Non-Differentiable Exact Solutions for the Nonlinear ODEs Defined on Fractal Sets

In the present paper, a family of the special functions via the celebrated Mittag–Leffler function defined on the Cantor sets is investigated. The nonlinear local fractional ODEs (NLFODEs) are presented by following the rules of local fractional derivative (LFD). The exact solutions for these problems are also discussed with the aid of the non-differentiable charts on Cantor sets. The obtained results are important for describing the characteristics of the fractal special functions.


INTRODUCTION
Fractional ordinary differential equations (FODEs) [1][2][3] have been successfully used to model the complexity in mathematics, physics and societies, such as the fractional evolution, 4 control, 5 circuits, 6-9 relaxation 10-14 and population dynamics. 15 Finding the solutions for these mentioned models, many technologies were proposed in Ref. 16. For example, the Adomian decomposition technology (ADT) 17 and its extended version 18 were proposed to solve the approximate solutions for the FODEs. The spectral element method (SEM) 19 and the finite difference method (FDM) 20 and the linear multiple step method (LMSM) 21 were discussed to handle the numerical solutions for the FODEs. The technologies involving the differential transform (DT) 22 and the fractional operational calculus (FOC) 23 technologies were reported in order to find the analytical and exact solutions for FODEs, respectively.
Recently, fractional calculus (FC) was considered to solve a class of the fractal problems in mathematical physics, 24-28 mechanics, [29][30][31] heat, 32 biology 33 and others. [34][35][36][37] There is an alternative operator (called local FC) to model the local FODEs in fractal electric circuits, 38 free damped vibrations, 39 shallow water surfaces 40 and populations. [41][42][43] The fractal partial differential equations (FPDEs) in mathematical physics were also discussed in Refs. [44][45][46][47][48]. The structure solutions for the nonlinear local fractional ordinary differential equations (NLFODEs) have not been sufficiently investigated. Motivated especially by the above idea, our aim in the present article is to structure the NLFODEs by means of a family of the special functions via the Mittag-Leffler function defined on the Cantor sets.
The structure of the paper is designed as follows. In Sec. 2, the basic definitions of the local fractional derivative (LFD) and special functions defined on Cantor sets are introduced. In Sec. 3, we present the NLFODEs with the use of the LFDs of the special functions defined on the Cantor sets. Finally, we give the conclusion in Sec. 4.

NONLINEAR LOCAL FRACTIONAL ODES
In this section, we apply the results of the LFDs of the special functions defined on Cantor sets in order to structure the NLFODEs. Defining the following special functions on Cantor sets: and where ϕ 1 and ϕ 2 are two parameters, we find from Table 2 that and so that we get the following NLFODE: When ϕ 1 = 1 and ϕ 2 = 1, from Eq. (8), we get the NLFODE as follows: Non-Differentiable Exact Solutions for the Nonlinear ODEs Table 1 The Expressions of the Fractal Special Functions.

Fractal Special Functions Expressions
where the non-differentiable solution has the form given by Similarly, by taking the following special functions defined on Cantor sets: and we have and 1740002-3

Fractal Special Functions LFDs
so that we present the form of the NLFODE as follows: Thus, we easily structure from Eqs. (8) and (15), the following NLFODE: where the non-differentiable solutions can be written as follows: In a similar manner, we consider the following special functions defined on Cantor sets: and In view of Eqs. (18) and (19), we have and Thus, we directly obtain the following NLFODE: where ν is a parameter and the non-differentiable solutions can be given as follows: In a similar manner, we can structure the following NLFODE: where the non-differentiable solution is represented by Making use of Eqs. (23) and (25), we can derive the following NLFODE: where the non-differentiable solutions are given by Let us define the following special functions on Cantor sets:
From Eqs. (29)-(32), we can establish the following formulas: and and (40) so that where the non-differentiable solutions are given as follows: From Table 1, we set up the following special function defined on Cantor sets: where ρ, ϕ 1 , ϕ 2 and ϕ 3 are parameters. From Eq. (43), we easily have For finding the LFD of Eq. (43), we present Upon substituting Eqs. (44) and (45) into Eq. (46), we have When κ = 1, we find from Eq. (47) that together with the non-differentiable solution given by Similarly, we propose the following special function defined on Cantor sets: which leads to In order to find the LFD of Eq. (49), we have where ρ, ϕ 1 , ϕ 2 and ϕ 3 are parameters. When κ = ϕ 1 = 1 and ϕ 3 = 1, we find from Eq. (54) that (see Ref. 41) where the non-differentiable solution becomes (see with the parameters ρ and ϕ 2 .
Let us now consider the following special functions defined on Cantor sets: and By finding the LFDs of Eqs. (57) and (58), we have which yield and From Eqs. (61) and (62), we get which lead us to the following NLFODE: Thus, from Eq. (65), we easily obtain the following NLFODE: ), (66) where the non-differentiable solutions are presented as follows: By a similar process, we present the special functions defined on Cantor sets as follows: and which lead us to the following formulas: and respectively. Therefore, we find from Eqs. (70) and (71) that and which deduce to and respectively.