Analytical Approximate Solutions of ( n + 1 )-Dimensional Fractal Heat-Like and Wave-Like Equations

Omer Acan 1,* , Dumitru Baleanu 2,3, Maysaa Mohamed Al Qurashi 4 and Mehmet Giyas Sakar 5 1 Department of Mathematics, Faculty of Art and Science, Siirt University, Siirt 56100, Turkey 2 Department of Mathematics, Faculty of Art and Sciences, Çankaya University, Ankara 06790, Turkey; dumitru@cankaya.edu.tr 3 Institute of Space Sciences, Magurele-Bucharest 077125, Romania 4 Department of Mathematics, Faculty of Art and Science, King Saud University, Riyadh 11495, Saudi Arabia; maysaa@ksu.edu.sam 5 Department of Mathematics, Faculty of Science, Yuzuncu Yil University, Van 65080, Turkey; giyassakar@yyu.edu.tr * Correspondence: acan_omer@siirt.edu.tr; Tel.: +90-484-212-1111


Introduction
The importance of fractional calculus and its popularity have increased during the past four decades, due to its applications in many fields of engineering and applied science.For example, analysis of entropy in fractional dynamical systems, entropy in thermodynamics, control theory of dynamic systems, probability and statistics, electrical networks, signal processing, optics, chemical physics, the electrochemistry of corrosion and so on can all be successfully modelled by fractional order differential equations .
In thermodynamics, entropy is known as a state function of a thermodynamic system.The production of entropy by fractional calculus was suggested in [4].The production of entropy rate for fractional diffusion processes was discussed in [6][7][8].The analysis of entropy in fractional dynamic systems was proposed in [10].However, these entropy processes are differentiable.Non-differentiable production of entropy in heat conduction of the fractal temperature field was studied in [13].The heat conduction equation was discussed by the help of local fractional derivative (LFD) [26].Some numerical methods are applied to many non-differentiable problems in Cantor sets by using LFD [21][22][23][24][25][26][27][28][29].
In our present study, the basic definitions of local fractional calculus are given in Section 2. Two-dimensional LFRDTM and (n + 1)-dimensional LFRDTM with the basic definitions and theorems are presented in Section 3. In Section 4, the applications of the new method, graphics of the solutions and discussion are given and finally, we put forth our conclusions in Section 5.

Main Results
In this section, we describe two-dimensional LFRDTM and (n + 1)-dimensional LFRDTM.

Two-Dimensional LFRDTM
In this subsection, we recall and review briefly the local fractional Taylor theorems, and then, we extend two-dimensional LFRDTM.
Using Definitions 3 and 4, Based on results of [28], the fundamental mathematical operations of the two-dimensional LFRDTM are presented in Table 2: Table 2. Basic operations of the two-dimensional LFRDTM.

Original Function
Transformed Function In Table 2, the lowercase ψ(x, t), π(x, t) and ϕ(x, t) represent the local fractional analytic original functions while the uppercase Ψ k (x), Π k (x) and Φ k (x) stand for LFRDT functions.a and b are constants.
Proof.From Definition 5, we have: The proof is thus completed.
Now solve this problem by using (n + 1)-dimensional LFRDTM.By taking the (n + 1)-dimensional LFRDT of (20), it can be obtained that: The (n + 1)-dimensional LFRDT of the IC in ( 21) is given by: By using ( 23) in ( 22), we can obtain the following Ψ k (x, y) values successively: From ( 24), the {Ψ k (x, y)} n k=0 values give the following approximation solution: Hence, from (25), ψ(x, y, t) is: This finding is the exact solution of the (2 + 1)-dimensional local fractional homogeneous HLE (20) on the Cantor set.The graph of this solution is given in Figure 1 for α = ln 2 ln 3 .
Using (n + 1)-dimensional LFRDTM, Equation ( 27) transforms to: From the IC (28), we write: From ( 30) and ( 29), the following Ψ k (x, y, z) values can be obtained: From (31), the following approximation solution can be written as: Hence, from (32), ψ(x, y, z, t) is: This result is the exact solution of the (3 + 1)-dimensional local fractional inhomogeneous HLE (27) on the Cantor set.The graph of this solution is given in Figure 2 for α = ln2/ln3.( This result is the exact solution of the (3 + 1)-dimensional local fractional inhomogeneous HLE (27) on the Cantor set.The graph of this solution is given in Figure 2 x y t x y with the ICs: , , x y t = ψ ψ .
We solve this problem by using (n + 1)-dimensional LFRDTM.By taking (n + 1)-dimensional LFRDT of (34), it can be obtained that: From the ICs (35), it can be written as follows: By using (37) in (36), we can obtain the following Ψ k (x, y) values successively: . . .

Conclusions
In this paper, a new technique, (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM), was presented to find the analytical approximate solutions of local fractional PDEs.Then, the new method was applied to (n + 1)-dimensional fractal HLEs and WLEs.In the applications, our method directly gave us the exact solution for the problems without any transformation, discretization and any other restrictions.Physical behaviors of the solutions on fractal spaces were illustrated using 3D graphics.The results showed that presented method gives good outcomes for solutions of (n + 1)dimensional local fractional PDEs.Hence, our results suggest that the new procedure (n + 1)-dimensional LFRDTM is reliable, useful and simplify for local fractional PDEs to solve many complicated fractal problems.