Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method
© Su et al.; licensee Springer 2013
Received: 23 January 2013
Accepted: 23 March 2013
Published: 10 April 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 .
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
if is satisfied the conditions and .
if is satisfied the conditions and .
with given conditions for .
for given condition .
3 The method
where is a general fractal Lagrange’s multiplier.
4 Solution of dissipative wave equation with a fractal string
5 Solution of damped wave equation with a fractal string
and so on.
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.
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).
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