LOCAL FRACTIONAL VARIATIONAL ITERATION METHOD FOR

LOCAL FRACTIONAL VARIATIONAL ITERATION METHOD FOR DIFFUSION AND WAVE EQUATIONS ON CANTOR SETS XIAO-JUN YANG1, DUMITRU BALEANU2,3,4*, YASIR KHAN5, S. T. MOHYUD-DIN6 1

Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, P. R. China, E-mail: [email protected] 2 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, 06530, Turkey, E-mail: [email protected] 3 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia 4 Institute of Space Sciences, Magurele, RO-077125, Bucharest, Romania 5 Department of Mathematics, Zhejiang University, Hangzhou, 310028, P. R. China, E-mail: [email protected] 6 Department of Basic Sciences, Heavy Industries Taxila Education City (HITEC) University, TaxilaCantt, 44000, Pakistan, E-mail: [email protected] Received May 8, 2013

In this work, the local fractional variational iteration method is employed to handle the sub-diffusion and wave equations and the analytical solutions are obtained. The present method is efficient and implicit to investigate the differential equations with the local fractional derivatives. Key words: Wave equation, sub-diffusion equation, Cantor set, local fractional operator.

1. INTRODUCTION The diffusion equations are important in many processes in science and engineering, e.g. the diffusion of a dissolved substance in the solvent liquids, neutrons in a nuclear reactor and Brownian motion, while wave equations characterize the motion of a vibrating string (see [1-4] and the references therein). Recently, fractional calculus theory has successfully applied to differential equations and dynamics [5-13]. Many authors tried to model diffusion and wave equations from the classical diffusion or wave equation by replacing the first or second-order time derivative by a fractional order derivative [14, 15]. There are several analytical methods for solving the differential equations, e.g. the Laplace transform method [16], the Fourier method [17], the homotopy analysis method [18], the homotopy perturbation method [19], the Adomian decomposition method [20, 21], the generalized differential transform [22], and the variational iteration method [23, 24]. Rom. Journ. Phys., Vol. 59, Nos. 1–2, P. 36–48, Bucharest, 2014

2

Local fractional variational iteration method for diffusion and wave equations

37

The diffusion equation on Cantor sets was recently described in [25] as

∂u ( x, t )

= c*

∂t

∂ 2α u ( x, t ) ∂x 2α

,

(1)

where c* is the fractal thermal capacity of the material per unit volume, and local fractional diffusion equation (called local fractional sub-diffusion equation) was described [26] as

∂ α u ( x, t ) ∂t α

∂ 2α u ( x, t )

= a 2α

∂x 2α

,

(2)

where a 2α denotes the fractal diffusion constant which is, in essence, a measure for the efficiency of the spreading of the underlying substance. Equation (2) in rescaled non-dimensional variables reads as

∂ α u ( x, t ) ∂t α

−

∂ 2α u ( x, t ) ∂x 2α

= 0,

(3)

where u ( x, t ) is the field variable. More recently, fractional diffusion-wave equations on Cantor sets

∂ 2α u ( x, t ) ∂t 2α

−D

∂ 2 u ( x, t ) ∂x 2

= 0,

(4)

is considered in the reference [27] while local fractional wave equation is written in the form [28-30]

∂ 2α u ( x, t ) ∂t 2α

= a 2α

∂ 2α u ( x, t ) ∂x 2α

.

(5)

Equation (5) in re-scaled non-dimensional variables can be structured as

∂ 2α u ( x, t ) ∂t 2α

−

∂ 2α u ( x, t ) ∂x 2α

= 0.

(6)

The local fractional Laplace operator is given by [28, 31]

∇ 2α =

∂ 2α ∂ 2α ∂ 2α + + . ∂x 2α ∂y 2α ∂z 2α

(7)

38

Xiao-Jun Yang et al.

3

and the local fractional partial derivatives are written below utα =

∂ α u ( x, t ) ∂t α

2α = , u yx

∂ 2α u ( x, t ) ∂xα ∂yα

(8)

.

We notice that the local fractional diffusion equation yields

∇ 2α u =

1 α ut , a 2α

(9)

and the local fractional wave equation has the form

∇ 2α u =

1 2α utt , a 2α

(10)

where 1/ a 2α is a constant. This equation describes vibrations in a fractal medium. The quantity u is interpreted as the local fractional deviation at the time t from the position at rest of the point with rest position given by x, y , z . The above fractal derivatives were considered as the local fractional operators [32-36]. The main purpose of this paper is to present solutions to sub-diffusion equation (2) and for wave equation (5) by using the local fractional variational iteration method [34-36]. The paper is organized as follows: In Section 2 we introduce the notions and the notations of local fractional calculus theory used in this paper. In Section 3 we give a brief survey of the local fractional variational iteration method. Section 4 presents the solutions for sub-diffusion and wave equations in Cantor-set conditions. Section 5 is devoted to our conclusions. 2. PRELIMINARIES For the convenience of the reader, we include some general results about local fractional calculus [28-30, 33, 35-36]. It is so-called local fractional continuous on the interval ( a, b ) , denoted by

f ( x ) ∈ Cα ( a, b ) .

(11)

Local fractional derivative of local fractional continuous function f ( x ) of order α at x = x0 is given by [28-30, 32-36]

Dx( ) f ( x0 ) = f ( α

α)

( x0 ) =

dα f ( x) dxα

x = x0

= lim x → x0

∆α ( f ( x ) − f ( x0 ) )

( x − x0 )

α

(12)

4


39

where ∆α ( f ( x ) − f ( x0 ) ) ≅ Γ (1 + α ) ∆ ( f ( x ) − f ( x0 ) ) .

(13)

We notice that [28-33]

f ( x ) ∈ Dx (α ) ( a, b )

(14)

f (α ) ( x ) = Dx (α ) f ( x )

(15)

if for any x ∈ ( a, b ) . Local fractional derivative of high order is written in the form [28-30] f

( kα )

k times

(α ) ( x ) = Dx ...Dx (α ) f ( x ) ,

(16)

and the local fractional partial derivative of high order is defined as [28-30] kα

∂ ∂x kα

k times

α ∂ ∂α f ( x ) = α ... α f ( x ) . ∂x ∂x

(17)

Let a function f ( x ) satisfying the condition (11). Local fractional integral of

f ( x ) of order α in the interval [ a, b] is given by [28-30, 35, 36] (α ) f ( x) = a Ib

j = N −1 b α 1 1 α f t dt = f ( t j )( ∆t j ) , (18) lim ( )( ) ∑ ∫ a t ∆ → 0 Γ (1 + α ) Γ (1 + α ) j =0

where a partition of the interval

[ a, b ]

is denoted as ∆t j = t j +1 − t j ,

∆t = max {∆t1 , ∆t2 , ∆t j ,...} and j = 0,..., N − 1 , t0 = a, t N = b .

Here, we consider a function on Cantor sets such that its fractal dimension is α .

Local fractional multiple integrals of local fractional continuous function f ( x ) is written in the following form

x0

Ix

( kα )

k times

(α ) f ( x ) = x0 I x ... x0 I x (α ) f ( x )

(19)

if (11) is valid for x ∈ ( a, b ) . We recall the following results [28-30]. (1) Suppose that f ( x ) , g ( x ) ∈ Dx (α ) ( a, b ) , then local fractional Leibniz product law reads [28-31]

40


(

5

)

(

)

Dx (α )  f ( x ) g ( x )  = Dx (α ) f ( x ) g ( x ) + f ( x ) Dx (α ) g ( x ) .

(20)

(2) Suppose that f ( x ) ∈ Dx (α ) ( a, b ) , then local fractional Leibniz formula can be written as [28-31] (α )

I

a x

Dx (α ) f ( x ) = f ( x ) − f ( a ) .

(21)

(3) Suppose that f ( x ) ∈ Dx (α ) ( a, b ) , then the local fractional integration by parts reads [28-31] a

I x (α ) f ( x ) g (α ) ( x ) =  f ( x ) g ( x ) 

x a

− a I x ( α ) f (α ) ( x ) g ( x ) .

(22)

(4) Suppose that f ( t ) is local fractional continuous on the interval [ a, b] , then we have a

(5) If f ( x ) =

Ix

(α)

a

Iτ

(α)

f (t ) = a I x

( x − t ) f (t ) , ( x ∈ [ a, b ]) . Γ (1 + α ) α

(α)

(23)

t kα , then we have by the local fractional Fubini’s formula Γ ( k α + 1) 0

I x (α) 0 I τ (α)

t kα t ( k + 2)α = . Γ ( k α + 1) Γ ( ( k + 2 ) α + 1)

(24)

(6) If y ( x ) = ( f D u )( x ) and u ( x ) = g ( x ) , then we have [28-31] d α y ( x) dx

where f

(α)

( g ( x ) ) and

α

= f

(α)

( g ( x )) ( g ( ) ( x )) 1

α

,

(25)

g (1) ( x ) exist.

3. LOCAL FRACTIONAL VARIATIONAL ITERATION METHOD

In this section, we present the main steps of the local fractional variational iteration method [34-36]. The method gives the solution in a local fractional series form that converges to the closed form solution if an exact solution exists. Consider the local fractional differential equation

L αu + N αu = g ( t ) ,

(26)

6


41

where L α and N α are linear and nonlinear local fractional operators, respectively, and g ( t ) is the source inhomogeneous term. The local fractional variational iteration method presents a correction local fractional functional for Eq. (26) in the following form [36]: u n +1 ( x ) = u n ( x ) + 0 I x

(α)

 λ ( s ) α   L α u n ( s ) + N α u n ( s ) − g ( s )   .   Γ (1 + α ) 

(27)

Taking the local fractional variation of Eq. (27) with respect to the independent variable we find that δ u n +1 ( x ) = δ u n ( x ) + 0 I x α

α

(α) δ α

 λ ( s ) α   L α u n ( s ) + N α u n ( s ) − g ( s )   .  Γ 1 + α)  ( 

The extremum condition of u n +1 requires that δ α u n +1 = 0 . This yields the stationary conditions:  λ (s) α   1−   Γ (1 + α )   

(α) s=x

 λ (s) α   = 0,   Γ (1 + α )   

s= x

 λ (s) α   = 0,   Γ (1 + α )   

( 2α ) s= x

= 0. (28)

This in turn gives λ (s)

(s − x) = . Γ (1 + α ) Γ (1 + α ) α

α

(29)

The function u 0 ( x ) should be selected by using the initial conditions as follows: u 0 ( x ) = u (0) +

xα u (α) ( 0 ) . Γ (1 + α )

(30)

We can obtain a correction local fractional functional, which reads u n +1 ( x ) = u n ( x ) + 0 I x

(s − x) {L α u n ( s ) + N α u n ( s ) − g ( s )} . Γ (1 + α ) α

(α)

(31)

Consequently, the solution is u = lim u n . n →∞

(32)

Here, this technique is called as local fractional variational iteration method. It is remarked that the classical variation is in the case of local fractional variation when fractal dimension is equal to 1 [28-30].

42


7

4. SOLUTIONS TO THE SUB-DIFFUSION AND WAVE EQUATION ON CANTOR SETS

In this section we discuss solutions to the sub-diffusion equation and wave equation on the Cantor-set conditions by using local fractional variational iteration method. 4.1. SOLUTION TO SUB-DIFFUSION EQUATION ON CANTOR SETS Let us start with sub-diffusion equation on the Cantor-set sections given by

∂ 2 α u ( x, t ) ∂x 2 α

−

1 ∂ α u ( x, t ) =0 a 2α ∂t α

(33)

and subject to the fractal value conditions ∂α u ( 0, t ) = 0, u ( 0, t ) = a 2 α E α ( t α ) . ∂t α

(34)

From Eq. (34) we take the initial value, which reads

u 0 ( x, t ) = a 2 α E α ( t α ) .

(35)

By using Eq. (31) we structure a local fractional iteration procedure as u n +1 ( x, t ) = u n ( x, t ) + 0 I x

(α)

 ( ξ − x ) α  ∂ 2 α T n ( ξ, t ) 1 ∂ α T n ( ξ, t )   −    . (36)  a 2α ∂ξ 2 α ∂t α  Γ (1 + α )   

Hence, we can derive the first approximation term u 1 ( x, t ) = u 0 ( x, t ) + 0 I x

(α)

 ( ξ − x ) α  ∂ 2 α T 0 ( ξ, t ) 1 ∂ α T 0 ( ξ, t )   −     Γ 1 + α)  ∂ξ 2 α ∂t α a 2α    (

 ( ξ − x ) α  = a 2α E α ( t α ) + 0 I x (α)  − E α ( t α ) ) (  Γ (1 + α )  =a

2α

 1 1  t 2kα . E α ( t )  ∑ 2kα  k =0 a Γ (1 + 2k α )    α

The second approximation can be calculated in the similar way, which is

(37)

8


43

u 2 ( x, t )  ( ξ − x ) α  ∂ 2 α T 1 ( ξ, τ ) 1 ∂ α T 1 ( ξ, τ )   = u 1 ( x, t ) + 0 I x ( α )  −    a 2α ∂ξ 2 α ∂τ α  Γ (1 + α )    =a

2α

(38)

 2 1  t 2kα . E α ( t )  ∑ 2kα  k =0 a Γ (1 + 2k α )    α

Proceeding in this manner, we get the third approximation as u 3 ( x, t )  ( ξ − x ) α  ∂ 2 α T 2 ( ξ, τ ) 1 ∂ α T 2 ( ξ, τ )   = u 2 ( x, t ) + 0 I x ( α )  −    a 2α Γ 1 + α )  ∂ξ 2 α ∂τ α    ( =a

2α

(39)

 3 1  x 2kα . E α ( t )  ∑ 2kα  k =0 a Γ (1 + 2k α )    α

Hence, we obtain u 0 ( x, t ) = a 2 α E α ( t α )  1 1  t 2kα  u 1 ( x, t ) = a 2 α E α ( t α )  ∑ 2 k α  k =0 a Γ (1 + 2k α )    2   1 t 2kα  u 2 ( x, t ) = a 2 α E α ( t α )  ∑ 2 k α  k =0 a Γ (1 + 2k α )    #

and so on. Thus, we have the local fractional series solution

u n ( x, t ) = a

2α

 n 1  x 2kα . E α ( t )  ∑ 2kα  k =0 a Γ (1 + 2k α )   

As a result, the final solution reads

α

(40)

44


9

u ( x, t ) = lim u n ( x, t ) n →∞

 ∞ 1  x 2kα  = a E α ( t )  ∑ 2kα  k =0 a Γ (1 + 2k α )    α x  = a 2 α E α ( t α ) cosh α  α  . a  2α

α

(41)

4.2. SOLUTION TO WAVE EQUATION ON CANTOR SETS Consider the following wave equation on Cantor sets

∂ 2 α T ( x, t ) ∂x 2 α

−

1 ∂ 2 α T ( x, t ) =0 ∂t 2α a 2α

(42)

subject to fractal value conditions given by ∂α T ( 0, t ) = a 2 α E α ( t α ) , T ( 0, t ) = 0. ∂t α

(43)

By using Eq. (34) we take an initial value as

u 0 ( x, t ) =

a 2α x α E α ( t α ) Γ (1 + α )

.

(44)

Applying Eq. (31) yields a local fractional iteration procedure:

u n +1 ( x, t ) = u n ( x, t ) + 0 I x

(α)

 ( ξ − x ) α  ∂ 2 α T n ( ξ, t ) 1 ∂ 2 α T n ( ξ, t )   −    . (45)  ∂ξ 2 α ∂t 2 α a 2α  Γ (1 + α )   

The first approximation term reads

10


45

u 1 ( x, t ) = u 0 ( x, t ) + 0 I x =

(α)

a 2α x α E α ( t α )

=a

Γ (1 + α )

3α

 ( ξ − x ) α  ∂ 2 α T 0 ( ξ, t ) 1 ∂ 2 α T 0 ( ξ, t )   − 2α     a Γ 1 + α )  ∂ξ 2 α ∂t 2 α    (

+ 0Ix

(α)

 ( ξ − x ) α  x α E α ( t α )    −     Γ (1 + α )  Γ (1 + α )  

(46)

 1  x ( 2 k +1)α 1 . E α ( t )  ∑ ( 2 k +1)α  k =0 a Γ (1 + ( 2k + 1) α )    α

In the same manner, the second approximation is given by u 2 ( x, t ) = u 1 ( x, t ) + 0 I x =

(α)

a 2α x α E α ( t α )

=a

Γ (1 + α )

3α

+

 ( ξ − t ) α  ∂ 2 α T 1 ( ξ, t ) 1 ∂ 2 α T 1 ( ξ, t )   −     2α  a 2α ∂t 2 α  Γ (1 + α )  ∂ξ   x 3α E α ( t α ) Γ (1 + 3α )

+ 0Ix

(α)

 ( ξ − t ) α  1 x 3α E α ( t α )   (47)  − 2 α   Γ (1 + 3α )    Γ (1 + α )  a 

 2  x ( 2 k +1)α 1 . E α ( t )  ∑ ( 2 k +1)α  k =0 a Γ (1 + ( 2k + 1) α )    α

The third approximation reads as u 3 ( x, t ) = u 2 ( x, t ) + 0 I x

=a

3α

(α)

α 2α 1 ∂ α T 2 ( ξ, τ )    ( ξ − x )  ∂ T 2 ( ξ, τ ) −     a 2α Γ 1 + α)  ∂ξ 2 α ∂τ α    (

 2  x ( 2 k +1)α 1 . E α ( t )  ∑ ( 2 k +1)α  k =0 a Γ (1 + ( 2k + 1) α )    α

Therefore, we obtain

(48)

46


u 0 ( x, t ) =

11

a 2α x α E α ( t α ) Γ (1 + α )

 1  x ( 2 k +1)α 1  u 1 ( x, t ) = a E α ( t )  ∑ ( 2 k +1)α  k =0 a Γ (1 + ( 2k + 1) α )    2   x ( 2 k +1)α 1  u 2 ( x, t ) = a 3α E α ( t α )  ∑ ( 2 k +1)α  k =0 a Γ (1 + ( 2k + 1) α )    3   x ( 2 k +1)α 1  u 3 ( x, t ) = a 3α E α ( t α )  ∑ ( 2 k +1)α  k =0 a Γ (1 + ( 2k + 1) α )    # 3α

α

(49)

and so on. Finally, the solution, in local fractional series form, reads n

u n ( x, t ) = a

3α

E α (t

α

x ( 2 k +1)α . k = 0 Γ (1 + ( 2k + 1) α )

)∑

(50)

Thus, the expression of the final solution is given by u ( x, t ) = lim u n ( x, t ) n →∞

n

= a 3α E α ( t α ) ∑ k =0

1 a ( 2 k +1)α

x ( 2 k +1)α Γ (1 + ( 2k + 1) α )

(51)

 xα  = a 3α E α ( t α ) sinh α  α  . a  5. CONCLUSIONS

The local fractional variational iteration method was applied to the subdiffusion equation and the wave equation defined on Cantor sets with the fractal conditions. The local fractional variational iteration method was proved to be effective and very reliable for analytic purposes. We noticed that the variational iteration method [23, 24, 37-42] and the fractional variational iteration method [4346] are applied to deal with differential equations of solutions belonging to Lipschitz mappings, while the present method is used to process local fractional

12


47

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