Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method
Fixed Point Theory and Applications volume 2013, Article number: 89 (2013)
In this paper, the local fractional variational iteration method is given to handle the damped wave equation and dissipative wave equation in fractal strings. The approximation solutions show that the methodology of local fractional variational iteration method is an efficient and simple tool for solving mathematical problems arising in fractal wave motions.
MSC:74H10, 35L05, 28A80.
The variational iteration method was effectively applied in various fields of science and engineering [1–15] and the references therein. It is in some cases, more powerful than the existing techniques, e.g., the fractional variational iteration method [6, 16, 17], the homotopy perturbation method [18, 19], the exp-function method [20, 21], the decomposition method [22–24], the homotopy analysis method [25, 26] and others . The wave equation was investigated within some differential methods [7–15, 18–26] and the references therein.
As it is known, the quantum behavior of microphysics in terms of a non-differentiable space-time continuum possesses and has fractal property. Also, it was shown by many authors that a time-space structure of microphysics is non-differentiable. The relativistic quantum mechanics in fractal time space was suggested in . It was pointed out that, while the zero set represents the Cantor point-like quantum particle, the empty set was the basic mathematical representation of the quantum wave . The exact solutions for a class of fractal time random walks were researched in . The questions of a philosophical nature about fractal spacetime and its implications for phenomenology and ontology were shown in . The fractal time-space structure for dealing with the non-differentiability and infinities of fractals derived from local fractional operators was presented in [32–34] and the references therein. A solution of the wave equation in fractal vibrating string by using the local fractional Fourier series was discussed in . The diffusion equation on Cantor time-space was reported in  while the diffusion problems on fractal space were suggested in . The heat conduction problem by local fractional variational iteration method was investigated in . The heat conduction equation in fractal time space was structured in . A relaxation equation in fractal space was set up in . The anomalous diffusion equation in the fractal time-space fabric was pointed out in . The Fokker-Planck equation in fractal time was considered in .
Recently, fractional calculus analysis and fractional dynamics are hot topics [42–48]. In this paper, we consider a general wave equation of a fractal string within the local fractional operators, namely
and where is a local fractional linear operator, which has low order local fractional partial derivatives with respect to ξ, subject to fractal initial conditions
Thus, we obtain
and we have the following dissipative wave equation in fractal time space:
subject to initial conditions
If we start with
then we obtain the following damped wave equation given by
where the damping force is proportional to the velocity, a and b are constants, subject to initial conditions, which are suggested by the following expression:
More recently, the local fractional variational iteration method, which was structured in , was applied to solve heat conduction equation on Cantor sets  and the local fractional Laplace equation . The purpose of this paper is to present the solutions of the damped wave equation and the dissipative wave equation in fractal strings equipped with fractal initial conditions.
2 Mathematical tools
with as , or
Suppose that is satisfied the condition (10) for , we can denote 
if is satisfied the conditions and .
if is satisfied the conditions and .
if there are conditions for a partition of the interval given by
with given conditions for .
for given condition .
3 The method
Let us consider a general wave equation (1) subject to initial conditions as
We can construct a correction local fractional iteration algorithm given below
where is a general fractal Lagrange’s multiplier.
By using the local fractional integration by parts , we obtain
For the determination of the fractal Lagrange multiplier, the extremum condition of lead us to . By making use of Eq. (25), we have
This yields to the stationary conditions listed below:
Thus, we conclude that
From Eq. (28), the recurrence relation becomes
The function is selected by using the fractal initial conditions given as below:
Thus, the approximation expression becomes
4 Solution of dissipative wave equation with a fractal string
The dissipative wave equation with local fractional differential operator has the form
subjected to the fractal initial conditions
Making use of Eq. (29), the recurrence relation reads as
If the expression from Eq. (30) is given, we can determine the fractal initial conditions, which are expressed through
The first iteration yields
Thus, the second iteration reads
In similar manner, the third iteration is described by
The fourth iteration is suggested by
The fifth approximation is written as follows:
Proceeding in this manner, we can derive the following formula:
Finally, the compact solution becomes
5 Solution of damped wave equation with a fractal string
The damped wave equation with local fractional differential operator can be written in the form
and it is subjected to the initial conditions described by
Applying Eq. (29), we arrive at the following iteration formula:
By using Eq. (35), we obtain
Therefore, we deduce the first approximation as
The second approximation has the form
By using the same procedure, the third approximation becomes
Thus, we have
and so on.
Finally, the solution is given by
In this manuscript, utilizing the local fractional differential operators, we investigated the damped and the dissipative wave equations in fractal strings. Based on the local fractional variational iteration method, the solutions of the damped and dissipative wave equations were presented. The iteration functions, which is local fractional continuous, is obtained easily within the fractal Lagrange multipliers, which can be optimally determined by the local fractional variational theory . It is shown that the local fractional variational iteration method is an efficient and simple tool for handling partial differential equations with local fractional differential operator.
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Dedicated to Professor Hari M Srivastava.
The authors would like to thank the editor and the referees for their useful comments and remarks. The work is supported by the Natural Science Foundation of Tianjin, China (No. 10JCZDJC25100).
The authors declare that they have no competing interests.
Authors contributed equally and in writing this article. Authors read and approved the final manuscript.
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Su, WH., Baleanu, D., Yang, XJ. et al. Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method. Fixed Point Theory Appl 2013, 89 (2013). https://doi.org/10.1186/1687-1812-2013-89
- local fractional variational iteration method
- damped wave equation
- dissipative wave equation
- local fractional operators
- fractal strings