Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis

R E S E A R C H Open Access

Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis

Xiao-Jun Yang^1,2,3*, Dumitru Baleanu^4,5,6and José António Tenreiro Machado⁷

*Correspondence:

dyangxiaojun@163.com

1Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China

2Institute of Software Science, Zhengzhou Normal University, Zhengzhou, 450044, China Full list of author information is available at the end of the article

Abstract

In this paper, we discuss the mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. The Schrödinger equation and Heisenberg uncertainty principles are structured within local fractional operators.

Keywords: Heisenberg uncertainty principle; local fractional Fourier operator;

Schrödinger equation; fractal time-space

1 Introduction

As it is known, the fractal curves [, ] are everywhere continuous but nowhere differ- entiable; therefore, we cannot use the classical calculus to describe the motions in Cantor time-space [–]. The theory of local fractional calculus [–], started to be considered as one of the useful tools to handle the fractal and continuously non-differentiable func- tions. This formalism was applied in describing physical phenomena such as continuum mechanics [], elasticity [–], quantum mechanics [, ], heat-diffusion and wave phenomena [–], and other branches of applied mathematics [–] and nonlinear dynamics [, ].

The fractional Heisenberg uncertainty principle and the fractional Schrödinger equa- tion based on fractional Fourier analysis were proposed [–]. Local fractional Fourier analysis [], which is a generalization of the Fourier analysis in fractal space, has played an important role in handling non-differentiable functions. The theory of local fractional Fourier analysis is structured in a generalized Hilbert space (fractal space), and some re- sults were obtained [, –]. Also, its applications were investigated in quantum me- chanics [], differentials equations [, ] and signals [].

The main purpose of this paper is to present the mathematical aspects of the Heisen- berg uncertainty principle within local fractional Fourier analysis and to structure a local fractional version of the Schrödinger equation.

The manuscript is structured as follows. In Section , the preliminary results for the local fractional calculus are investigated. The theory of local fractional Fourier analysis is introduced in Section . The Heisenberg uncertainty principle in local fractional Fourier analysis is studied in Section . Application of quantum mechanics in fractal space is con- sidered in Section . Finally, the conclusions are presented in Section .

©2013 Yang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Mathematical tools

2.1 Local fractional continuity of functions Definition [–, –] If there is

f(x) –f(x)<ε^α (.)

with|x–x|<δ, forε,δ>  andε,δ∈R. Nowf(x) is called a local fractional continuous atx=x, denoted bylim_x→xf(x) =f(x). Thenf(x) is called local fractional continuous on the interval (a,b), denoted by

f(x)∈C_α(a,b). (.)

The functionf(x) is said to be local fractional continuous atx_from the right iff(x_) is defined, and

x→xlim⁺_f(x) =f x⁺_

.

The functionf(x) is said to be local fractional continuous atxfrom the left iff(x) is defined, and

x→xlim^–_f(x) =f x^–_

.

Suppose thatlim_x→x⁺

f(x) =f(x⁺_),lim_x→x^–_f(x) =f(x^–_) andf(x⁺_) =f(x^–_), then we have the following relation:

x→xlim⁺_f(x) = lim

x→x^–_f(x) = lim

x→x

f(x).

For other results of theory of local fractional continuity of functions, see [–, –].

2.2 Local fractional derivative and integration

Definition [–, –] Settingf(x)∈C_α(a,b), a local fractional derivative off(x) of orderαatx=xis defined by

f^(α)(x_) =d^αf(x) dx^α

x=x

= lim

x→x

^α(f(x) –f(x))

(x–x)^α , (.)

where^α(f(x) –f(x_))∼=( +α)(f(x) –f(x_)) with a gamma function( +α).

Definition [–, –] Settingf(x)∈C_α(a,b), a local fractional integral off(x) of orderαin the interval [a,b] is defined as

aI_b^(α)f(x) =  ( +α)

b a

f(t)(dt)^α=  ( +α) lim

t→

j=N–

j=

f(t_j)(t_j)^α, (.)

wheret_j=t_j+–t_j,t=max{t_,t_,t_j, . . .}and [t_j,t_j+],j= , . . . ,N– ,t_=a,t_N=b, is a partition of the interval [a,b].

Their fractal geometrical explanation of local fractional derivative and integration can be seen in [, , –].

Iff(x)∈C_α[a,b], then we have [, ]

_aI_b^(α)f(x)≤aI_b^(α)f(x) (.)

withb–a> .

Lemma [, ]

–∞I_∞^(α)f(x)g(x)

≤

–∞I_∞^(α)g(x)^

–∞I_∞^(α)g(x)^g(x)

. (.)

Proof See [, ].

3 Theory of local fractional Fourier analysis

In this section, we investigate local fractional Fourier analysis [–], which is a gener- alized Fourier analysis in fractal space. Here we discuss the local fractional Fourier series, the Fourier transform and the generalized Fourier transform in fractal space. We start with a local fractional Fourier series.

3.1 Local fractional Fourier series

Definition [, , –] The local fractional trigonometric Fourier series off(t) is given by

f(t) =a+ ∞

i=

aksin_α k^αω^α_t^α

+ ∞

i=

bkcos_α k^αω_^αt^α

. (.)

Then the local fractional Fourier coefficients can be computed by

⎧⎪

⎪⎨

⎪⎪

⎩

a=_T^αT

 f(t)(dt)^α, a_k= (_T^)^α_T

 f(t)sin_α(k^αω^αt^α)(dt)^α, bk= (_T^)^αT

 f(t)cos_α(k^αω^α_t^α)(dt)^α.

(.)

The Mittag-Leffler functions expression of the local fractional Fourier series is described by [, , –]

f(x) = ∞ k=–∞

CkE_α

π^αi^α(kx)^α l^α

, (.)

where the local fractional Fourier coefficients are

Ck=  (l)^α

_l

–l

f(x)E_α

–π^αi^α(kx)^α l^α

(dx)^α withk∈Z. (.)

The above is generalized to calculate the local fractional Fourier series.

3.2 The Fourier transform in fractal space

Definition [, , –] Suppose thatf(x)∈C_α(–∞,∞), the Fourier transform in fractal space, denoted byF_α{f(x)} ≡f_ω^F,α(ω), is written in the form

F_α f(x)

= 

( +α) _∞

–∞E_α

–i^αω^αx^α

f(x)(dx)^α, (.)

where the latter converges.

Definition [, , –] IfF_α{f(x)} ≡f_ω^F,α(ω), its inversion formula is written in the form

f(x) =  (π)^α

_∞

–∞E_α i^αω^αx^α

f_ω^F,α(ω)(dω)^α, x> . (.) 3.3 The generalized Fourier transform in fractal space

Definition [, ] The generalized Fourier transform in fractal space is written in the form

F_α f(x)

= 

( +α) _∞

–∞f(x)E_α

–i^αhx^αω^α

(dx)^α, (.)

whereh_=_(+α)^(π)α with  <α≤.

Definition [, ] The inverse formula of the generalized Fourier transform in fractal space is written in the form [, ]

f(x) =  ( +α)

_∞

–∞f_ω^F,α(ω)Eα

i^αhx^αω^α

(dω)^α, (.)

whereh_=_(+α)^(π)α with  <α≤.

3.4 Some useful results

The following formula is valid [, ].

Theorem [, ]

F_α f^(α)(x)

=i^αhω^αF_α f(x)

. (.)

Proof See [, ].

Theorem [, ] If F_α{f(x)}=f_ω^F,α(ω),then we have _∞

–∞

f(x)^(dx)^α= _∞

–∞

f_ω^F,α(ω)^(dω)^α. (.)

Proof See [, ].

Theorem [, ] If F_α{f(x)}=f_ω^F,α(ω),then we have _∞

–∞f(x)g(x)(dx)^α= _∞

–∞f_ω^F,α(ω)g_ω^F,α(ω)(dω)^α. (.)

Proof See [, ].

4 Heisenberg uncertainty principles in local fractional Fourier analysis Theorem  Suppose that f ∈L_,α[],F_α{f(x)}=f_ω^F,α(ω),then we have

^( +α)

h^_ ≤ 

(+α)

_∞

–∞[f(x)x^α]^(dx)^α

(+α)

_∞

–∞|f(x)|^(dx)^α

· 

(+α)

_∞

–∞[f_ω^F,α(ω)ω^α]^(dω)^α

(+α)

_∞

–∞|f(x)|^(dx)^α

, (.)

with equality only if f(x)is almost everywhere equal to a constant multiple of C_E_α

–x^α K

, (.)

with K> and a constant C_. Proof Considering the equality

 ( +α)

_∞

–∞

f_ω^F,α(ω)hω^α

(dω)^α=  ( +α)

_∞

–∞

f^(α)(x)

(dx)^α, (.)

we have 

( +α) _∞

–∞

f(x)x^α

(dx)^α 

( +α) _∞

–∞

f_ω^F,α(ω)hω^α

(dx)^α

= 

( +α) _∞

–∞

f(x)x^α

(dx)^α 

( +α) _∞

–∞

f^(α)(x)

(dx)^α

≥  ( +α)

_∞

–∞

f(x)f^(α)(x)x^α(dx)^α

≥ 

( +α) _∞

–∞f(x)f^(α)(x)x^α(dx)^α

^. (.)

When ^f(x)x_K^α =f^(α)(x), then we havef(x) =CE_α(–^x_K^α) with a constantC. Since

f(x)^(α)

=

f(x)f(x)(α)

=f^(α)(x)f(x) +f(x)f^(α)(x) (.) and

 ( +α)

_∞

–∞f^(α)(x)f(x)x^α(dx)^α =

 ( +α)

_∞

–∞f(x)f^(α)(x)x^α(dx)^α

, (.)

we have

 ( +α)

_∞

–∞

f(x)^(α)

x^α(dx)^α

=f(x)^ x^α_∞

–∞– _∞

–∞

f(x)^(dx)^α

= – _∞

–∞

f(x)^(dx)^α, (.)

when (|f(x)|^)x^α→,x→ ∞.

Hence, there is _∞

–∞

f(x)^(dx)^α

= 

( +α) _∞

–∞

f(x)^_(α)

x^α(dx)^α

= 

( +α) _∞

–∞f^(α)(x)f(x)x^α(dx)^α+  ( +α)

_∞

–∞x^αf(x)f^(α)(x)(dx)^α

≤  ( +α)

_∞

–∞f^(α)(x)f(x)x^α(dx)^α

(.)

such that 

( +α) _∞

–∞

f(x)^(α)

x^α(dx)^{α }

= _∞

–∞

f(x)^(dx)^{α }

=^( +α) 

( +α) _∞

–∞

f(x)^(dx)^{α }

≤ 

( +α) _∞

–∞f^(α)(x)f(x)x^α(dx)^α 

≤ 

( +α) _∞

–∞

f(x)f^(α)(x)x^α(dx)^α 

≤ 

( +α) _∞

–∞

f(x)x^α

(dx)^α 

( +α) _∞

–∞

f_ω^F,α(ω)h_ω^α

(dω)^α

. (.)

Therefore, we deduce to ^( +α)

 ( +α)

_∞

–∞

f(x)^(dx)^{α }

≤ 

( +α) _∞

–∞

f(x)x^α

(dx)^α 

( +α) _∞

–∞

f_ω^F,α(ω)hω^α

(dω)^α

= h^_ 

( +α) _∞

–∞

f(x)x^α

(dx)^α 

( +α) _∞

–∞

f_ω^F,α(ω)ω^α

(dω)^α

. (.)

Hence, this result is obtained.

As a direct result, we have two equivalent forms as follows.

Theorem  Suppose that f ∈L_,α[]and f^(α)(x) =^dα_dx^f(x)α ,then we have ^( +α)

 ≤

 (+α)

_∞

–∞[f(x)x^α]^(dx)^α

(+α)

_∞

–∞|f(x)|^(dx)^α

· 

(+α)

_∞

–∞[f^(α)(x)]^(dx)^α

(+α)

_∞

–∞|f(x)|^(dx)^α

, (.)

with equality only if f(x)is almost everywhere equal to a constant multiple of C_E_α(^–x_K^α), with K> and a constant C.

Proof Applying Theorem , we have

 ( +α)

_∞

–∞

f^(α)(x)

(dx)^α=  ( +α)

_∞

–∞

f_ω^F,α(ω)hω^α

(dω)^α (.)

such that ^( +α)

 ≤

 (+α)

_∞

–∞[f(x)x^α]^(dx)^α

(+α)

_∞

–∞|f(x)|^(dx)^α

· 

(+α)

_∞

–∞[f^(α)(x)]^(dx)^α

(+α)

_∞

–∞|f(x)|^(dx)^α

. (.)

Hence, Theorem  is obtained.

The above results [, ] are different from the results in fractional Fourier transform [, ] based on the fractional calculus theory.

5 The mathematical aspect of fractal quantum mechanics 5.1 Local fractional Schrödinger equation

We structure the non-differential phase of a fractal plane wave as a complex phase factor using the relations

ψα=AE_α i^α

k^αr^α–ω^αt^α

=AE_α i^α

h_α

P_αr^α–E_αt^α

, (.)

where the Planck-Einstein and De Broglie relations are in fractal space

⎧⎨

⎩

E_α=h_αω^α,

P_α=h_αk^α. (.)

We can realize the local fractional partial derivative with respect to fractal space

∇^αψα= i^α h_αP_αAE_α

i^α h_α

P_αr^α–E_αt^α

= i^α

h_αP_αψα (.)

and fractal time

∂^αψα

∂t^α = –i^α h_αE_αAE_α

i^α h_α

P_αr^α–E_αt^α

= –i^α

h_αE_αψα, (.)

where∇^α=_∂x^∂ααi^α+_∂y^∂ααj^α+_∂z^∂ααk^α[] withr^α=x^αi^α+y^αj^α+z^αk^α[].

From (.) we have

–i^αh_α∇^αψα=P_αψα (.)

such that –h^_α

m∇^αψα= 

mP_α·P_αψα, (.)

where∇^α=_∂x^∂α_α +_∂y^∂α_α +_∂z^∂α_α withr^α=x^αi^α+y^αj^α+z^αk^α. From (.) we have

i^αh_α∂^αψα

∂t^α =E_αψα. (.)

We have the energy equation

E_α= 

mP_α·P_α+V_α

=H_α (.)

such that

E_αψα=H_αψα, (.)

and

E_αψα= –h^_α

m∇^αψα+V_αψα,

whereH_αis the local fractional Hamiltonian in fractal mechanics.

Hence, we have that

i^αh_α∂^αψα

∂t^α = –h^_α

m∇^αψα+V_αψα. (.)

Therefore, we can deduce that the local fractional energy operator is

E_α=i^αh_α∂^α

∂t^α (.)

and that the local fractional momentum operator is

P_α=i^αh_α∇^α. (.)

Therefore, we get the local fractional Schrödinger equation in the form of local fractional energy and momentum operators

H_αψα= 

mP_α·P_αψα+V_αψα, (.)

where the local fractional Hamiltonian is H_α= 

mP_α·P_α+V_α. (.)

We also deduce that the general time-independent local fractional Schrödinger equation is written in the form

i^αh_α∂^αψα

∂t^α =H_αψα, (.)

which is related to the following equation:

∂^αS_α(qi,t)

∂t^α =H_α

qi,∂^αS_α

∂q^α_i ,t

, (.)

whereS_α is non-differential action,H_αis the local fractional Hamiltonian function, and q^α_i (i= , , ) are generalized fractal coordinates.

5.2 Solutions of the local fractional Schrödinger equation

.. General solutions of the local fractional Schrödinger equation

The general solution of the local fractional Schrödinger equation can be seen in the fol- lowing. For discretek, the sum is a superposition of fractal plane waves:

ψα(r,t) = ∞

n=

A_nE_α i^α

k_n^αr^α–ω^α_nt^α

= ∞

n=

AnE_α i^α

h_α

P_αr^α–E_αt^α

= ∞

n=

A_nE_α i^α

h_α

P_αr^α– P^_α

mt^α

(.)

and

E_α= P^_α

m. (.)

If we considerP_α=p_xαi^α+p_yαj^α+p_zαk^α≡p_xαi^α andr^α=x^αi^α+z^αj^α+z^αk^α, we have fractal plane waves:

ψα(x,t) =ψα(P_xα,t)

= ∞

n=

AnE_α i^α

h_α

p_xαx^α–p^_xα

mt^α

. (.)

.. Fractal complex wave functions

The meaning of this description can be seen in the following. Similar to the classical wave mechanics, we prepareN atoms independently, in the same state, so that when each of them is measured, they are described by the same wave function. Then the result of a position measurement is described as the fractal probability density, and we wish it is not the same for all. The set of impacts is distributed in space with the probability density

φα P(r),t

=ψα(r,t)^. (.)

In view of (.), we have

φα(x,t) =ψα(x,t)^. (.)

The set of N measurements is characterized by an expectation valuerα and a root mean square dispersion (r)^α,

rα=  ^( +α)

_∞

–∞r^αψα(r,t)^(dr)^α. (.)

Similarly, the square of the dispersion (r)^αis defined by (r)^α=

x^

α– xα

=

x^α–xα

α

= 

^( +α) _∞

–∞

r^α–rαψα(r,t)^(dr)^α. (.)

If the physical interpretation of a particle in fractal space is that the probability

dP(r) = 

^( +α)ψα(r,t)^(dr)^α, (.)

the integral of this quantity over all fractal space is

P(r) =  ^( +α)

_∞

–∞

ψα(r,t)^(dr)^α

= . (.)

For (.) we have

ψα(r,t) = ∞

n=

A_nE_α i^α

k_n^αr^α–ω^αnt^α

= ∞

n=

AnE_α i^α

h_α

P_αr^α–E_αt^α

= ∞

n=

AnE_α i^α

h_α

P_αr^α– P^_α

mt^α

(.)

such that

 = 

^( +α) _∞

–∞

ψα(r,t)^(dr)^α. (.)

.. Probabilistic interpretation of fractal complex wave function of one variable In (.), we have

φα(x,t) =ψα(x,t)^ (.)

and

P(x) =  (.)

such that

 ( +α)

_∞

–∞

ψα(x,t)^(dx)^α

= 

( +α) _L

–L

AE_α i^α

h_α

p_xαx^α–p^_xα

mt^{α }(dx)^α

= A^L^α ( +α)

= , (.)

where

ψα(x,t) =

⎧⎨

⎩

AE_α(_h^iα_α(p_xαx^α–^p_m^xαt^α)), x∈L,

, x∈/L. (.)

We have an expectation valuexαand a root mean square dispersion (x)^α,

xα=  ( +α)

_∞

–∞x^αψα(x,t)^(dx)^α (.)

and

(x)^α= x^

α– xα

=

x^α–xα

α

= 

( +α) _∞

–∞

x^α–xαψα(x,t)^(dx)^α. (.)

For a given fractal mechanical operatorA, we have an expectation valueAαand a root mean square dispersion (A)^α,

Aα=  ( +α)

_∞

–∞Aψα(x,t)^(dx)^α (.)

and

(A)^α=

A–Aα

α

= A^

α– Aα

= 

( +α) _∞

–∞

A–Aαψα(x,t)^(dx)^α. (.)

5.3 The Heisenberg uncertainty principle in fractal quantum mechanics Suppose that

 ( +α)

_∞

–∞

ψα(x,t)^(dx)^α= , (.)

we have a fractal positional operator expectation value xα= 

( +α) _∞

–∞i^αx^αψα(x,t)^(dx)^α=  (.)

and a root mean square dispersion of positional operator (x)^α= 

( +α) _∞

–∞i^αx^αψα(x,t)^(dx)^α. (.) Similar to the fractal positional operator, we have a fractal momentum operator expec- tation value

Pxα=

i^αh_α ∂^α

∂x^α

α

= 

( +α) _∞

–∞i^αh_α ∂^α

∂x^αψα(x,t)^(dx)^α=  (.) and a root mean square dispersion of positional operator

(Px)^α=  ( +α)

_∞

–∞i^αh^_α ∂^α

∂x^αψα(x,t)^(dx)^α. (.) Considering

(x)^α=  ( +α)

_∞

–∞i^αx^αψα(x,t)^(dx)^α, (.) (P_x)^α= 

( +α) _∞

–∞i^αh^_α ∂^α

∂x^αψα(x,t)^(dx)^α, (.)

and

 ( +α)

_∞

–∞

ψα(x,t)^(dx)^α= , (.)

by using Theorem , we have that ^( +α)

 ≤

 ( +α)

_∞

–∞x^αψα(x,t)^(dx)^α 

( +α) _∞

–∞

∂^α

∂x^αψα(x,t)^(dx)^α

such that ^( +α)

 ≤(x)^α(Px)^α

h^_α . (.)

Hence, we have that ^( +α)h^_α

 ≤(x)^α(Px)^α (.)

such that ( +α)hα

 ≤(x)^α(Px)^α, (.)

where

(x)^α=

(x)^α=

 ( +α)

_∞

–∞i^αx^αψα(x,t)^(dx)^α (.) and

(Px)^α=

(Px)^α=

 ( +α)

_∞

–∞i^αh^_α ∂^α

∂x^αψα(x,t)^(dx)^α. (.) Suppose that

h_α= h

π α

, (.)

then we have ( +α)(_π^h )^α

 ≤(x)^α(Px)^α (.)

and

i^α h

π

α∂^αψα

∂t^α = –(_π^h )^α

m ∇^αψα+V_αψα, (.)

where∇^α=_∂x^∂α_α +_∂y^∂α_α +_∂z^∂α_α [].

The above equation (.) differs from the results presented in [, ]. Also, Eq. (.) is different from the ones reported in [–, , ].

Below we define the local fractional energy operator

E_α=i^α h

π α ∂^α

∂t^α (.)

and the local fractional momentum operator

P_α=i^α h

π α

∇^α, (.)

where∇^α=_∂x^∂ααi^α+_∂y^∂ααj^α+_∂z^∂ααk^α[].

Thus, we get the Planck-Einstein and de Broglie relations are in fractal space as

⎧⎨

⎩

E_α= (_π^h )^αω^α,

P_α= (_π^h )^αk^α, (.)

wherehis Planck’s constant.

6 Conclusions

In this manuscript, the uncertainty principle in local fractional Fourier analysis is sug- gested. Since the local fractional calculus can be applied to deal with the non-differentiable functions defined on any fractional space, the local fractional Fourier transform is impor- tant to deal with fractal signal functions. The results on uncertainty principles could play an important role in time-frequency analysis in fractal space. From Eq. (A.) we conclude that there is a semi-group property for the Mittag-Leffler function on fractal sets. Mean- while, uncertainty principles derived from local fractional Fourier analysis are classical uncertainty principles in the case ofα = . We reported the structure the local fractional Schrödinger equation derived from Planck-Einstein and de Broglie relations in fractal time space.

Appendix We have [, ]

γ^α[F,a,b] +γ^α[F,b,c] =γ^α[F,a,c] (A.)

such that

S^α_F(y) –S^α_F(x) =γ^α[F,x,y] = (y–x)^α

( +α), (A.)

whereS^α_F(y) is a fractal integral staircase function. We have the relation [–]

H^α

F∩(γ, )

= –γ^α (A.)

such that

S^α_F(y) –S^α_F(x) =γ^α[F,x,y] =H^α

F∩(x,y)

= (y–x)^α

( +α). (A.)

Inversely we obtain

S^α_F(x) –S_F^α(y) =γ^α[F,y,x] =H^α

F∩(y,x)

= –(y–x)^α

( +α). (A.)

Hence, bothS^α_F(x) =_(+α)^xα andS_F^α(y) =_(+α)^yα are seen in [, ].

In view of Eq. (A.), we easily obtain that E_α

i^αx^α

=cos_αx^α+i^αsin_αx^α (A.)

and E_α

x^α+y^α

=E_α (x+y)^α

=E_α x^α

E_α y^α

, (A.)

whereE_α(x^α) =_∞

k= x^αk

(+kα),sin_αx^α=_∞

k=(–)^kx^(k+)α

[+(k+)α],cos_αx^α=_∞

k=(–)^kx^αk

(+αk) andi^αis a fractal unit of imaginary number [–, ].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Authors contributed equally in writing this article. Authors read and approved the final manuscript.

Author details

1Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China.²Institute of Software Science, Zhengzhou Normal University, Zhengzhou, 450044, China.

3Institute of Applied Mathematics, Qujing Normal University, Qujing, 655011, China.⁴Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, 06530, Turkey.⁵Institute of Space Sciences, Magurele, Bucharest, Romania. ⁶Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia.⁷Department of Electrical Engineering, Institute of Engineering of Polytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida, 431, Porto, 4200-072, Portugal.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank to the referees for their very useful comments and remarks.

Received: 29 January 2013 Accepted: 2 May 2013 Published: 20 May 2013

References

1. Mandelbrot, BB: The Fractal Geometry of Nature. Freeman, New York (1982)

2. Falconer, KJ: Fractal Geometry-Mathematical Foundations and Application. Wiley, New York (1997) 3. Zeilinger, A, Svozil, K: Measuring the dimension of space-time. Phys. Rev. Lett.54(24), 2553-2555 (1985) 4. Nottale, L: Fractals and the quantum theory of space-time. Int. J. Mod. Phys. A4(19), 5047-5117 (1989) 5. Saleh, AA: On the dimension of micro space-time. Chaos Solitons Fractals7(6), 873-875 (1996)

6. Maziashvili, M: Space-time uncertainty relation and operational definition of dimension. (2007) arXiv:0709.0898 7. Caruso, F, Oguri, V: The cosmic microwave background spectrum and an upper limit for fractal space dimensionality.

Astrophys. J.694(1), 151-156 (2009)

8. Calcagni, G: Geometry and field theory in multi-fractional spacetime. J. High Energy Phys.65(1), 1-65 (2012) 9. Kong, HY, He, JH: A novel friction law. Therm. Sci.16(5), 1529-1533 (2012)

10. Kong, HY, He, JH: The fractal harmonic law and its application to swimming suit. Therm. Sci.16(5), 1467-1471 (2012) 11. Kolwankar, KM, Gangal, AD: Fractional differentiability of nowhere differentiable functions and dimensions. Chaos

6(4), 505-513 (1996)

12. Jumarie, G: Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Appl. Math. Lett.22, 378-385 (2009)

13. Parvate, A, Gangal, AD: Calculus on fractal subsets of real line - I: formulation. Fractals17(1), 53-81 (2009) 14. Chen, W: Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractals28, 923-929 (2006) 15. Adda, FB, Cresson, J: About non-differentiable functions. J. Math. Anal. Appl.263, 721-737 (2001)

16. Balankin, AS, Elizarraraz, BE: Map of fluid flow in fractal porous medium into fractal continuum flow. Phys. Rev. E85(5), 056314 (2012)

17. He, JH: A new fractal derivation. Therm. Sci.15, 145-147 (2011)

18. Yang, XJ: Local fractional integral transforms. Prog. Nonlinear Sci.4, 1-225 (2011)

19. Yang, XJ: Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher, Hong Kong (2011) 20. Yang, XJ: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York (2012) 21. Carpinteri, A, Chiaia, B, Cornetti, P: Static-kinematic duality and the principle of virtual work in the mechanics of

fractal media. Comput. Methods Appl. Mech. Eng.191, 3-19 (2001)

22. Carpinteri, A, Cornetti, P: A fractional calculus approach to the description of stress and strain localization in fractal media. Chaos Solitons Fractals13(1), 85-94 (2002)

23. Yang, XJ: The zero-mass renormalization group differential equations and limit cycles in non-smooth initial value problems. Prespacetime J.3(9), 913-923 (2012)

24. Kolwankar, KM, Gangal, AD: Local fractional Fokker-Planck equation. Phys. Rev. Lett.80, 214-217 (1998)

25. Wu, GC, Wu, KT: Variational approach for fractional diffusion-wave equations on Cantor sets. Chin. Phys. Lett.29(6), 060505 (2012)

26. Zhong, WP, Gao, F, Shen, XM: Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral. Adv. Mater. Res.461, 306-310 (2012)

27. Yang, XJ, Baleanu, D: Fractal heat conduction problem solved by local fractional variation iteration method. Therm.

Sci. (2012). doi:10.2298/TSCI121124216Y

28. Hu, MS, Agarwal, RP, Yang, XJ: Local fractional Fourier series with application to wave equation in fractal vibrating string. Abstr. Appl. Anal.2012, Article ID 567401 (2012)

29. Yang, XJ, Baleanu, D, Zhong, WP: Approximation solution to diffusion equation on Cantor time-space. Proc. Rom.

Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. (2013, in press)

30. Hu, MS, Baleanu, D, Yang, XJ: One-phase problems for discontinuous heat transfer in fractal media. Math. Probl. Eng.

2013, Article ID 358473 (2013)

31. Babakhani, A, Gejji, VD: On calculus of local fractional derivatives. J. Math. Anal. Appl.270(1), 66-79 (2002) 32. Chen, Y, Yan, Y, Zhang, K: On the local fractional derivative. J. Math. Anal. Appl.362, 17-33 (2010) 33. Kim, TS: Differentiability of fractal curves. Commun. Korean Math. Soc.20(4), 827-835 (2005)

34. Parvate, A, Gangal, AD: Fractal differential equations and fractal-time dynamical systems. Pramana J. Phys.64(3), 389-409 (2005)

35. Yang, XJ, Liao, MK, Wang, JN: A novel approach to processing fractal dynamical systems using the Yang-Fourier transforms. Adv. Electr. Eng. Syst.1(3), 135-139 (2012)