Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

Abstract

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

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Ming-Sheng Hu 1 and Ravi P. Agarwal 2 and Xiao-Jun Yang 3

Recommended by Lan Xu

1, Institute of Software Science, Zhengzhou Normal University, Zhengzhou 450044, China
2, Department of Mathematics, Texas A and M University, Kingsville, TX 78363-8202, USA
3, Department of Mathematics and Mechanics, China University of Mining and Technology, Jiangsu, Xuzhou 221008, China

Received 14 August 2012; Revised 25 October 2012; Accepted 8 November 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Fractional calculus arises in many problems of physics, continuum mechanics, visco-elasticity, and quantum mechanics, and other branches of applied mathematics and nonlinear dynamics have been studied [ 1- 7]. In general, the fractional analogues are obtained by changing the classical time derivative by a fractional one, which can be Riemann-Liouville, Caputo, or another one. Many classical partial differential equations possess a fractional analogue, like the fractional diffusion-wave equation [ 8- 12], the fractional diffusion equation [ 13- 16], the fractional wave equation [ 17, 18], the fractional Schrödinger equation [ 19, 20], the fractional heat equation [ 21], the fractional KdV equation [ 22], the fractional Fokker-Planck equations [ 23], the fractional Fick's law [ 24], the fractional evolution equation [ 25], the Fractional Heisenberg equation [ 26], the fractional Ginzburg-Landau equation [ 27], Fractional hydrodynamic equation [ 28], the fractional seepage flow equation [ 29], and the fractional KdV-Burgers equation [ 30].

There also are other methods for solving fractional differential equations, for example, the fractional variational iteration method [ 31, 32] and the fractional complex transform [ 33- 37]. In all of the methods mentioned above, the solutions of the fractional differential equations should be analytical if the fractional derivative is in the Caputo or Riemann-Liouville sense. However, some solutions to ordinary and partial differential equations are fractal curves. As a result, we cannot employ the classical Fourier series, which requires that the defined functions should be differentiable, to describe some solutions to ordinary and partial differential equations in fractal space. However, based on the modified Riemann-Liouville derivative, Jumarie structured a Jumurie's calculus of fractional order [ 38] (which is one of useful tools to deal with everywhere continuous but nowhere differentiable functions) and its applications were taken into account in Probability calculus of fractional order [ 39], Laplace transform of fractional order via the Mittag-Leffler function (in convenient Hilbert space) [ 40], and adomian decomposition method for nonsmooth initial value problems [ 41]. Local fractional calculus is revealed as one of useful tools to deal with everywhere continuous but nowhere differentiable functions in areas ranging from fundamental science to engineering [ 42- 57]. For these merits, local fractional calculus was successfully applied in the local fractional Laplace problems [ 53, 54], local fractional Fourier analysis [ 53, 54], local fractional short time transform [ 53, 54], local fractional wavelet transform [ 53- 55], fractal signal [ 55, 56], and local fractional variational calculus [ 57].

In this paper we introduce a local fractional wave equation in fractal vibrating string which is described as [figure omitted; refer to PDF] with fractal boundary conditions [figure omitted; refer to PDF] where ^{∂ 2 α} u ( x ,t ) / ∂ ^{t 2 α} , ^{∂ 2 α} u ( x ,t ) / ∂ ^{x 2 α} , ^{∂ α} u ( l ,0 ) / ∂ ^{x α} , and ^{∂ α} u ( l ,t ) / ∂ ^{x α} are local fractional partial differential operator, and where u ( x ,t ) is local fractional continuous (for more details, see [ 53, 54]). We study the technique of the local fractional Fourier series for treating the local fractional wave equation in fractal vibrating string. This paper is organized as follows. In Section 2, we specify and investigate the concepts of local fractional calculus and local fractional Fourier series. In Section 3, we present the solving process for local fractional wave equation with local fractional derivative. In Section 4, we study the expression solution with Mittag-Leffler functions in fractal space. Finally, Section 5is conclusions.

2. Preliminaries

In this section we start with local fractional continuity of functions, and we introduce the notions of local fractional calculus and local fractional Fourier series.

2.1. Local Fractional Continuity of Functions

In order to discuss the local fractional continuity of nondifferential functions on fractal sets, we first consider the following results.

Lemma 2.1 (see [ 57]).

Let F be a subset of the real line and be a fractal. If f : ( F ,d ) [arrow right] ( ^{Ω [variant prime]} , ^{d [variant prime]} ) is a bi-Lipschitz mapping, then there are for constants ρ , τ >0 , and F ⊂R , [figure omitted; refer to PDF] such that for all _{x 1} , _{x 2} ∈F , [figure omitted; refer to PDF]

As a direct result of Lemma 2.1, we have, [ 57], [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where α is fractal dimension of F . The result that is directly deduced from fractal geometry is related to fractal coarse-grained mass function ^{γ α} [ F ,a ,b ] , which reads, [ 57], [figure omitted; refer to PDF] with [figure omitted; refer to PDF] where ^{H α} is α dimensional Hausdorff measure.

Notice that we consider the dimensions of any fractal spaces (e.g., Cantor spaces or like-Cantor spaces) as a positive number. It looks like Euclidean space because its dimension is also a positive number. The detailed results had been considered in [ 53, 54, 57].

Definition 2.2.

If there exists, [ 53, 57], [figure omitted; refer to PDF] with | x - _{x 0} | < δ , for [straight epsilon] , δ >0 and [straight epsilon] , δ ∈R , then f ( x ) is called local fractional continuous at x = _{x 0} , denoted by _{lim x [arrow right] x 0} f ( x ) =f ( _{x 0} ) . f ( x ) is called local fractional continuous on the interval ( a ,b ) , denoted by [figure omitted; refer to PDF] if ( 2.7) is valid for x ∈ ( a ,b ) .

Definition 2.3.

If a function f ( x ) is called a nondifferentiable function of exponent α , 0 < α ...4;1 , which satisfies Hölder function of exponent α , then for x ,y ∈X such that, [ 54, 57], [figure omitted; refer to PDF]

Definition 2.4.

A function f ( x ) is called to be continuous of order α , 0 < α ...4;1 , or shortly α continuous, when we have that, [ 54, 57], [figure omitted; refer to PDF]

Remark 2.5.

Compared with ( 2.10), ( 2.7) is standard definition of local fractional continuity. Here ( 2.9) is unified local fractional continuity [ 57].

2.2. Local Fractional Derivatives and Integrals

Definition 2.6 (let f ( x ) ∈ _{C α} ( a ,b ) ).

Local fractional derivative of f ( x ) of order α at x = _{x 0} is given, [ 53- 57], [figure omitted; refer to PDF] where ^{Δ α} ( f ( x ) -f ( _{x 0} ) ) [congruent with] Γ ( 1 + α ) Δ ( f ( x ) -f ( _{x 0} ) ) .

For any x ∈ ( a ,b ) , there exists, [ 53- 57], [figure omitted; refer to PDF] denoted by [figure omitted; refer to PDF] Local fractional derivative of high order is derived as, [ 57], [figure omitted; refer to PDF] and local fractional partial derivative of high order, [ 57], [figure omitted; refer to PDF]

Definition 2.7 (let f ( x ) ∈ _{C α} ( a ,b ) ).

Local fractional integral of f ( x ) of order α in the interval [ a ,b ] is given by, [ 53- 57], [figure omitted; refer to PDF] where Δ _{t j} = _{t j +1} - _{t j} , Δt =max { Δ _{t 1} , Δ _{t 2} , Δ _{t j} , ... } , and [ _{t j} , _{t j +1} ] , j =0 , ... ,N -1 , _{t 0} =a , _{t N} =b , is a partition of the interval [ a ,b ] .

For convenience, we assume that [figure omitted; refer to PDF] For any x ∈ ( a ,b ) , we get, [ 53, 54, 57], [figure omitted; refer to PDF] denoted by [figure omitted; refer to PDF]

Remark 2.8.

If f ( x ) ∈ ^{D x ( α )} ( a ,b ) , or ^{I x ( α )} ( a ,b ) , we have that, [ 46, 47, 50], [figure omitted; refer to PDF]

2.3. Special Functions in Fractal Space

Definition 2.9.

The Mittag-Leffler function in fractal space is defined by, [ 53, 57], [figure omitted; refer to PDF]

Definition 2.10.

The sine function in fractal space is given by the expression, [ 54, 57], [figure omitted; refer to PDF]

Definition 2.11.

The cosine function in fractal space is given, [ 54, 57], [figure omitted; refer to PDF] The following rules hold [ 54, 57]: [figure omitted; refer to PDF]

Remark 2.12.

^{i α} is fractal imaginary unit, for more details, see [ 53- 57].

2.4. Local Fractional Fourier Series

Definition 2.13.

Suppose that f ( x ) ∈ _{C α} ( - ∞ , ∞ ) and f ( x ) be 2l -periodic. For k ∈Z , local fractional Fourier series of f ( x ) is defined as, [ 53- 55], [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are the local fractional Fourier coefficients.

For local fractional Fourier series ( 2.25), the weights of the fractional trigonometric functions are calculated as [figure omitted; refer to PDF]

Definition 2.14.

Suppose that f ( x ) ∈ _{C α} ( - ∞ , ∞ ) and f ( x ) be 2l -periodic. For k ∈Z , complex generalized Mittag-Leffler form of local fractional Fourier series of f ( x ) is defined as, [ 53, 54], [figure omitted; refer to PDF] where the local fractional Fourier coefficients is [figure omitted; refer to PDF] The above generalized forms of local fractional series are valid and are also derived from the generalized Hilbert space [ 53, 54].

For local fractional Fourier series ( 2.28), the weights of the Mittag-Leffler functions are written in the form [figure omitted; refer to PDF] Above is generalized to calculate local fractional Fourier series.

3. Solutions to Wave Equation with Fractal Vibrating String

Now we look for particular solutions of the form [figure omitted; refer to PDF] and arrive at the equations [figure omitted; refer to PDF] with the boundary conditions [figure omitted; refer to PDF]

Equation has the solution [figure omitted; refer to PDF] According to ( 3.4), for x =0 and x =l we derive as [figure omitted; refer to PDF]

Assuming that _{C 2} ...0;0 , since otherwise [varphi] ( x ) is identically zero, we find that [figure omitted; refer to PDF] where n is an integer; we write [figure omitted; refer to PDF] For ^{λ α} = ^{λ n α} equation ( 3.3) leads to [figure omitted; refer to PDF] and therefore [figure omitted; refer to PDF]

To solve our problem, we form the local fractional Fourier series [figure omitted; refer to PDF] and require that [figure omitted; refer to PDF] A calculation of local fractional Fourier coefficients of f ( x ) and g ( x ) with respect to the system { si _{n α} ^{n α ( πx / l ) α} } is given by [figure omitted; refer to PDF] But ^{∫ 0 l sin α 2 n α ( πx /l ) α ( dx ) α} = ^{l α} / 2 and therefore [figure omitted; refer to PDF]

Thus, the solution of our problem is given by formula ( 3.11), where local fractional Fourier coefficients are determined. From ( 3.14) and ( 3.16), we get the harmonic vibrations [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

4. Expression Solutions with Mittag-Leffler Functions in Fractal Space

Taking into account the relations, [ 57], [figure omitted; refer to PDF] we obtain the harmonic vibration with the Mittag-Leffler functions in fractal space [figure omitted; refer to PDF] where its coefficients are [figure omitted; refer to PDF]

Hereby, we always find that [figure omitted; refer to PDF] with | x - _{x 0} | < δ , for [straight epsilon] , δ >0 and [straight epsilon] , δ ∈R .

Hence the boundary conditions are fractal and solution with Mittag-Leffler functions in fractal space is given by [figure omitted; refer to PDF] where its coefficients are derived as [figure omitted; refer to PDF]

5. Conclusions

We applied the technique of the local fractional Fourier series to treat with the local fractional wave equation in fractal vibrating string. When contrasted with other analytical methods, such as the heat-balance integral method, the homotopy perturbation method [ 11], the variational iteration method [ 29], the exp-function method [ 58], the fractional variational iteration method [ 31, 32], the fractional complex method [ 33- 37], and others [ 59- 61], the present method combines the following two advantages. The boundary conditions to the governing equations are local fractional continuous (the functions are nondifferential functions in fractal space) because we employ the local fractional Fourier series, derived from local fractional calculus, to deal with them. The governing equations with fractal behaviors in media are structured based on the local fractional calculus. The way plays a crucial role in local fractional calculus. This technique is efficient for the applied scientists to process these differential equations with the local fractional differential operators in fractal space. This paper that is an outstanding example of application of local fractional Fourier series to the local fractional differential operators is given to elucidate the solution processes and reliable results.

Acknowledgments

This paper is sponsored by the National Natural Science Foundation of China (NSFC, Grant U1204703), the Key Scientific and Technological Project of Henan Province (122102310004), the Fundamental Research Funds for the Central Universities (HUST: 2012QN087, 2012QN088), and the Innovation Scientists and Technicians Troop Construction Projects of Zhengzhou City (10LJRC190, 121PRKXF658-4).

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