Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel
© The Author(s) 2017
Received: 24 November 2016
Accepted: 17 February 2017
Published: 28 February 2017
This work presents the homotopy perturbation transform method for nonlinear fractional partial differential equations of the Caputo-Fabrizio fractional operator. Perturbative expansion polynomials are considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding the exact solutions of the fractional equations proposed, for the special case when the limit of the integral order of the time derivative is considered.
Fractional calculus (FC) has become an alternative mathematical method to describe models with nonlocal behavior. In the last decade, considerable interest in fractional differential equations has been stimulated due to their numerous applications in the areas of physics and engineering [1–15]. Several numerical and analytical methods have been developed to study the solutions of nonlinear fractional partial differential equations, fractional sub-equation methods [16–18], the homotopy perturbation methods [19–21], the variational iteration methods [22–27], homotopy perturbation transform methods [28, 29], Adomian descomposition methods [30–33], and other analytical approaches that could be of interest for the reader are presented in [34–38]. It worth noting that there exist only two main definitions of the fractional derivative; the first was proposed by Riemann and Liouville and is the derivative of the convolution of a given function and a power law kernel, the second one was suggested by Caputo and it is the convolution of the local derivative of a given function with power law function .
Due to the fact that the power law cannot be used to model all the physical problems, Caputo and Fabrizio  have suggested an alternative concept of differentiation using the exponential decay as kernel instead of the power law. This new differentiation has also attracted attention of many scholars but was also disqualified for classification as a fractional derivative due to the fact that the kernel was not nonlocal; however, it is clear that many problems in nature also follow the exponential decay law which indeed has no singularity; therefore this derivative is significantly useful in modeling such real world problems [41–44].
The homotopy analysis method (HAM), proposed by Liao, has been successfully applied to solving many problems in physics and science [45–48], this method transforms a problem into an infinite number of linear problems without using the perturbation techniques. The Laplace homotopy analysis method (LHAM) is a combination of HAM and Laplace transform [49, 50]. The homotopy perturbation method is also combined with the well-known Laplace transformation method and the variational iteration method to produce a highly effective technique (homotopy perturbation transform method) for handling many nonlinear problems .
In this paper, we use the homotopy perturbation transform method (HPTM) to solve nonlinear fractional partial differential equations using the fractional operator of Caputo-Fabrizio type. The basic definitions of fractional calculus are given in Section 2, several test problems that show the effectiveness of the proposed method are given in Section 3, and finally the conclusion is given in Section 4.
2 Basic tools
3 General description of the method using the operator of Caputo-Fabrizio type
We present the solutions obtained by the application of the HPTM with Caputo-Fabrizio fractional operator for some NFPDEs.
4.1 Convergence and stability analysis
If the series (12) converges \((n=0,1,2,\ldots, n)\), where \(\Theta (x,s )\) is governed by (7), it must be the solution of equation (4). Overall, the results show that the proposed approach is unconditionally stable and convergent. The method provides a simple way to control the convergence region of the solution by introducing (11) and our approximate results agree well with exact solutions and numerical ones.
Madani in  have compared the approximate solutions obtained by means of HPTM in a wide range of the problem’s domain with those results obtained from the exact analytical solutions and the HAM. This comparison shows precise agreement between the HPTM and exact results. The HPTM solution is valid for a wide range of time and this suggests that the HPTM method can solve non-homogeneous equations with a high degree of accuracy by considering only few terms in the perturbed solution. On the other hand the relative error for the HAM is dramatically increased as the time value t increases, so the HAM solution validity range is restricted to a short region.
Therefore the HPTM method is a powerful new method which needs less computation time and is much easier and more convenient than the HAM, because the Laplace transform allows one in many situations to overcome the deficiency mainly caused by unsatisfied boundary or initial conditions that appear in other semi-analytical methods such as HAM .
In this paper, the HPTM method was developed to solve fractional nonlinear differential equations using the Caputo-Fabrizio operator. With the polynomials expansion considered in the HTPM method we obtained an infinite series solution for the fractional partial differential equations. Based on the HPTM, a general scheme was developed to obtain approximate solutions of fractional equations and the solutions are given in a series form, which converges rapidly. The methodology presented has become an important mathematical tool, motivated by the potential use for physicists and engineers working in various areas of the natural sciences.
This work shows that the HPTM method is an efficient tool for solving nonlinear fractional partial differential equations considering the fractional operator of Caputo-Fabrizio type. The HPTM yields a rapidly convergent series solution by using a few iterations [53, 54]. In this paper a Mathematica program has been used for computations and programming.
We would like to thank Mayra Martínez for the interesting discussions. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014.
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