^{1,2}

^{3,4,5}

^{6}

^{2}

^{7}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

The mappings for some special functions on Cantor sets are investigated. Meanwhile, we apply the local fractional Fourier series, Fourier transforms, and Laplace transforms to solve three local fractional differential equations, and the corresponding nondifferentiable solutions were presented.

Special functions [

The Mittag-Leffler function had successfully been applied to solve the practical problems [

Recently, based on the Mittag-Leffler functions on Cantor sets via the fractal measure, the special integral transforms based on the local fractional calculus theory were suggested in [

The paper is organized as follows. In Section

In order to give the mappings for special functions on Cantor sets, we first recall some basic definitions about the fractal measure theory [

Let Lebesgue-Cantor staircase function be defined as [

Following (

In this way, we define some real-valued functions on Cantor sets as follows [

The Cantor staircase function is defined as [

Graph of

The Mittag-Leffler functions on Cantor sets are given by [

Graph of

The sine on Cantor sets is defined by [

Graph of

The cosine on Cantor sets is [

Graph of

Hyperbolic sine on Cantor sets is defined by [

Graph of

Hyperbolic cosine on Cantor sets is defined as [

Graph of

Following (

If for

In this section, we introduce the conceptions of special integral transforms within the local fractional calculus concluding the local fractional Fourier series and Fourier and Laplace transforms. After that, we present three illustrative examples.

We here present briefly some results used in the rest of the paper.

Let

Let

For more details of special integral transforms via local fractional calculus, see [

We now present the powerful tool of the methods presented above in three illustrative examples.

Let us begin with the local fractional differential equation on Cantor set in the following form:

Following (

We now consider the following differential equation on Cantor sets:

Application of local fractional Fourier transform gives

Let us find the solution to the differential equation on Cantor sets

Taking the local fractional Laplace transform, from (

In this work, we investigated the mappings for special functions on Cantor sets and special integral transforms via local fractional calculus, namely, the local fractional Fourier series, Fourier transforms, and Laplace transforms, respectively. These transformations were applied successfully to solve three local fractional differential equations, and the nondifferentiable solutions were reported.