundary conditions. Since many appication probems in science and engineering invove Neumann boundary conditions [9] [20],[5], such as zero fow or specified fow fux condition. Thus, it is very desirabe to use high-order agorithms for efficient computations of the numerica soution of this kind of probem. This motivates us to consider the impicit finite difference approximation for spatia discretization. The main purpose of this paper is to construct a mathematica mode for time fractiona heat conduction in an infinite pate and discuss the anomaous diffusion, sub-diffusion and aso discuss the effect of imiting vaue of time fractiona constant by taking a particuar exampe. The rest of the paper is organized as foows. Section deas fundamenta of fractiona cacuus, time fractiona heat conduction equation and a mathematica mode of a boundary vaue probem governing the process of time fractiona heat conduction in an infinite pate of finite thickness whose both faces are subjected to boundary conditions of second kind. An impicit finite difference scheme is given in section 2. Stabiity and convergence of the scheme is given in section 3. Section 4 contains numerica computation and discussion. The concusion and future research pan is given in section 5... Fundamenta of fractiona cacuus The Grunwad Letnikov fractiona derivative [8] a D t g(t), of order > 0, of a function g(t) is defined as ad t g(t) = m k=0 g k (a)(t a) ( +k) Γ( + k + ) + Γ( + k + ) t a (t τ) m g m+ (τ)dτ () This definition of fractiona derivative is defined under the assumption that functions are m + - times continuousy differentiabe in the cosed interva [ a, t ] where, m is an integer number satisfying the condition m >. But such type of functions are very narrow. Let [a, t] be a finite interva on the rea axis R. The Riemann-Liouvie (R-L) fractiona derivative [8] RL a Dt g(t), of order > 0, of a function g(t) is defined by RL a Dt g(t) = dn dt n [ Γ(n ) ] t a (t τ) n g(τ)dτ, (n < n) (2) The Riemann-Liouvie approach have payed an important roe in the deveopment of theory of fractiona cacuus due to their appications in pure mathematics. However, R-L approach ead to initia conditions containing the imit vaues at the ower termina t = a as im t a RL a D j t g(t) = c j, j =, 2,...n. where, c j are given constants. The above initia vaue probems with such initia conditions can be successfuy soved mathematicay but their soutions have no known physica interpretation due to which they are useess.

3 Internationa Journa of Appied Mathematica Research 37 The Caputo fractiona derivative [8] C a D t g(t), of order, of a function g(t) is defined as C a Dt g(t) = Γ( n) t a (t τ) n g n (τ)dτ, n < n (3) Under natura conditions on the function g(t), for n the Caputo derivative becomes a conventiona n-th derivative of the function g(t). The main advantage of Caputo s approach is that the initia conditions for fractiona differentia equations with Caputo derivatives takes on the same form as for integer-order differentia equations, i.e. contain the imit vaues of integer-order derivatives of unknown functions at the ower termina t = a, due to this in present study we take Caputo fractiona derivative..2. Time fractiona heat conduction equation The cassica theory of heat conduction is based on Fourier aw, q = T, reating to the heat fux vector ( q) to the temperature gradient ( T ), where, therma conductivity K is a geometry-independent coefficient and mainy depends on the composition and structure of the materia and the temperature. In many one dimensiona systems with tota momentum conservation, the heat conduction equation does not obey the Fourier aw and heat conductivity depends on the system size [8]. Such size dependent therma conductivity has been observed in theoretica modes, such as the harmonic chains [22], the FPU-β mode [0] and the hard point gas mode [5]. The heat conduction equation is a combination of conservation equation of energy and conduction aw. The energy conservation is expressed ocay by q(t) = ρc b T t (4) where, c b, is the specific heat capacity at constant pressure and ρ, is the density. The theory of heat conduction proposed by Norwood [6], the corresponding generaization of Fourier aw, t q(t) = K 0 a(t τ) T (τ)dτ (5) where, a(t τ) is the kerne. The time non oca dependence between the heat fux vector and the temperature gradient with ong tae power kerne [9], q(t) = K Γ() t 0 T (τ) (t τ) dτ, 0 < (6) t Considering the definition of Caputo fractiona derivative for 0 < c 0D t g(t) = t Γ( ) 0 Now equation (6) becomes, g (τ) dτ, 0 < (7) (t τ) q(t) = K( c 0D t T ) (8) In uni-dimensiona case, the gradient of temperature in orthogona curviinear coordinate system is given by [7] T = x h T x (9) here,the coefficients h is caed the scae factors and can be evauated by, h 2 =

4 38 Internationa Journa of Appied Mathematica Research The heat fux vector q becomes, q = K( c 0Dt x T h x ) (0) Therefore divergence of heat fux vector q in orthogona curviinear coordinate system is given by q = [ h x K(c 0Dt T h x ) () Then, the fractiona heat conduction equation in orthogona curviinear coordinate system is given by substituting the resut of Eq. () into Eq. (4) i.e. [ h x K(c 0Dt T h x )] = ρc T b t.3. Mathematica mode of heat conduction in an infinite pate of finite thickness The transport process in an infinite pate of finite thickness 2R, with the ong memory where, the rate of heat conduction in the body is inconsistent with the cassica Brownian motion mode, is considered. At initia instant, the temperature of the body is a function of space coordinate and the faces are subjected to boundary condition of second kind. The mathematica mode describing this anomaous heat conduction in presence of interna heat source, in dimensioness form can be written as: θ F 0 = 2 θ x 2 + P 0(x, ) (2) ( ) j θ x = k i j ( ), x = ( ) j, j =, 2 (3) θ(x, 0) = f(x), > 0 (4) Nomencature, dimensiona variabe and simiarity criteria a q therma diffusivity (m 2 s ) f(x) dimensioness initia temperature b time dependent coefficient (s K) Fourier number, aqt R 2 q(τ)r c b specific heat capacity (J/kg K) k i ( ) Kirpichev number, K(T c T 0) adt Grunwad Letnikove fractiona derivative k i ( ) dimensioness heat fux at boundary x=- RL a Dt Riemann-Liouvie fractiona derivative k i2 ( ) dimensioness heat fux at boundary x= C a Dt Caputo fractiona derivative P 0 Pomerantsev number, Q(x,τ)R 2 K(T c T 0) b R 2 a q T g(t) a continuous differentiabe function of t P d Predvoditeev number, r h scae factor x dimensioness spatia coordinate, R T T K therma conductivity (W/m K) θ dimensioness temperature, 0 T c T 0 k tempora grid size q(t) heat fux (W/ m 2 ) Q heat source (J) 2R thickness of the infinite pate (m) T temperature (K) T 0 initia temperature (K) T c ambient temperature (K) T temperature gradient (K/m) x spatia grid size (= h) order of fractiona derivative in (n-, n] ρ density (kg m 3 ) a, t ower and upper terminas of fractiona order 2. Impicit finite difference scheme To estabish the numerica approximation scheme, et x = h = 2 X > 0 and k = M be the grid size in space and time direction respectivey. The grid points in the space interva [, ] are the numbers x i = ih, i =, 2, 3...p and grid points in the time interva [0, ] are abeed n = nk, n =, 2,.., m. The vaues of the functions θ and f

5 Internationa Journa of Appied Mathematica Research 39 at grid points are denoted by θi n = θ(x i, n ) and f i = f(x i ), respectivey. In the differentia Eq. (2), we have adopted a symmetric second difference quotient in space at eve = n+ for approximating the second order space derivative. The time fractiona derivative term can be approximated by the foowing scheme: θ(x i, n+ ) F 0 = Γ( ) F 0n+ 0 θ(x i, s) (n+ s) ds = Γ( ) n j=0 (j+)k jk [ θj+ i θ j i ] {(n + )k s} ds k = Γ( ) ( ) n j=0 ([ θj+ i θ j i ][(n j + ) (n j) ])k k = Γ( ) ( ) By taking σ,k = L h,kθ(x i,f0n+ ) = σ,k We have, θ(x i, n+ ) F 0 i θ j i ) + Γ( ) ( ) k (θj+ Γ( ) ( ) n j=0 n k (θ j+ i θ j i )[(n j + ) (n j) ] k and shifting indices w j = (j) (j ), and define w j (θ n+ j i L h,kθ(x i,f0n+ ) θ n j i ) ( ) n F 0j+ j=0 j θ(x i, ξ) ξ θ(x i, j+ ) θ(x i, j ) k dξ (n+ ξ) C n Γ( ) k F 0j+ j=0 j dξ (n+ ξ) C Γ( ) k F 0j+ 0 dξ (n+ ξ) θ(x i, n+ ) F 0 L h,kθ(x i,f0n+ ) C k (5) where, C and C are constants. On using finite centra difference formua and approximating derivatives at x 0 and x n, the differentia Eqs. (2)-(4), reduces in the vector-matrix form as foows: AΘ = Θ 0 + P (6)

6 40 Internationa Journa of Appied Mathematica Research AΘ n+ = c Θn + c 2 Θn c n Θ + d n+ Θ0 + P n, n (7) Θ 0 = f (8) where, Θ n = Θ n (σ,k ) Θ n = ( ) θ n θ2 n θ3 n.... θp 2 n θp n t a 38r 9r r b r r b r... 0 A = 0... r b r r b r r 38r a (p ) (p ) P n = ( P n + 20hk i, r P n 2 P n P n p 3 P n p 2 P n p + 20hk i,2 r ) t f = ( f(x ) f(x 2 ) f(x 3 ).... f(x p 3 ) f(x p 2 ) f(x p ) ) t c j = (wj w j+ ), d n = wn, a = (σ,k + 29r ), b = (σ,k + 2r), r = 2h, r = 2 h 2 matrix X. and X t be the transpose of From Eqs. (6)-(8) we observe that at each time step, we get a trianguar system of inear equations which utiize a the history of the computed soution up to that time. In present anaysis, Mathematica software MATLAB 7.0 has been used to obtain dimensioness temperature Θ n. 3. Stabiity and convergence In this section we wi anayze the stabiity and convergence of the impicit finite difference scheme for time fractiona heat conduction equation. From Eqs. (6)-(8), and by using the property of the function g(x) = x, ( x) and = w > w 2 > w 3 > w 4 >... 0 we have the foowing, n c j = wn+ c j =, 2 > = c > c 2 > c Matrix A is stricty diagonay dominant with positive diagona term and non positive off-diagona terms. Hence the Eqs. (6)-(8) can be soved. The soution θ n i (i =, 2, 3...p ; n =, 2, 3...m) possess non negativity, if θn 0 is non negative. Soution θ j i is conservative, i.e i= θ 0 i < = i= θ j i = i= θ 0 i, j ℵ

7 Internationa Journa of Appied Mathematica Research 4 Now we suppose θ j i is the approximate soution of Eqs. (6)-(8), then the error ɛj i = θ j i θj i satisfies, AE = E 0 AE n+ = c Ē n + c 2 Ē n + c 3 Ē n c n Ē + d n+ Ē 0, n where, Ē n = E n (σ,k ) and E n = ( ɛ n ɛ n 2 ɛ n ɛ n p 2 ɛ n p ) t. Hence, by the mathematica induction foowing resut can be proved. Theorem 3. E n E 0 Proof : For n=, rɛ i + (σ,k + 2r)ɛ i rɛ i+ = σ,k ɛ 0 i Let ɛ = max }{{} ɛ i then we have i p ɛ r ɛ + (σ,k + 2r) ɛ r ɛ + rɛ + (σ,k + 2r)ɛ rɛ + = σ,k ɛ 0 E E 0 Therefore, E E 0. Suppose that E j E 0, j=,2,3...n. Let ɛ n+ = max }{{} ɛ n+ i, then we aso have, i p ɛ n+ r ɛ n+ + (σ,k + 2r) ɛ n+ r ɛ n+ + rɛ n+ + (σ,k + 2r)ɛ n+ rɛ n+ + n = c ɛ n + c j+ ɛ n j + d n+ ɛ 0 n c ɛ n + c j+ ɛ n j + d n+ ɛ 0 n E n+ c E n + c j+ E n j + wn+ E 0 n c + c j+ + wn+ E0 = E 0 E n+ E 0 Hence, the foowing theorem is obtained. Theorem 3.2 The time fractiona impicit difference approximation defined by (6) and (8) are unconditionay stabe.

8 42 Internationa Journa of Appied Mathematica Research 3.. Convergence To discuss the convergence of impicit finite difference approximation for time fractiona heat conduction equation under non-homogeneous boundary conditions of second kind, et θ(x i, n ) be the exact soution of the time fractiona heat conduction Eqs. (2)-(4) at mesh point (x i, n ). Define e n i = θ(x i, n ) θi n, i =, 2, 3...p ; n =, 2, 3...m and e n = (σ,k e n, σ,k e n 2, σ,k e n 3...σ,k e n p ) t. Using e 0 = 0, and substitution into (6)-(8) eads to, re i + (σ,k + 2r)e i re i+ = R i n re n+ i + (σ,k + 2r)e n+ i re n+ i+ = c e n i + c j+ e n j i + R n+ i where, R n+ i = σ,k θ(x i, n+ ) σ,k θ(x i, n ) + σ,k n wj [ θ(xi, n+ j ) θ(x i, n j ) ] r [ θ(x i+, n+ ) 2θ(x i, n+ ) + θ(x i, n+ ) ] P n+ i = σ,k n j=0 w j [ θ(xi, n+ j ) θ(x i, n j ) ] r [ θ(x i+, n+) 2θ(x i, n+) + θ(x i, n+) ] P n+ i (9) From Eq. (5), we have n σ,k wj j=0 [ θ(xi, n+ j ) θ(x i, n j )] = θ(x i, n+ ) F 0 n+ + C k (20) and θ(x i+, n+ ) 2θ(x i, n+ ) + θ(x i, n+ ) h 2 = 2 θ(x i, n+ ) x 2 + C 2 h 2 (2) Therefore from Eqs. (9), (20) and (2) we get, R n+ i C(k + + k h 2 ), i =, 2, 3...p ; n =, 2, 3...m. where, C is a constant. Theorem 3.3 e n C(w n) (k + + k h 2 ), n =, 2,...m where, e n = max }{{} e n i and C is a constant. i p Proof : We wi prove the above theorem by mathematica induction method. For n =, et e = σ,k e = σ,k e = max }{{} i p σ,k e i = σ,k max }{{} i p e i i.e e = e = max }{{} i p e i, hence we have e r e + + (σ,k + 2r) e r e re + + (σ,k + 2r)e re = R i C(w ) (k + + k h 2 ) Suppose that e j C(wj ) (k + + k h 2 ),,2,..n and σ,k e n+ = max }{{} σ,k e i i.e en+ i p Here we note that (wj ) (wn),,2,3...n. Then we have e n+ r e n+ + + (σ,k + 2r) e n+ r e n+ re n+ + + (σ,k + 2r)e n+ re n+ }{{} i p = max e i.

9 Internationa Journa of Appied Mathematica Research 43 n = c e n + c j+ e n j + R n+ i n c e n + c j+ e n j + C(k + + k h 2 ) n c e n + c j+ e n j + C(k + + k h 2 ) n c + c j+ + wn+ (wn+) C(k + + k h 2 ) = (w n+) C(k + + k h 2 ) Since, (w im n+) n (n + ) = im (n + ) n (n + 2) (n + ) (n + ) = im n ( + n+ ) (n + ) = im n ( )(n + ) =, hence, there is a constant C,such that e n C(n + ) (k + + k h 2 ). Since, nk is finite, we obtain the foowing resut. Theorem 3.4 Let θ n i be the approximate vaue of θ(x i, n ) computed by use of the difference scheme (6)-(8) then there is a positive constant C such that θ n i θ(x i, n ) C(k + h 2 ), i = i, 2, 3...p, n =, 2, 3...m. 4. Numerica computation and discussion Consider the heat conduction equation in an infinite pate of finite thickness with the boundary and initia condition: θ F 0 = 2 θ x 2 (22) ( ) j θ x = k i j ( ), x = ( ) j, j =, 2 (23) θ(x, 0) = 0, > 0 (24) where k i ( ) = k i2 ( ) = k i0 exp( P d ) In this study the computation has been made and the resuts are presented in ten Figures. On the Figures presented in this study, ony the parameters whose vaues different from the reference vaue are indicated. The seected reference vaues incude h =, k = 64, k i 0 = k i20 =.0. Variation of dimensioness temperature θ with space coordinate x is given in Fig.. The temperature decreases as x increases and attains a minimum vaue at centre i.e x = 0 and then increases symmetricay. Further we observe that for = 75, temperature decreases as increases as shown in Fig. (a) whereas for =.50, temperature increases as increases as shown in Fig. (b). From Fig. 2, it is evident that, dimensioness temperature increases and attains a maximum vaue with increase in time fractiona order. A cose examination of Figs. 2 and 3 revea that:

10 44 Internationa Journa of Appied Mathematica Research 0.9 =0.00 =0.0 = =5 = =0.00 =0.0 = =5 =5 =.0 0 Space (a) At P d =.0, = Space (b) At P d =.0, =.50 Fig. : Variation of dimensioness temperature (θ) with space (x) (a) At x=0, P d =.0, = (b) At x=0, P d =.0, =.25 Fig. 2: Variation of dimensioness temperature (θ) with =75 =90 =90 =06 0 (a) At x=0, P d =.0 0 (b) At x=0, P d =0.9 Fig. 3: Variation of dimensioness temperature (θ) with

11 Internationa Journa of Appied Mathematica Research 45 Vaue of at which Θ max ( * ) At x=0, P d =.0 Fig. 4: Variation of with at maximum temperature =0.00 =0.002 =0.0 =0.02 =0.0 =0.2 = = =0.7 = (a) At x=0, P d = (b) At x=0, P d =.0 Fig. 5: Variation of dimensioness temperature (θ) with For 75 (say critica time F 0 ) and P d =.0, dimensioness temperature decreases for any sma increase in time fractiona order at x = 0. For > 75 and P d =.0, dimensioness temperature increases and attains a maximum vaue for certain time fractiona order (say ) and decreases further with increase of as shown in Fig. 3(a). When P d decreases, the critica time F 0 increases as shown in Fig. 3(b). When increases from 75, for fixed P d =.0, the vaue of at which maximum temperature attains, increases, as shown in Fig. 4. Aso from Fig. 4 we observe that the vaue of at which dimensioness temperature is maximum, increases with, for fixed P d. Variation of dimensioness temperature with for different is given in Fig. 5. For sma time fractiona order, dimensioness temperature decreases with increase of at fixed P d =.0, as shown in Fig. 5(a). A cose examination of Fig. 5 reveas that temperature increases and attains a maximum vaue for certain F0 and then decreases with increase of. From Fig. 6(a) it is evident that dimensioness temperature increases then decreases graduay with, whereas, for i.e. in case of norma diffusion, dimensioness temperature increases with increase in Fourier number, as shown in Fig. 6(b). From Fig. 7, it is evident that dimensioness temperature decreases with increase of P d for any fixed and any fixed time fractiona order (0, ]. At the same time, as P d increases, for sma and sma very cose to zero, dimensioness temperature decreases with increase in as shown in Fig. 7(a).

12 46 Internationa Journa of Appied Mathematica Research =4 =5 =6 =7 0.5 Dimensioness temperature (θ) =0.990 =0.993 =0.996 =0.999 = (a) At x=0, P d =.0 (b) At x=0, P d =.0 Fig. 6: Variation of dimensioness temperature (θ) with 0.9 =0.000 =0.00 =0.0 = =0.00 =0.006 =0.02 =0.05 P d (a) At x=0, =75 0 P d (b) At x=0, =37 Fig. 7: Variation of dimensioness temperature (θ) with P d =0.000 =0.00 =0.0 = =0.90 =0.93 =0.96 =0.99 =.0 0. P d (a) At x=0, =.50 P d (b) At x=0, =.50 Fig. 8: Variation of dimensioness temperature (θ) with P d

13 Internationa Journa of Appied Mathematica Research x=0 =0. =0.0 x= and x=0 =0 =0 0 x= and (a) At P d =.0, = (b) At P d =.0, =.50 Fig. 9: Variation of dimensioness temperature (θ) with CPU time (second) Time grid size Fig. 0: Variation of CPU time with the time grid size Fig. 8 reveas that the dimensioness temperature increases for very sma increase of at =.50, which shows that there is a state of and at which dimensioness temperature decreases then further increases with increase of P d, as shown in Fig. 7(b). For very sma in the range of (0, 0 2 ) dimensioness temperature decreases as shown in Fig. 9(a) and for in the range (0, 0 0 ) dimensioness temperature decreases and then takes a fixed vaue for any fixed P d and any fixed, which indicate the stabiity of temperature as shown in Fig. 9(b). Fig. 0 shows the executing time of our method in 2.3 GH i3 processor with 2 GB ram and Windows 7 operating system. 5. Concusions A mathematica mode describing time fractiona heat transfer (in the sense of Caputo), in an infinite pate of finite thickness whose faces are subjected to non-homogeneous boundary condition of second kind has been anayzed using a fuy impicit unconditionay stabe finite difference scheme. This impicit finite difference scheme is unconditionay stabe and convergent. Our simuation shows that for F0, θ decreases as increases. Further, after > F0, θ increases as, and for >, θ decreases as increases. This is because of Fourier number is a measure of rate of heat conduction in comparison with the heat storage in the given voume eement. Larger the Fourier number, deeper is the penetration of heat into a body over a given period of time. The Predoditeev number is defined as, P d = br2 a(t c T 0), due to which, F 0 increases as P d decreases. For any fixed and P d, when time fractiona order

14 48 Internationa Journa of Appied Mathematica Research 0, corresponding temperature tends to a fixed vaue showing the stabiity of scheme. This technique can aso be appied to sove the time fractiona heat conduction equation under most generaized boundary conditions. Acknowedgement The authors are acknowedge to the Prof. Umesh Singh, Coordinator DST-Center for Interdiscipinary Mathematica Sciences (CIMS), BHU, for providing necessary faciities. References [] D. Benson, S. Wheatcraft, M. Meerschaert, Appication of fractiona advection-dispersion equation, Water Resour. Res., 36 (6) (2000) [2] C. Chen, F. Liu, K. Burrage, Finite difference methods and a Fourier anaysis for the fractiona reaction-sub-diffusion equation, Appied Mathematics and Computation, 98 (2) (2008) [3] C. Chen, F. Liu, I. Turner, V. Anh, A Fourier method for the fractiona diffusion equation describing sub-diffusion, J. Comput. Phys., 227 (2) (2007) [4] H. Ding, Y. Zhang, Notes on impicit finite difference approximation for time fractiona diffusion equations, Int. J. Compu. Math. App., 6 (20) [5] P. Grassberger, W. Nader, L. Yang, Heat conduction and entropy production in a one-dimensiona hard-partice gas, Physica Rev. Lett., 89 (8) (2002) [6] F. Huang, F. Liu, The time fractiona diffusion and advection dispersion equation, ANZIAM J. 46 (2005) [7] X. Jiang, M. Xu, The time fractiona heat conduction equation in the genera orthogona curviinear coordinate and the cyindrica coordinate systems, Physica A, 389 (200) [8] F. Khanna, D. Matrasuov, Non-inear Dynamics and Fundamenta Interactions, Nato Pubic Dipomacy Division, [9] T. Langands, B. Henry, The accuracy and stabiity of an impicit soution method for the fractiona diffusion equation, J. Comput. Phys. 205 (2005) [0] S. Lepri, R. Livi, A. Poiti, On the anomaous therma conductivity of one-dimensiona attices, Europhys. Lett., 43 (3) (998) [] F. Mainardi, M. Raberto, R. Goreno, E. Scaas, Fractiona cacuus and continuous-time finance II: the waiting-time distribution, Physica A, 287 (2000) [2] M. Meerschaert, D. Benson, B. Baeumer, Operator evy motion and mutiscaing anomaous diffusion, Physica Rev. E, 63 (200) [3] M. Meerschaert, D. Benson, H. Scheer, B.Baeumer, Stochastic soution of space-time fractiona diffusion equations, Physica Rev. E 65 (2002) [4] D. Murio, Impicit finite difference approximation for time fractiona diffusion equations, Int. J. of Compu. Math. App., 56 (2008) [5] M. Naber, Distributed order fractiona sub-diffusion, Fractas, 2 (23) (2004) [6] F. Norwood, Transient therma waves in the genera theory of heat conduction with finite wave speeds, J. App. Mech., 39 (3) (972) [7] M. Ozisik, Heat Conduction, John Wiey & Sons, Inc, New York, 993. [8] I. Podubny, Fractiona Differentia Equations, Academic Press, San Diego, 999. [9] Y. Povstenko, Fractiona heat conduction equation and associated therma stresses in an infinite soid with spherica cavity, Quartery Journa of Mechanics and Appied Mathematics, 6 (4) (2008) [20] Y. Povstenko, Signaing probem for time fractiona diffusion-wave equation in a haf-space in the case of anguar symmetry, Noninear Dyn., 59 (200)

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