Tarig Projected Differential Transform Method to Solve Fractional Nonlinear Partial Differential Equations

Recent advances in nonlinear analyzes and fractional calculus is to address the challenges arise in the solution methodology of nonlinear fractional partial differential equations. This paper presents a hybrid technique to solve nonlinear fractional partial differential equations, which is a combination of Tarig transform and Projected Differential Transform Method (TPDTM). The effectiveness of the method is examined by solving three numerical examples that arises in the field of heat transfer analyzes. In this proposed scheme the solution obtained as a convergent series and using the result the hyper diffusive process with pre local information regarding the heat transfer for different values of fractional order are analyzed. To validate the results and to analyze the computational efficiency of the proposed method a comparative study has been carried out with the solution obtained by the Laplace Adomian Decomposition Method (LADM) and Homotophy Pertubation Method (HPM) and observed good agreement. Also, the computational time in each method is calculated using CPU and the results are presented. It was observed that the proposed technique provide good results with less computational time than homotophy pertubation technique. Even though there is a uniformity between the solutions obtained by TPDTM and LADM, the proposed hybrid technique overcome the complexity of manupulation of Adomian polynomials in LADM and evaluation of integrals in HPM respectively. The methodology and the results presented in this paper clearly reveals the computational efficiency of the present method. The TPDTM, due to its computational efficiency has the potential to be used as a novel tool, not only for solving nonlinear fractional differential equations but also to analyse the prelocal information of the system.


Introduction
Linear and nonlinear fractional partial differential equations have attracted many researchers due to their enormous applications in various fields of engineering like vibration analysis, fluid flow and heat transfer analysis etc.In recent years, most of the physical and biological problems are modeled as nonlinear fractional partial differential equations and have been analyzed by different semi analytical solution techniques like Adomian decomposition, homotopy perturbation and variational iteration method etc.
The emergence of new mathematical method is to reduce the computational complexity in solving nonlinear fractional differential equations.In literature, He (1999) introduced topology based homotopy perturbation method (HPM), which has been used to solve all types of differential equations.Varsha Daftardar Gejji and Jafari (2005) proposed Adomian decomposition method for solving linear and nonlinear fractional differential equations, where the computations are based on the Adomian polynomials.Abbasbandy (2006) applied homotopy analysis method (HAM) to solve nonlinear heat transfer problems.Unlike homotopy perturbation method (HPM), homotophy analysis method is applicable for all kind of auxiliary parameter.Eltayeb and Cman (2007) applied double convolution method in Mellin transform to solve the partial differential equations.Soliman (2008) adopted modified extended direct algebraic (MEDA) method to solve different kinds of nonlinear partial differential equations.Elzaki and Ezaki (2011) introduced "Elzaki transform" for solving partial differential equations.Like the Laplace and Fourier transform, the "Elzaki transform" converts a time domain function into a frequency domain function with a different kernel.Moitsheki and Harley (2011) analysed the heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient and proposed a closed form solution.Elzaki (2012) combined both Elzaki transform and differential transform method (DTM) and applied this hybrid technique to solve nonlinear partial differential equations.Elzaki and Ezaki (2013) developed a new transformation called "Tarig transform" to solve linear system of integro-differential equations.Zhuo-Jia Fu et al., (2013) developed Laplace transformed boundary particle method for solving time fractional diffusion equation and studied the long time-history of fractional diffusion systems.Butera and Paola (2014) have proposed a solution methodology using complex Mellin transform for solving multi order fractional differential equations.
Sobhan Mosayebidorcheha et al., (2014) applied differential transform method (DTM) to solve PDE for the heat transfer analysis along fins.Rawashdeh andMaitama (2014, 2015) proposed natural decomposition method (NDM) to solve coupled system of nonlinear partial differential equations and ordinary differential equations, which is a combination of natural transform method (NTM) and Adomian decomposition method.Elzaki and Alamri (2014) combined Elzaki transform and projected DTM to solve nonlinear partial differential equations.Rabie and Elzaki (2014) used Adomian with modified decomposition method for solving systems of nonlinear partial differential equations.Hilal and Elzaki (2014) applied Laplace and variational iteration method for the solution of system of nonlinear partial differential equations.Deshna Loonker (2014) adopted Tarig transform to solve nonhomogeneous fractional order differential equations.Elzaki (2015) applied projected differential transform method to solve nonlinear, space and time fractional partial differential equations.DiMatteoa and Pirrottaa (2015) applied differential transform method to solve linear and nonlinear boundary value problems of fractional order.Wei et al., (2015) developed a mesh free local radial basis function method for solving two dimensional time fractional diffusion equations, where the spatial and temporal discretization is based on the local collocation nodes with implicit time-marching.Zhuo-Jia Fu et al., (2015) proposed method of approximate particular solutions to analyze the constant and variable order fractional diffusion models, which is based on the linear combination of the particular solutions of the non homogeneous equations and radial basis functions.Pang et al (2015) applied Kansa method to solve space fractional advection dispersion equations.Pang et al (2016) carried out a comparative study between the Finite Element and Finite Difference Methods to solve two Dimensional Space Fractional Advection Dispersion Equation.Recently, Chen and Pang (2016) newly-defined fractional Laplacian for modeling and to analyse the power law behaviors of three-dimensional nonlocal heat conduction.
Many of the researchers have focused on the solution methodology of either fractional differential equations or nonlinear differential equations.This research work, makes a useful contribution in fractional calculus and in the nonlinear analyzes of physical and biological problems.In particular, this paper proposes a hybrid technique to solve nonlinear fractional partial differential equations (NLF-PDE) called Tarig projected differential transform method (TPDTM).This method has been applied to solve linear, nonlinear fractional partial differential equations in heat transfer analysis and the temperature distribution functions are obtained as a convergent series for different values of fractional order and nonlinearity.The solution obtained by TPDTM has been compared with HPM, LADM solution and the results are validated with the integer order solution and are represented graphically.The TPDTM is an effective tool for solving nonlinear fractional differential equations and more adaptable than the other methods.
The paper is organized as follows.The basic definitions and useful mathematical results which are necessary for the present analysis are discussed in section 2. A detail description of the present hybrid methodology to solve NLFPDE is proposed in section 3. The proposed method is clearly illustrated by solving few example problems and the solutions are presented in section 4. The results are briefly discussed in section 5 with the conclusion in section 6.

Preliminaries
In this section we provide some basic definitions of fractional calculus, which will be used in this study.
Definition 2. Subjected to variable limit the Riemann integral on the half axis can be expressed as Similarly, the left and right handed Riemann-Liouville fractional derivatives of order α, 0 < α < 1, in the interval [a, b] are defined as (2.5) Definition 4. Caputo fractional derivative of order α is defined as where m − 1 < α ≤ m, m ∈ N .
Definition 5.The Mittag-Leffler function which is a generalization of exponential function is defined as The fundamental definition of Tarig transform, Projected differential transform method and few theorems are provided in the next consecutive subsections.

Tarig transform
If f (t) is any time domain function then, the Tarig transform of f (t) is defined as where v is the frequency domain variable.Let f (t), g(t) be any two time domain functions and its frequency domain functions under Tarig transform are F (v), G(v) respectively.The Tarig transform of standard functions and few basic properties are given in Appendix [A].
As in [Deshna Loonker and Banerji, 2014], if F (v) is the Tarig transform of f (t), then the Tarig transform of Fractional integral of f (t) of order α is (2.9) Similarly, the Tarig transform of fractional derivative of f (t) of order α is (2.10)

Projected differential transform method
The projected differential transform method is a modified technique of the differential transform method [Elzaki and Alamri ( 2014 where, 12) The basic theorems obtained by the PDTM, which are useful for our study are listed below.Consider u(x 1 , x 2 , ..., x n ), v(x 1 , x 2 , ..., x n ) be any two multi variable functions and u(x 1 , x 2 , ..., x n−1 , k), v(x 1 , x 2 , ..., x n−1 , k) are the transformed functions of u and v respectively.Let c be a constant.
The basic idea of the hybrid technique to solve fractional nonlinear partial differential equation is described clearly in the next section.

Tarig Projected Differential Transform Method
Consider a fractional time nonlinear partial differential equation with initial condition as, where D α is the fractional order differential operator D α = ∂ α ∂t α , R is the linear differential operator, N is the nonlinear differential operator and g(x, t) is the source term.By applying the Tarig transform (denoted throughout this paper by T ) on both sides Using the differentiation property of Tarig transform 2.10, on equation 3.1, we have Applying the inverse of the Tarig transform on both sides of equation 3.3 imply where G(x, t) represent the term arise from the source term and the prescribed initial conditions.Using PDTM as in Elsaki and Alamri (2014), the nonlinear terms can be easily decomposed as The exact solution of equation 3.1 can be computed in series form as where each term u(x, m) is obtained as a function of x and t.
Unlike the discretization of derivative and complex computation of nonlinear terms the exact or accurate solution can be easily obtained in series form for nonlinear fractional partial differential equations using the present approach.

Error calculation and Convergence of TPDTM
It is essential to test the convergence of the series solution obtained in equation 3.6 by TPDTM.The approximate solution of equation 3.1 can be obtained as where eu k (x, t) is the error function.Generally, the absolute error is defined as Eu k (x,t) < 1 for k < p.Using the following algorithm, the convergence of the iterative solution u app(k) (x, t) to the exact solution u(x, t) can be shown as follows.
This algorithm is applied in this paper to prove the convergence of the series solution obtained by TPDTM.

Numerical Examples
In this section, few example problems are solved to illustrate the proposed hybrid technique and to show the computational efficiency.Consider a fractional time linear heat conduction problem along a rod of length l with sinusoidal initial temperature and constant thermal conductivity.
Here k, c and ρ are the thermal conductivity, specific heat and density of the rod respectively.By applying Tarig transform on both sides of equation 4.1 yield The inverse of the Tarig transform imply that Using PDTM, we express From the relation in equation 4.4, we obtain , etc. Now, the exact solution of equation 4.1 is Using Mittag Leffler function the solution can be expressed as, Using TPDTM, the temperature distributions are predicted at different points along the rod for different values of the fractional order α = 0.5, 0.75, 1 and is shown in Table 1.  2 shows the absolute error at some particular points along the length of the rod for α = 0.75 at time t = 80 seconds (Erwin Kreyszig, 2008).This proves the convergence of the series solution of equation 4.1 and Figure 1 Using TPDTM, the solution of equation in terms of Mittag Leffler function is obtained as . (4.9) The influence of the source term is also analyzed in the temperature distribution and depicted graphically.It is observed that as M → 0 the effect of the source term becomes small and the result exactly coincides with the solution obtained in the previous section in equation 4.7.

Heat conduction with variable thermal conductivity
Consider a fractional time linear heat conduction problem as in equation 4.1 with variable thermal conductivity as k = k(1 + β(u − u 0 )), where β is any constant and u 0 is the atmospheric temperature.

.10)
Let A = κ(1 − βu 0 ), B = κβ to reduce complexity.Then, equation 4.10 can be expressed as, Using the proposed hybrid method, we obtain u(x, t) = 100sin πx l The approximate closed form solution to the time fractional nonlinear heat conduction equation is obtained by considering the first two terms of the series, which give more accurate result for the nonlinearity.Using TPDTM, the temperature distributions are predicted at different points along the rod for different values of nonlinearity (variable thermal conductivity with respect to β) β = 0.05, 0.10 when α = 0.75 and is presented in Table 3.

System of fractional nonlinear PDE
Consider a system of fractional nonlinear coupled differential equations.
The closed form solution of the coupled system of nonlinear fractional differential equations using Mittag Leffler can be expressed as, If α = 1, the results obtained in equation 4.20 are reduced to their exact solution U (x, t) = e x−t , W (x, t) = e −x+t respectively.Table 4 shows the temperature distribution for a coupled system and Table 5 shows the absolute error when α = 0.75 at time t = 0.2.This proves the convergence of the series solution of the coupled system of fractional differential equations 4.14 and 4.15.Also, Figure 2 depicts the comparison of the approximate absolute error for different sequence of partial sums.Applying TPDTM on both sides imply By assuming the unknown value as y ′ (0) = b, we obtain the accurate series solution to the boundary value problem as Using the boundary condition y(1) = 1, we obtain the value b = 0.777.Only few terms in the series are enough to produce the accurate solution to the boundary value problem and when β = 2 the result is identical with the exact solution y(t) = 1.387e x + 0.613e −x − (x 2 + 2) shown in Figure 8.

Results and Discussion
The temperature distribution is predicted along a rod for a linear fractional time heat conduction problem (Example 1) by using the Tarig Projected differential transform method with the parameters κ = 1.158 and length of the rod l = 80 cm.The Table 1 represent the temperature distribution along a rod during the heat conduction process for different values of fractional order α = 0.25, 0.50, 0.75.The Figure 3        In order to show the effectiveness of TPDTM, for every numerical examples a comparative study has been carried out with the solution obtained by the LADM and HPM for a particular value of fractional order at some specified points and their computational time is calculated using CPU and the results are presented in Tables 6-9 including the graphical representation from Figures 9-12.An excellent agreement is observed between TPDTM, LADM and acceptable deviation with HPM.It is observed that, the CPU time is comparatively less in TPDTM than the other methods for both integer and fractional order derivatives in all the numerical examples.Even though there is a uniformity between the solutions obtained between TPDTM and LADM it is worth mentioning that, the proposed hybrid technique avoids the difficulty of manupulation of Adomian polynomials and evaluation of integrals in HPM.In the present method transformation technique is employed with projected differential transform method to overcome the difficulty and the results presented in this paper clearly reveals the computational efficiency and easiest adaptability of TPDTM.T (f ′ (t))

Definition 1 .
The Riemann-Liouville fractional integrals [Samko et al. (1993)] of the left and right sided are defined for any function ϕ(x) ∈ L 1 (a, b) as,

4. 1 .
Heat conduction in the absence of source term with constant thermal conductivity

. 6 )
The series solution of fractional time equation 4.1 obtained in terms Mittag Leffler function in equation 4.6 approaches to the exact solution shown in equation 4.7, when α = 1.

Figure 2 :
Figure 2: The comparison of absolute error (TPDTM) (a), (b) and Figure 4 (a) depict the instability phenomenon during the heat conduction and provides the prelocal information about the heat transfer process.The Figure 4 (b) presents the temperature distribution for alpha=1, where the comparison shows well agreement between the result obtained by TPDTM and the exact integer order solution.The influence of the source term is also analyzed (Example 2) for a small value of M and the results are graphically shown in Figure 5.The TPDTM is applied to solve nonlinear fractional time heat conduction problem with variable thermal conductivity (Example 3) and the temperature distributions are presented for the case beta=0.05and 0.10 when alpha=0.75.The Figure 6 (a), (b) provides the nonlinear behavior of temperature distribution of the system for fractional time and temperature dependent thermal conductivity.

Table 1 :
Example 1:Numerical values of temperature distribution using TPDTM at different time intervals Table

Table 2 :
depicts the comparison of the approximate absolute error for different sequence of partial sums.
Absolute error calculation for the temperature distribution u(x, t)

Table 5 :
Absolute error calculation for the functions U (x, t) and W (x, t)