Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

Abstract This work presents a numerical comparison between two efficient methods namely the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) to solve a certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equations with variable coefficients. These two methods provided an accurate and efficient tool for solving this type of equations. To show the efficiency and capability of the proposed methods we have solved some numerical examples. The results show that there is an excellent agreement between the series solutions obtained by these two methods. However, the FNVIM has an advantage over FNHPM because it takes less time to solve this type of nonlinear problems without using He’s polynomials. In addition, the FNVIM enables us to overcome the diffi-culties arising in identifying the general Lagrange multiplier and it may be considered as an added advantage of this technique compared to the FNHPM.


Introduction
In mathematics, the fractional calculus is a branch of the analysis, which studies the generalization of the derivation and integration of integer order n (ordinary) to the non-integer order (fractional). It has turned out that many phenomena in engineering, physics and other sciences can be described very successfully by models using mathematical tools from fractional calculus. Recently, The nonlinear fractional partial differential equations appeared in many branches of physics, engineering and applied mathematics including fluid mechanics, viscoelasticity, aerodynamics, electrodynamics, rheology, mathematical biology and so on (see [6], [7], [11], [13], [16], [17]). Hence, it is important to solve nonlinear fractional partial differential equations. In general, there exists no method that yields an exact solution for nonlinear fractional partial differential equations due to the computational complexities of nonlinear parts involving them. Therefore, several different and powerful methods for solving fractional partial differential equations have been proposed in order to obtain the approximate solutions. The most commonly used ones are: the adomian decomposition method (ADM) [4], variational iteration method (VIM) [15] homotopy analysis method (HAM) [12], homotopy perturbation method (HPM) [5], fractional reduced differential transform method [10], and fractional residual power series method (FRPSM) [9].
The main objective of this paper is to introduce a numerical comparison of two powerful methods, the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) for solving certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equation with variable coefficients of the form ( [8], [9]) with the initial conditions where D α t is the Caputo fractional derivative operator of order α, 1 < α ≤ 2 and v is a function of (X, t) ∈ R n ×R + , F 1ij , G 1i i, j ∈ {1, 2, ..., n} are nonlinear functions of X, t and v, F 2ij , G 2i i, j ∈ {1, 2, ..., n} , are nonlinear functions of derivatives of v with respect to x i and x j i, j ∈ {1, 2, ..., n} , respectively. Also H, S are nonlinear functions and k, m, p are integers.

Definitions and properties
We present some definitions and important properties of the fractional calculus theory and natural transform that will be widely used in this paper.
For α = 1, E α (z) reduces to e z . A further generalization of (2.1) is given in the form Theorem 2.5 ( [8]). Let n ∈ N * and α > 0 be such that n − 1 < α ≤ n and R + (s, u) be the natural transform of the function f (t), then the natural transform denoted by R + α (s, u) of the Caputo fractional derivative of the function f (t) of order α, is given by Proof. To prove the above theorem, firstly we define the nonlinear operators

FNVIM for nonlinear Caputo time-fractional wave-like equations
Then, the equation (1.1) is written in the form The methodology consists of applying the natural transform first on both sides of (3.1) and using the Theorem 2.5, we have Operating the inverse natural transform on both sides of (3.2), we get where L(X, t) is a term arising from the source term and the prescribed initial conditions. After that, let us take the first partial derivative with respect to t of the equation (3.3), to obtain According to the variational iteration method ([3]), we can construct a correct functional as follows where λ(τ ) is a general Lagrange multiplier which can be identified optimally via the variational theory and integration by parts. The subscript n denotes the n th -order approximation, v n is considered as a restricted variation (i.e. δ v n = 0). Making the above correction functional stationary, and noting that δ v n = 0, we obtain the following stationary conditions Therefore, the Lagrange multiplier can be easily identified as Substituting equation (3.5) into the correction functional equation (3.4), we get the iterative formula for n = 0, 1, 2, ..., as follows Finally, the exact solution of the equations (1.1) and (1.2) is given as a limit of the successive approximations v n (X, t), n = 0, 1, 2, ..., in other words v(X, t) = lim n→∞ v n (X, t). Proof. Similarly like in the proof of Theorem 3.1, we have

FNHPM for nonlinear Caputo time-fractional wave-like equations
Now, applying the homotopy perturbation method ( [2]), we can assume that the solution can be expressed as a power series in p as given below where the homotopy parameter p is considered as a small parameter p ∈ [0, 1].
The nonlinear terms can be decomposed as where H n (v), K n (v) and J n (v) are He's polynomials ( [14]), and it can be calculated by the formulas given below Substituting the equalities (4.2) and (4. . Using the coefficient of the like powers of p in (4.5), the following approximations are obtained

Numerical applications
In order to evaluate the advantages and the accuracy of the FNVIM and FNHPM for the resolution of nonlinear Caputo time-fractional wave-like equations with variable coefficients, we will consider the following three numerical examples. All the results are calculated using Matlab (version 7.9.0.529 (R2009b)).
with the initial conditions where D α t is the Caputo fractional derivative operator of order α, 1 < α ≤ 2 and v is a function of (x, y, t) ∈ R 2 × R + . Application of the FNVIM. Following the description of the FNVIM presented in Section 3, we obtain the iteration formula as follows v n+1 = e xy + te xy Then, the general term in successive approximation is given by v n (x, y, t) = n k=0 (−1) k t kα Γ(kα + 1) Therefore, the exact solution of the equations (5.1) and (5.2) using Mittag- Application of the FNHPM. Following the description of the FNHPM presented in Section 4, gives where H n (v) and K n (v) are He's polynomials that represents the nonlinear terms, ∂ 2 ∂x∂y (v xx v yy ) and ∂ 2 ∂x∂y (v x v y ) respectively.
Using (4.4), the first few components of He's polynomials, are given by Equating the coefficients of corresponding power of p on both sides in (5.4), we get p 0 : v 0 (x, y, t) = (1 + t)e xy , . . .
Numerical comparison of FNVIM and FNHPM for solving a certain type of...
with the initial conditions where D α t is the Caputo fractional derivative operator of order α, 1 < α ≤ 2 and v is a function of (x, t) ∈ ]0, 1[ × R + . Application of the FNHPM. Following the description of the FNVIM presented in Section 3, we obtain the iteration formula as follows Then, the general term in successive approximation is given by Therefore, the exact solution of the equations (5.6) and (5.7) using Mittag- So, the solution of the equations (5.6) and (5.7) using Mittag-Leffler functions can be expressed as Taking α = 2 in equalities (5.8) and (5.10), the solution will be as follows v(x, t) = E 2 (t 2 ) + tE 2,2 (t 2 ) e x = e x+t , which is exactly the same solution obtained by FNDM ( [8]) and FRPSM ( [9]).
with the initial conditions where D α t is the Caputo fractional derivative operator of order α, 1 < α ≤ 2 and v is a function of (x, t) ∈ ]0, 1[ × R + . Application of the FNVIM. Following the description of the FNVIM presented in Section 3, we obtain the iteration formula as follows . . .
Then, the general term in successive approximation is given by Therefore, the exact solution of the equations (5.11) and (5.12) using Mittag- So, the solution of the equations (5.11) and (5.12) using Mittag-Leffler function can be expressed as ...

Numerical results and discussion
In Figures 1,2  These figures afirm that when the order of the fractional derivative α tends to 2, the approximate solutions obtained by FNVIM and FNHPM tends continuously to the exact solutions. In Tables 1, 2 and 3, we compute the absolute errors for differences between the exact solutions and the 3 th order approximate solution by FNVIM and the 4−term approximate solution by FNHPM at α = 2. The absolute errors obtained by the FNVIM are the same results obtained by FNPHM.

Conclusion
In this work, we compared the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) as applied to nonlinear Caputo time-fractional wave-like equations with variable coefficients. For illustration purposes, we consider three different numerical examples. The results show that FNVIM has advantages over FNHPM, it is that it takes less time to solve this type of nonlinear problems without using He's polynomials and enables us to overcome the difficulties arising in identifying the general Lagrange multipliers. However, there is the high agreement of the numerical results obtained between the FNVIM and the FNHPM. Therefore, it may be concluded that both methods are powerful and efficient techniques for finding exact as well as approximate solutions for wide classes of nonlinear fractional partial differential equations.