Abstract
In this article, the homotopy analysis method (HAM) is applied to solve the fractional cable equation by the RiemannLiouville fractional partial derivative. This method includes an auxiliary parameter h which provides a convenient way of adjusting and controlling the convergence region of the series solution. In this study, approximate solutions of the fractional cable equation are obtained by HAM. We also give a convergence theorem for this equation. A suitable value for the auxiliary parameter h is determined and results obtained are presented by tables and figures.
Keywords:
cable equation; fractional differential equations; fractional cable equation; homotopy analysis method1 Introduction
Fractional calculus has a very long history. However, this field lagged behind classic analysis. In fact, the basis of fractional calculus depended on classic analysis. Especially, in recent years fractional differential equations were used in fluid mechanics, viscoelasticity, biology, pharmacy, physics, chemistry and biochemistry, hydrology, medicine, finance, and engineering. The fractionalorder models are more useful than integerorder models in many cases. Structures having fractional order are more useful in the studies that have been done by developing technology.
However, the analytic solutions of most fractional differential equations generally cannot be obtained. Thus, fractional differential equations have been solved by many approximate methods. Examples are the homotopy perturbation method [1,2], the method of separating variables [3], the iteration method [4], the decomposition method [5], and the homotopy analysis method [6].
In this study, we will consider the cable equation that has been used in modeling
the ion electro diffusion at the neurons. The cable equation occurred due to anomalous
diffusion and this equation is one of the most fundamental equations for modeling
neuronal dynamics [7]. The cable equation can be derived from the NernstPlanck equation for electrodiffusion
in smooth homogeneous cylinders [8]. In recent years, studies were conducted on various biological and physical systems.
In this equation, the diffusion rate of species cannot be characterized by the single
parameter of the diffusion constant [7]. The anomalous diffusion is characterized by a scaling parameter γ as well as the diffusion constant D and the mean square displacement of diffusing species
Consider the following fractional cable equation:
where
In the literature, there are few treatments of approximate solutions of the fractional cable equation in terms of (1.1). Equation (1.1) has been solved by implicit numerical methods (INM) [9], the implicit compact difference scheme (ICFDS) [10], and explicit numerical methods [11].
Here, we will use the HAM, which is an approximate solution to solve this equation. The HAM method was developed in 1992 by Liao in [12]. This method has been successfully applied by many authors [1317]. The HAM contains the auxiliary parameter h which provides us with a simple way to adjust and control the convergence region of solution series for large or small values of x and t.
2 Preliminaries and notations
We give some basic definitions and properties of the fractional calculus theory, which are used further in this paper.
Definition 2.1 The Euler Gamma function
where
Definition 2.2 The RiemannLiouville fractional integral operator of order
and properties of the operator
3 Homotopy analysis method
We consider the following differential equation:
where N is a nonlinear differential operator, x and t denote independent variable;
where
respectively. The solution
where
The convergence of the series (3.4) depends upon the auxiliary parameter h. If it is convergent at
According to (3.6), the governing equation can be deduced from the zerothorder deformation equation (3.2). Define the vector
Differentiating (3.2) m times with respect to the embedding parameter q and then setting
where
and
It should be emphasized that
4 Numerical applications and comparison
Consider the following initial and boundary problem of the fractional cable equation:
where
We choose the linear operator
with the property
Therefore we establish the zerothorder deformation equation
In (4.6),
So we obtain the mthorder deformation equation
where
and
Now the solution of the mthorder deformation equation (4.8) for
Instead of
can be written. The auxiliary function
Rearrangement of (4.12) gives the mthorder deformation equation
Therefore, some of the symbolically computed components are found as
and so on.
As a result, the mthorder approximation of
Theorem 4.1 (Convergence Theorem)
As long as the series
Proof If the series
converges, then we can write
and we have
Using definition (4.13), we get
Since
From (4.9), we have
From the initial
Therefore, according to the above expressions,
We get the following tables and figures by using a series solution obtained with HAM of (4.1).
5 Conclusion
In this paper, we have achieved approximate solutions of the fractional cable equation that involve two RiemannLiouville fractional derivatives by means of the homotopy analysis method. We tried to find an approximate solution of this equation by HAM, which is a semianalytical method. It is not possible to find the analytical solutions of fractional partial differential equations in most cases. In addition, there is an approximate solution of the fractional cable equation that we have considered just with the finite difference method. The HAM results were given by Tables 14 and Figures 15.
Table 1. Absolute errors obtained when
Table 2. Comparison of the HPM, HAM, exact solution (ES) and absolute errors results of
Table 3. Comparison of the HPM, HAM, exact solution (ES) and absolute errors results of
Table 4. Comparison of the HPM, HAM, exact solution (ES) and absolute errors results of
Figure 1. Thehcurves of 5thorder and 10thorder approximate solutions obtained by the HAM for
Figure 2. The 10thorder approximate solution of
Figure 3. The 10thorder approximate solution of
Figure 4. Comparison of the HPM, HAM and Exact solution for 5thorder approximate when
Figure 5. Comparison of the HPM, HAM and Exact solution for 10thorder approximate when
The range of convergence control parameter h was determined by taking a different number of terms of the series solution in Figure 1. We showed that convergent results can be obtained by selecting the appropriate values
of x and t of the convergence parameter
An approximate solution that was obtained for different values of the parameter h, the fractionalorder derivatives
A comparison between HPM, HAM, and the analytical solution, when
The absolute errors that were obtained by the implicit numerical method [9], implicit compact finite difference method [10], and HAM can be seen in Table 1. In this table
A comparison between HPM, HAM, and the analytical solution for
Although convergent results for almost every value of the independent variables and convergent control parameter h have been obtained in HAM; the approximate solution diverged at some small and large values of independent variables in HPM. Namely, it is possible to find results that converge rapidly to the analytical solution by HAM.
Consequently HAM is a recommended method for obtaining an approximate solution of
the fractional cable equation with
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
References

Wang, Q: Homotopy perturbation method for fractional KdV equation. Appl. Math. Comput.. 190, 1795 (2007). Publisher Full Text

Golmakhaneh, AK, Golmakhaneh, AK, Baleanu, D: On nonlinear fractional KleinGordon equation. Signal Process.. 91, 446 (2011). Publisher Full Text

Chen, J, Liu, F, Anh, V: Analytical solution for the time fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl.. 338, 1364 (2008). Publisher Full Text

Momani, S, Odibat, Z, Alawneh, A: Variational iteration method for solving the space and timefractional KdV equation. Numer. Methods Partial Differ. Equ.. 24(1), 262 (2008). Publisher Full Text

Momani, S: An explicit and numerical solutions of the fractional KdV equation. Math. Comput. Simul.. 70, 110 (2005). Publisher Full Text

Inc, M: On numerical solution of Burgers’ equation by homotopy analysis method. Phys. Lett. A. 372, 356 (2008). Publisher Full Text

Langlands, TAM, Henry, B, Wearne, S: Solution of a fractional cable equation: Finite case. Preprint, Submitted to Elsevier Science http://www.maths.unsw.edu.au/applied/filed/2005/amr0533.pdf (2005)

Keener, J, Sneyd, J: Mathematical Physiology, Springer, Berlin (1991)

Liu, F, Yang, Q, Turner, I: Stability and convergence of two new implicit numerical methods for fractional cable equation. In: IDETC/CIE, San Diego, California, USA. (2009)

Hu, X, Zhang, L: Implicit compact difference scheme for the fractional cable equation. Appl. Math. Model.. 36(9), 4027 (2012). Publisher Full Text

QuintanaMurillo, J, Yuste, SB: An explicit numerical method for the fractional cable equation. Int. J. Differ. Equ. (2011). Publisher Full Text

Liao, SJ: Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton (2003)

Inc, M: On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method. Phys. Lett. A. 365, 412 (2007). Publisher Full Text

Abbasbandy, S: Soliton solutions for the FitzhughNagumo equation with the homotopy analysis method. Appl. Math. Model.. 32, 2706 (2008). Publisher Full Text

Abbasbandy, S, Shivanian, E: Series solution of the system of integrodifferential equations. Z. Naturforsch. A. 64, 811 (2009)

Yinping, L, Zhibin, L: The homotopy analysis method for approximating the solution of the modified Kortewegde Vries equation. Chaos Solitons Fractals. 39, 1 (2009). Publisher Full Text

Jafari, H, Tajadodi, H, Biswas, A: Homotopy analysis method for solving a couple of evolution equations and comparison with Adomian decomposition method. Waves Random Complex Media. 21(4), 657–667 (2011). Publisher Full Text

Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations (2006)

Podlubny, I: Fractional Differential Equations, Academic Press, San Diego (1999)

Oldham, KB, Spanier, J: The Fractional Calculus, Academic Press, New York (1974)