Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional PDE

Abstract We apply homotopy perturbation transformation method (combination of homotopy perturbation method and Laplace transformation) and homotopy perturbation Elzaki transformation method on nonlinear fractional partial differential equation (fpde) to obtain a series solution of the equation. In this case, the fractional derivative is described in Caputo sense. To avow the adequacy and authenticity of the technique, we have applied both the techniques to Fractional Fisher’s equation, time-fractional Fornberg-Whitham equation and time fractional Inviscid Burgers’ equation. Finally, we compare the results obtained from homotopy perturbation transformation technique with homotopy perturbation Elzaki transformation.


Introduction
Most of the real world problems arising in the eld of biology, uid mechanics, ecology and thermodynamics etc. are modelled as nonlinear partial di erential equation(PDE). The fractional calculus is important tool to rene the description of most of the natural phenomenon. Fractional di erential equations have attracted considerable interest of many researcher because of their successive appearance in diverse elds of science and engineer-*Corresponding Author: Prince Singh, Department of Mathematics, Lovely Professional University, Phagwara, Punjab-144411,India, E-mail: princeshersingh@gmail.com Dinkar Sharma, Department of Mathematics, Lyallpur Khalsa College, Jalandhar, Punjab-144001, India ing. Many numerical and semi-analytical techniques are used to obtain the solution of linear and nonlinear partial di erential equations. Dr. Ji Huan He in 1999, proposed homotopy perturbation method (HPM) [11,13] which is coupling of homotpoy method and classical perturbation technique, has been successfully implemented on linear and nonlinear problems like nonlinear wave equation [15,16], fractional diffusion equation [3], fractional convection-di usion equation [19], space-time fractional advection-dispersion equation [35], fractional Zakharov-Kuznetsov equations [34], fractional partial di erential equations in uid mechanics [33], fractional Schrödinger equation [32]. The signi cance of HPM is that it doesn't require a small parameter in the equation , so it overcome the impediments of classical perturbation technique. The other semianalytical techniques such as HAM (Homotopy analysis method) [18], Laplace homotopy analysis method [20] Adomian decomposition method [4], HPTM(Homotopy perturbation transformation method) [10,17,24,26,28] and HPSTM(Homotopy perturbation Sumudu transform method) [23,27], we can always obtain better result than the numerical one for partial di erential equation. In recent years, many researchers have used numerical and analytical technique [1,[29][30][31] for the solution of fractional partial di erential equation. In [9], author suggested a new form of fractional di erentiation to model comlex physical problems. In this work, we apply HPTM [17,24,25] technique (which is combination of homotopy perturbation method and Laplace transformation) and HPETM [5][6][7][8]21] (Homotopy perturbation method with Elzaki transform) to nd the solution of Fractional Fisher's equation,timefractional Fornberg-Whitham equation and time fractional Inviscid Burgers' equation and we get a power series solution in the form of a rapidly convergent series and only a few iterations lead to high accurate solutions. In these techniques, there is no need of algorithm like discritizing the problem, no linearization is required for nonlinear problem , only few iterations will lead to the solution which can be easily calculated. There are many sym-bolic computation software like Maple, Mathematica etc. with which we can easily calculate more terms very easily, hence it reduces the computational cost for solving such complex problem. Finally, we compare the result obtained by these methods.

Basic de nitions and properties
De nition 2.2. The Caputo fractional derivative of g(τ) [2,22] is de ned as Here ∂ α ∂τ α and Γ denotes Caputo derivative operator and the Gamma function respectively. [5,7] of g(τ) is de ned as

. Properties
Elzaki transform of the Caputo fractional derivative is

Homotopy perturbation method (HPM)
Consider the nonlinear partial di erential equation where L and N are linear and nonlinear di erential operator and f (r) is an analytic function. Ji-huan He [13], [14] construct a homotopy of eq. (3.1) as H : Ω × [ , ] → R which satis es where p ∈ [ , ] is an embedding parameter and w is an initial approximation which satis es the boundary conditions. Clearly, from (3.3), we have The process of changing of p from zero to unity is that v varies from w to w(x, t). The basic assumption for this method is that the solution of (3.1) can be expressed as The solution of (3.1) is given by

Homotopy perturbation Elzaki transform method (HPETM)
Consider the following general fractional nonlinear partial di erential equation here, ∂ α ∂t α , is the fractional Caputo derivative with respect to t, L and N are linear and non linear di erential operators respectively which satisfy Lipschitz condition, f (x, t) is the source term. Now applying Elzaki transform, we get Applying the inverse Elzaki transform, we have By applying HPM, we get where , n = , , , , . . .
Comparing the coe cients of like powers of p, we have

Convergence analysis
In this section, we emphasis on the condition of convergence of the proposed method for the series solution of eq. (4.1).
with initial condition w(x, ) = ( +e x ) . By applying HPETM on (6.1), we have where Hn(w) represents He's polynomial . The rst few components of He's polynomial are given by Comparing the like powers of p on both sides of (6.2), we have , , Hence the solution is The above solution obtained is equivalent to the closed form solution when α = i.e w(x, t) = ( +e x− t ) up to fourth order term approximation.
Comparing the like powers of p on both sides of (6.5), we have Hence, the solution is By applying HPETM on eq.(6.7), we have On comparing the like powers of p on both sides of (6.8), we have p : wo = x; Hence the solution of (6.7) is where E α, (−t α ) in eq. (6.9) is Mittag-Le er function dened in (2.5). When α = , the exact solution of (6.7) is w(x, t) = x + t.

Homotopy perturbation transformation method (HPTM)
Now we present the solution of (4.1) using Laplace transformation, Using (2.4), we have Operating inverse Laplace transform , we get By applying HPM, we get hence, the approximate solution is obtained as p → w(x, t) = w + w + w + . . . . where Hn(w) are He's polynomial represents nonlinear terms. The rst few components of He's polynomial are given by

Application
Comparing the like powers of p on both sides of (8.2), we have . . .  Hence, the solution is given as  where Hn(w) are He's polynomial represents nonlinear terms. The rst few components of He's polynomial are given by Comparing the like powers of p on both sides of (8.5), we have ; Hence, the solution is given as By applying HPTM on eq.(8.7), we have
In this work, we intend to study two semi-analytical techniques to solve nonlinear fractional partial di erential equations: homotopy perturbation with Elzaki transform and homotopy perturbation transformation method.So, from above analysis, we conclude that the Elzaki transformation and its properties could be derive from Laplace transformation. This is the reason that either we use HPTM or HPETM, we come out with same series solution of nonlinear PDE or fractional PDE. Fig. 1-4, represent the surface graph of approximate solution of (8.1) for various estimations of α and the exact solution for α = and we nd that approximate solution up to order 4 converges to exact solution for α = , in Table 1, the condition of convergence is veri ed i.e. we analyse that ||w || < ||w || < ||w ||. Moreover, from Fig.5, we conclude that with the decrease in the value of α, the value of w(x, t) increases. On the other hand, Fig. 6-9 and Fig. 11-14 represents the surface graph of (8.4) and (8.7) for various estimations of α and the exact solution for α = , the approximate solution of w(x, t) converges to exact solution when α = , but by slightly decreasing the value of α, the value of w(x, t) also decreases which is shown in the Fig. 10 and Fig.15 . We have applied both the techniques (i.e HPTM and HPETM) on nonlinear homogeneous and non homogenous fractional PDE and the outcome exhibit the e ciency, simplicity and high rate of accuracy of the suggested methodologies to solve this type of complex equation.