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We investigate solutions of the Helmholtz equation involving local fractional derivative operators. We make use of the series expansion method and the variational iteration method, which are based upon the local fractional derivative operators. The nondifferentiable solution of the problem is obtained by using these methods.

The Helmholtz equation is known to arise in several physical problems such as electromagnetic radiation, seismology, and acoustics. It is a partial differential equation, which models the normal and nonfractal physical phenomena in both time and space [

Fractional calculus theory [

Local fractional calculus theory [

The main objective of the present paper is to solve the Helmholtz equation involving the local fractional derivative operators by means of the local fractional series expansion method and the variational iteration method. The structure of the paper is as follows. In Section

The Helmholtz equation involving local fractional derivative operators was proposed.

Let us denote the local fractional derivative as follows [

Using separation of variables in nondifferentiable functions, the three-dimensional Helmholtz equation involving local fractional derivative operators was suggested by the following expression [

In this case, the two-dimensional Helmholtz equation involving local fractional derivative operators is expressed as follows (see [

The two-dimensional local fractional inhomogeneous Helmholtz equation is considered as follows (see [

The previous local fractional Helmholtz equations with local fractional derivative operators are applied to describe the governing equations in fractal electromagnetic radiation, seismology, and acoustics.

Let us consider a given local fractional differential equation

By the local fractional series expansion method [

In view of (

Let us consider the following local fractional operator equation:

By using the local fractional variational iteration method [

Following (

Let us consider the following Helmholtz equation involving local fractional derivative operators:

We now consider (

Applying the iterative relation equation (

So, we get

The result is the same as the one which is obtained by the local fractional series expansion method. The nondifferentiable solution is shown in Figure

Graph of

In this work, the nondifferentiable solution for the Helmholtz equation involving local fractional derivative operators is investigated by using the local fractional series expansion method and the variational iteration method. By using these two markedly different methods, the same solution is obtained. These two approaches are remarkably efficient to process other linear local fractional differential equations as well.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (nos. 11126213 and 61170317), and the National Natural Science Foundation of the Hebei Province (nos. A2012209043 and E2013209215).