Received: 24 August 2017
Local fractional analytical methods for solving wave
equations with local fractional derivative
A. A. Hemeda E. E. Eladdad I. A. Lairje
Department of Mathematics, Faculty of
Science, Tanta University, Tanta 31527,
Atif A. Hemeda, Department of
Mathematics, Faculty of Science, Tanta
University, Tanta 31527, Egypt.
Communicated by: H. Bakouch
MSC Classification: 34K50; 34A12; 34A30;
In this article, we introduce the local fractional integral iterative method and
the local fractional new iterative method for solving the local fractional differ-
ential equations. Also, we perform a comparison between the results obtained
by these 2 local fractional methods with the results obtained by some other local
fractional methods. The obtained results illustrate the significant features of the
2 methods that are both very effective and straightforward for solving the differ-
ential equations with local fractional derivative compared with the other local
local fractional integral iterative method, local fractional new iterative method, local fractional
laplace equation, local fractional wave equations
Many problems of applied mathematics, physics, biology, and engineering are formulated by means of fractional differen-
tial equations with arbitrary orders.
We mention but few, fractional diffusion and wave,
and fractional Fisher equations.
The fractional differential equations were considered in the sense of the Caputo derivative and the Riemann-Liouville
However, they do not deal with the nondifferentiable functions defined on Cantor sets. Local fractional
is the best method for describing the nondifferentiable problems defined on Cantor sets. For example, the
heat equations arising in fractal transient conduction
and the Helmholtz and diffusion equations on the Cantor sets
within local fractional derivative.
Several analytical and numerical methods were successfully used to deal with local fractional differential equations,
such as the local fractional variational iterative method,
the Yang-Fourier transform,
the Yang-Laplace transform,
the local fractional Adomian decomposition method,
the local fractional function decomposition method (LFFDM),
and the local fractional Laplace variational iteration method.
Also, for the local fractional derivative and integral
transformations, you can see Yang et al.
The local fractional Laplace differential and wave equations are two of the most important equations in mathematics
and physics fields owing to their wide using and variety applications of these equations in many fields. The local fractional
Laplace differential equation is solved by Wang et al
by means of the LFFDM, and it is solved by Yan et al
by means of
the local fractional Adomian decomposition and function decomposition methods. The local fractional wave equation is
solved by Baleanu et al
using the local fractional variational iteration and decomposition methods and Wang et al
using the LFFDM.
Our goal, in this article, is to continue in the study of the local fractional methods by introducing 2 powerful methods:
the local fractional integral iterative method (LFIIM) and the local fractional new iterative method (LFNIM) for solving
the local fractional Laplace differential and wave equations. Moreover, we aim to compare the results obtained by these 2
Math Meth Appl Sci. 2018;41 2515–2529. wileyonlinelibrary.com/journal/mma Copyright © 2018 John Wiley & Sons, Ltd. 2515: