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Ternaryfractional differential transform schema: theory and application
Advances in Difference Equations volume 2019, Article number: 197 (2019)
Abstract
In this article, we propose a novel fractional generalization of the threedimensional differential transform method, namely the ternaryfractional differential transform method, that extends its applicability to encompass initial value problems in the fractal 3D space. Several illustrative applications, including the Schrödinger, wave, Klein–Gordon, telegraph, and Burgers’ models that are fully embedded in the fractal 3D space, are considered to demonstrate the superiority of the proposed method compared with other generalized methods in the literature. The obtained solution is expressed in a form of an α̅fractional power series, with easily computed coefficients, that converges rapidly to its closedform solution. Moreover, the projection of the solutions into the integer 3D space corresponds with the solutions of the classical copies for these models. This reveals that the suggested technique is effective and accurate for handling many other linear and nonlinear models in the fractal 3D space. Thus, research on this trend is worth tracking.
Introduction
Over recent years, it has been demonstrated that many physical phenomena can be successfully reformulated by means of noninteger order differential equations due to the inefficiency of the integerorder differential equations in modeling certain issues. For instance, when modeling chaotic thermodynamic systems, it is indispensable to use a noninteger model because the separation of timescales of classical physics does not work adequately [1]. The noninteger order derivatives are commonly called fractional order derivatives or simply fractional derivatives. To mention a few of the phenomena that can be modeled by fractional derivatives: the electromagnetic transient phenomenon in transmission lines is governed by the fractional diffusion model [2], the damping properties of the viscoelastic material are related to the fractional Kelvin–Voigt model [3], and the interaction of solitons in a collisionless plasma is simulated by the fractional Klein–Gordon model [4].
The fractional derivatives are in nonlocal nature, unlike the integerorder derivatives. Consequently, it is often stated that the physical interpretation of the fractional derivative order α is hard to pin down or does not exist at all. However, many researchers have attempted to attribute physical meanings to fractional derivatives, see, for example, [5, 6]. It has been demonstrated in certain circumstances that the nonlocality feature of the fractional derivatives makes them appropriate for describing the memory and hereditary features of various materials. Thus, the fractional derivative order α can be physically described as an index of memory [7,8,9,10,11,12,13].
Differential transform method (DTM) is a powerful transformation technique that can be easily applied to (non)linear problems to achieve much more accurate analytical or numerical solutions than the existing ones in the literature. One of the distinguishing features of this method is its capability to reduce the size of computational work. The concept of the onedimension DTM was introduced for the first time by Zhou [14] in 1986 to solve problems related to engineering models in electric circuit analysis. Posteriorly, Chen and Ho [15] developed this method to solve twodimensional PDEs. Later, the threedimensional DTM was introduced by Ayaz [16].
Over the past few years, the DTM has been successfully implemented in the area of partial differential equations of fractional order. Arikoglu and Ozkol [17] proposed an analytical technique, called the fractional differential transform method (FDTM), for solving (non)linear differential equations endowed with one memory index α. Very recently, Jaradat et al. [18, 19] developed this method to address (non)linear differential equations endowed with two memory indices \(\alpha _{1}\) and \(\alpha _{2}\). It is worth mentioning here that some recent advancements in analytical methods can be also found in [20,21,22,23,24]. In this article, we present a novel fractional generalization of the threedimensional DTM to extend the application of the DTM to (non)linear differential equations endowed with three memory indices \(\alpha _{1}\), \(\alpha _{2}\), and \(\alpha _{3}\). Comparing with the complexity of these equations, the suggested generalization is efficacious and easily applicable. Several examples were carried out to demonstrate the efficiency of the proposed technique.
The rest of this article proceeds as follows. We amalgamate the DTM with a new ternaryfractional power series in Sect. 2 to handle physical differential equations in the fractal 3D space. Then, in Sect. 3, we employ the new formulation of DTM to provide a full fractional solution of several wellknown physical models in the fractal 3D space. In Sect. 4, we offer a potential interpretation for the fractional derivative order. Finally, we present concluding remarks in Sect. 5.
Ternaryfractional differential transform schema
In this section, we introduce and investigate a developed analytical scheme, derived from a novel fractional version of the Taylor series expansion, to handle various (non)linear differential equations in the fractal 3D space. This new technique generalizes the classical threedimensional differential transform ideas in the fractal 3D space and provides new insights for analytically studying the combined effects of three distinct memory indices. The ternaryfractional power series is defined as follows.
Definition 2.1
([25])
An α̅fractional power series (α̅FPS) around \((0,0,0)\) is a fractional power series in the following Cauchy form:
where \(\overline{\pmb{\alpha }}=(\alpha _{1}, \alpha _{2}, \alpha _{3}) \in (0,1)^{3}\), x, y, t are nonnegative variables of indeterminate, and \(a_{i,j,k}\)’s are real constant coefficients.
It is readily verified that expansion (2.1) can be expressed as
Definition 2.2
The ternaryfractional differential transform (ternaryFDT) of a function \(w(x,y,t)\) is
for \((i,j,k)\in \mathbb{N}_{*}^{3} \).
Remark 1
We should point out here that the fractional derivative adopted in this work is the Caputo sense, although our work requires only the fractional derivative of the power function which almost all the existing fractional derivative definitions agree with. The fractional derivative of the power function is given by
It is worth mentioning here that different representations of fractional derivatives have recently been proposed based on the exponential law [26] and on the Mittag—Leffler function [27]. Some remarkable works in these trends can be found in [28,29,30].
Remark 2
In case of \({\overline{\pmb{\alpha }}}\rightarrow (1,1,1)\), then ternaryFDT (2.3) reduces to the classical threedimensional differential transform [16].
Definition 2.3
The ternaryfractional differential inverse transform of is defined as
Remark 3
Definition 2.3 evinces that the concept of ternaryFDT is derived from the α̅FPS.
With the aid of equations (2.3) and (2.5), some fundamental properties for ternaryFDT are shown in the following theorem.
Theorem 2.4
Let , and be the ternaryFDT of \(w(x,y,t)\), \(u(x,y,t)\), and \(v(x,y,t)\) respectively, and c be an arbitrary constant. Then the following properties hold true.
 (i):

If \(w(x,y,t)=u(x,y,t)\pm c v(x,y,t)\), then .
 (ii):

If \(w(x,y,t)=u(x,y,t)\cdot v(x,y,t)\), then .
 (iii):

If \(w(x,y,t)= {\frac{\partial ^{n\alpha _{1}} [u(x,y,t) ]}{\partial t^{n\alpha _{1}}}}\), then .
 (iv):

If \(w(x,y,t)= {\frac{\partial ^{n\alpha _{1}+m\alpha _{2}} [u(x,y,t) ]}{\partial t^{n\alpha _{1}}\partial x^{m\alpha _{2}}}}\), then .
 (v):

If \(w(x,y,t)= {\frac{\partial ^{n\alpha _{1}+m\alpha _{2}+l\alpha _{3}} [u(x,y,t) ]}{\partial t^{n\alpha _{1}}\partial x^{m\alpha _{2}}\partial y^{l \alpha _{3}}}}\), then
Proof
(i) Follows immediately from Definition 2.2 and the linearity of the fractional derivatives.
(ii)
as desired.
(iii)
(iv) and (v) The proof can be concluded by using the same manner of proof as for (iii). □
Some ternaryFDT for some basic functions around the origin are listed in Table 1.
The application side of the suggested method
In this section, we show the worthiness of the suggested technique by solving various wellknown partial differential equations that are viewed in the fractal 3D space. The resulting solutions generalized the existing solutions when these equations were projected into the integer 3D space. In all examples, the fractional derivative parameters are assumed to be in \((0,1)\) and \(t, x, y\geq 0\).
Example 1
Consider the following Schrödinger model embedded in the fractal 3D space:
contingent on the initial condition
By implementing the ternaryFDT and harnessing Theorem 2.4, we attain the following transformed recurrence equation of (3.1):
with initial transform coefficients
By recursively solving equation (3.3) with utilizing the initial transform coefficients (3.4), we acquire the following generic transform coefficients:
Therefore, the α̅memory exact solution of the Schrödinger model (3.1)–(3.2) has the form
We remark here that for \({\overline{\pmb{\alpha }}}\rightarrow (1,1,1)\), we attain the closedform solution \(w(x,y,t)=e^{\mathbf{i}t} (\sin (x)+\sin (y) )\) for the integer copy of the Schrödinger model (3.1)–(3.2).
Example 2
Consider the following wave model embedded in the fractal 3D space:
contingent on the initial conditions
By implementing the ternaryFDT and harnessing Theorem 2.4, we attain the following transformed recurrence equation of (3.7):
with initial transform coefficients
and
By recursively solving equation (3.9) with utilizing the initial transform coefficients (3.10)–(3.11), we acquire the following generic transform coefficients:
Therefore, the α̅memory exact solution of the wave model (3.7)–(3.8) has the form
We remark here that for \({\overline{\pmb{\alpha }}}\rightarrow (1,1,1)\), we attain the closedform solution \(w(x,y,t)=\cos (2t)\sin (x)\sin (y)\) for the integer copy of the wave model (3.7)–(3.8).
Example 3
Consider the following nonhomogeneous Klein–Gordon model embedded in the fractal 3D space:
contingent on the initial conditions
By implementing the ternaryFDT and harnessing Theorem 2.4, we attain the following transformed recurrence equation of (3.14):
with initial transform coefficients
and
By recursively solving equation (3.16) with utilizing the initial transform coefficients (3.17)–(3.18), we acquire, besides the initial transforms, the following generic transform coefficients:
Therefore, the α̅memory exact solution of the Klein–Gordon model (3.14)–(3.15) has the form
We remark here that for \({\overline{\pmb{\alpha }}}\rightarrow (1,1,1)\), we attain the closedform solution \(w(x,y,t)=x^{2}+y^{2}+ty^{2}+t^{2}+ \frac{1}{3}t^{3}\) for the integer copy of the Klein–Gordon model (3.14)–(3.15).
Example 4
Consider the following homogeneous linear telegraph model embedded in the fractal 3D space:
contingent on the initial conditions
By implementing the ternaryFDT and harnessing Theorem 2.4, we attain the following transformed recurrence equation of (3.21):
with initial transform coefficients
By recursively solving equation (3.23) with utilizing the initial transform coefficients (3.24), we acquire the following generic transform coefficients:
Therefore, the α̅memory exact solution of the telegraph model (3.21)–(3.22) has the form
We remark here that for \({\overline{\pmb{\alpha }}}\rightarrow (1,1,1)\), we attain the closedform solution \(w(x,y,t)=e^{x+y3t}\) for the integer copy of the telegraph model (3.21)–(3.22).
Example 5
Consider the following nonlinear Burgers’ model embedded in the fractal 3D space:
contingent on the initial condition
By implementing the ternaryFDT and harnessing Theorem 2.4, we attain the following transformed recurrence equation of (3.27):
with initial transform coefficients
By recursively solving equation (3.29) with utilizing the initial transform coefficients (3.30), we acquire the following relation among the transform coefficients:
where is given recursively by
and for \(i\geq 2\)
Therefore, the α̅memory solution of Burgers’ model (3.27)–(3.28) is
We should point out here that as \({\overline{ \pmb{\alpha }}}\rightarrow (1,1,1)\). Thus, the closedform solution for the integer copy of (3.27)–(3.28) is
Graphical analysis and discussions
In the present part, we choose Burgers’ model (3.27)–(3.28) to illustrate and comprehend the impact of the Caputo fractional derivative. Figure 1 shows a sample of the crosssections attitude for the 8th series solution of (3.34) for diverse values of the fractional parameters in particular domains. In all situations, we have continuous successive behavior as long as the fractional parameters get closer to the integer derivative order. Therefore, the role of ordervariation of the fractional derivative is to preserve a homotopy mapping from the present value of the solution “\({\overline{\pmb{\alpha }}}\rightarrow (0,0,0)\)” to its instantaneous rate of change “\({\overline{\pmb{\alpha }}}\rightarrow (1,1,1)\)”. This interpretation is made by the phenomena of a sequentialasymptotic propagation of the solution as the fractional parameters vary from 0 to 1 as depicted in Fig. 1. Further, the crosssections of the integer derivative order reconcile with their associates when \({\overline{ \pmb{\alpha }}}\rightarrow (1,1,1)\). This shows the generality of α̅Burgers’ model.
Conclusions
In this work, we have viewed several wellknown partial differential equations in the fractal 3D space and provided their solutions analytically in terms of an α̅FPS. An adaptation of the fractional differential transform method by means of a new α̅FPS representation is developed and used to obtain a complete fractional solution form of these new models. The projections of these solutions into the integer 3D space reconcile with the solutions of the classical copies of these models. Moreover, the proposed method has been autonomously constructed without the need to convert the equation into solvable and perturbation or linearization terms. In summary, we have successfully provided a comprehensive analytic study of partial differential equations that are fully embedded in the fractal 3D space.
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The authors thank the editor and anonymous reviewers for their valuable suggestions, which substantially improved the quality of the paper.
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The work of the first, third, and fourth authors was supported by the Deanship of Scientific Research at the University of Jordan.
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Yousef, F., Alquran, M., Jaradat, I. et al. Ternaryfractional differential transform schema: theory and application. Adv Differ Equ 2019, 197 (2019). https://doi.org/10.1186/s136620192137x
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MSC
 26A33
 34A25
 35R11
Keywords
 Fractional derivative
 PDEs in fractal 3D space
 Ternaryfractional differential transform