Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method

In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to be one of the best analytical techniques, in order to solve fractional-order linear and nonlinear Partial deferential equations, particularly fractional-order heat and wave equation.


Introduction
The idea of entropy and fractional calculus are attractive, further prevalent for investigating the dynamics of complex schemes. In recent years, fractional calculus (FC) has been progressively applied in various fields of science. Natural development identified with viscoelasticity, models of porous electrodes, thermal stresses, electromagnetism, propagation of energy in dissipative systems, relaxation vibrations and thermoelasticity are effectively portrayed by fractional differential equations (FDE's) [1]. The knowledge of entropy was presented in the field of thermodynamics by Clausius (1862) and Boltzmann (1896) and was further applied by Shannon (1948) and Jaynes (1957) in information theory. Newly, more universal entropy measures have been suggested for applications in numerous varieties of complex systems outstanding for the relaxation of the additives axiom [2]. The concept of entropy for analyzing the dynamics of multi-particle systems with an integer and fractional order behavior. The entropy production rate for the fractional diffusion procedure was considered in [3,4]. In [5], it has been shown that the total spectral entropy can be used as a measure of the data comfortable in a fractional order model of anomalous diffusion. Entropies based on fractional calculus [6], integer and fractional dynamical systems can be solved by entropy analysis [7], nonlinear partial differential equations and third-order dispersive [8,9] in entropy and convexity. Bifurcation and recurrent analysis of memristive circuits [10], density analysis of multi-wing and multi-scroll chaotic systems [11], ∂ γ υ ∂t γ = g(x, y, z)υ x,x + h(x, y, z)υ y,y + k(x, y, z)υ z,z with initial condition υ(x, y, z, 0) = u(x, y, z), υ t (x, y, z, 0) = p(x, y, z).
Natural transform and Adomian decomposition methods are two powerful methods that have been used to develop the natural transform decomposition method. Many physical phenomena which are modeled by PDE and FPDEs are solved by using NTDM, such as the analytical solution of a couple of systems of nonlinear PDE's is suggested in [28], the solution nonlinear ODE's are successfully presented in [29], nonlinear PDEs [30], fractional unsteady flow of a polytropic gas model [31], fractional telegraph equations [32], fractional Fokker-Plank equation and Schrödinger equation [33]. The accuracy of the proposed method is compared with the solutions obtained by HPM and Modified homotopy perturbation method (MHPM). The comparisons has shown that the proposed has a higher rate of convergence than HPM and MHPM. The rest of the article is structured as: in Section 2, we recall several basic properties and definitions from natural transform and fractional calculus. In Section 3, we present the idea of the natural transform decomposition method. In Section 4, we explain many problems with maintaining the accuracy and efficiency of the proposed method, while the last section is devoted to conclusions.

Preliminaries
Definition 1. The natural transform of g(t) is defined as [34,35]: where s and u are the transform variables.

Definition 2. The inverse natural transform of a function is defined by
where s and u are the natural transform variables and p is a real constant and the integral is taken along s = p in the complex plane s = x + iy. Definition 3. Natural Transform of nth Derivative If g n (t) is the nth derivative of function g(t) is given by where Γ denotes the gamma function defined by In this study, Caputo et al. suggested a revised fractional derivative operator in order to overcome inconsistency measured in the Riemann-Liouville derivative. The above mathematical statement described a Caputo fractional derivative operator of initial and boundary conditions for fractional as well as integer order derivatives [36,37]. Definition 5. The Caputo operator of order γ for a fractional derivative is given by the following mathematical expression for n ∈ N, x > 0, g ∈ C t , t ≥ −1 [38]:

Idea of the Fractional Natural Transform Decomposition Method
In this section, the natural transform decomposition method to find the general solution fractional-order equations: where D γ = ∂ γ ∂t γ the Caputo Operator γ, m ∈ N, where L and N are linear and nonlinear functions, and q is the source function.
The initial condition is Applying the natural transform to Equation (1), we have and using the differentiation property of natural transform, we get The NTDM solution υ(x, t) is represented by the following infinite series: and the nonlinear terms (if any) in the problem are defined by the infinite series of Adomian polynomials, substitution Equation (5) and Equation (6) in Equation (4), we get Applying the linearity of the natural transform, Applying the inverse natural transform, in Equation (9),

Example 1.
Consider the one-dimensional fractional heat equation [19]: with initial condition Taking the natural transform of Equation (11), Applying inverse natural transform, Using the ADM procedure, we get The subsequent terms are The NTDM solution for Example 1 is when γ = 1, then the NTDM solution is This result is calculated to get the exact solution in a closed form:

Example 2.
Consider the two-dimensional fractional heat equation [19]: with initial condition Taking the natural transform of Equation (17), Applying inverse natural transform, Using the ADM procedure, we get The subsequent terms are , , The NTDM solution for Example 2 is This result is calculated to get the exact solution in a closed form: υ(x, y, t) = x 2 sinh t + y 2 cosh t
Taking the natural transform of Equation (22), Applying inverse natural transform Using the ADM procedure, we get The subsequent terms are , , The NTDM solution for Example 3 is when γ = 1, then the NTDM solution is This result is calculated to get the exact solution in a closed form: υ(x, y, z, t) = (e t − 1)x 4 y 4 z 4 .

Example 4.
Consider the one-dimensional fractional heat equation [19]: with initial condition Taking natural transform of Equation (28), Applying inverse natural transform, Using the ADM procedure, we get The subsequent terms are . .

(32)
The NTDM solution for Example 4 is when γ = 2, then NTDM solution is This result is calculated to get the exact solution in a closed form:  Example 5. Consider the two-dimensional fractional wave equation [19]: with initial condition Taking natural transform of Equation (34), Applying inverse natural transform Using the ADM procedure, we get The subsequent terms are υ 2 (x, y, t) = N − u γ s γ N + y 2 12 The NTDM solution for Example 5 is This result is calculated to get the exact solution in a closed form: υ(x, y, t) = x 4 cosh t + y 4 sinh t.   Example 6. Consider the three-dimensional fractional wave equation [19]: with initial condition Taking natural transform of Equation (39), Applying inverse natural transform, Using the ADM procedure, we get The subsequent terms are . .

Conclusions
In this paper, the analytical solutions of fractional-order heat and wave equations are determined, using NTDM. The NTDM solutions are obtained at fractional and integer orders for all problems. The results revealed the highest agreement with the exact solutions for the problems. The NTDM solutions for some numerical examples have shown the validity of the proposed method. It is also investigated that the fractional order solutions are convergent to the exact solution for the problems as fractional order approaches to integer order. The implementation of NTDM to illustrative examples have also confirmed that the fractional order mathematical model can be the best representation of any experimental data as compared to integer order model. In the future, NTDM can be used to find the analytical solution of other nonlinear FPDEs, which are frequently used in science and engineering. NTDM solutions for fractional order problems will prove better understanding of the real world problems represented by FPDEs.