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

Abstract

Abstract:In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is named the (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM). First the theories, their proofs and also some basic properties of this procedure are given. To understand the introduced method clearly, we apply it on the (n + 1)-dimensional fractal heat-like equations (HLEs) and wave-like equations (WLEs). The applications show that this new technique is efficient, simply applicable and has powerful effects in (n + 1)-dimensional local fractional problems.

On the (N+1)‐dimensional local fractional reduced differential transform method and its applications

“…In this section, we firstly recall the basic definitions of the local fractional derivative (also called fractal derivative), and the local fractional Taylor's theorem, which once were presented in these articles. [32][33][34][35][36][37]…”

“…Definition 5 (other studies [32][33][34][35][36] ). The local fractional partial derivative operator of f(x, t) of order with respect to t at the point (x, t 0 ) is given as follows:…”

“…In this subsection, we firstly describe for the (1+1)-dimensional LFEDTM on the knowledge of the above stated. [32][33][34]36 Considering a function of u(x, t), and suppose that it can be written as a product of two single-variable function, that is, u(x, t) = f(x)g(t). Based on the basis of the properties of one-dimensional differential transform, the function u(x, t) has the form of…”

“…Definition 6 (other studies [32][33][34]36 ). The (1+1)-dimensional LFRDT U k (x) of the local analytical function u(x, t) in the field Σ can be given by…”

“…[20][21][22] Recently, this method also applied to this type differential equation of local fractional derivative. 32,33 More recently, this method was further developed to solve higher order partial differential equations with local fractional derivative by Acan et al 33 However, they only gave limit properties and theorems for this method under the high-order partial differential equations.…”