Recent Theory and Applications on Numerical Algorithms and Special FunctionsView this Special Issue
On the Fractional Nagumo Equation with Nonlinear Diffusion and Convection
We presented the Nagumo equation using the concept of fractional calculus. With the help of two analytical techniques including the homotopy decomposition method (HDM) and the new development of variational iteration method (NDVIM), we derived an approximate solution. Both methods use a basic idea of integral transform and are very simple to be used.
The Nagumo equation with linear or nonlinear diffusion and convection has broadly been useful to population dynamics, ecology, neurophysiology, chemical reactions, and flame propagation [1–3]. In particular, the case where the equation involves degenerate nonlinear diffusion is of considerable interest [4–8]. In this case, a travelling wave front solution of sharp type is known to exist for exactly one value of the wave speed. Such wave fronts, for instance, represent collective motion of populations in particular collective cell spreading, invasion in ecology, and concentration in chemical reactions [9–15]. However, it has been showing that the real world problems describe via fractional order derivative gives better prediction [16–20]. It is, therefore, important; further extend the nonlinear Nagumo equation using the concept of fractional derivative order.
It is something very difficult to obtain the exact solution of nonlinear equation with fractional order derivative. Many scholars sometimes, to avoid this difficulty, solve this class of problem numerically. However, even with numerical scheme it is also difficult to provide a numerical solution for nonlinear equations. Thus, many scholars, to access the behaviour of the solution of the real problem under study, present an approximate solution of this type of equations.
In the literature, there exist several analytical techniques [21–25] to deal with nonlinear equations including fractional type. The purpose of this work is to present an approximate solution for the generalized nonlinear Nagumo equation withnonlinear diffusion and convection via the well-known variational iteration method (VIM) and the homotopy decomposition method (HDM). The nonlinear fractional Nagumo equation considered here is given below as where , and are constants, and for the sake of simplicity in this paper, we consider . The concept of fractional order is not well by some scholars; in order to accommodate those, we present in the next section the basic information regarding this concept.
2. Basic Information about Fractional Calculus
There exists a vast literature on different definitions of fractional derivatives. The most popular ones are the Riemann-Liouville and the Caputo derivatives. For Caputo we have the following.
Definition 1 (see [26–30]). A real function , is said to be in the space , if there exists a real number , such that , where , and it is said to be in space if , .
Definition 2 (partial derivatives of fractional order [31–34]). Assume now that is a function of variables , also of class on . We define partial derivative of order for with respect to , the function where is the usual partial derivative of integer order .
Definition 3 (see [26–30]). The Riemann-Liouville fractional integral operator of order , of a function , , is defined as Properties of the operator can be found in [31–38]; we mention only the following.
3. Basic Information of the HDM and VIM
In this section, we shall present the basic information of the chosen analytical methods: the homotopy decomposition method and variational iteration method; we shall start with the homotopy decomposition method.
3.1. Information about the HDM
The interested reader can find the full detail of the methodology of the homotopy decomposition method in [39–42]. This relatively new method was recently used to solve some nonlinear fractional partial differential equations. However, since the new development of the variational iteration method using the Laplace transform was recently introduced, we shall present its methodology in the following subsection.
3.2. Some Information about the Variational Iteration Method
In its initial development, the essential nature of the method was to construct the following correction functional for (2) when is a natural number: where is the so-called Lagrange multiplier  and is the -approximate solution. However, this development was not suitable for equations with fractional order derivative . Therefore, in their work, they apply the new development of the VIM proposed in  to find the Lagrange multiplier. In this new VIM, the first step of the basic character of the method is to apply the Laplace transform on both sides of (2) to obtain The recursive formula of (8) can now be used to put forward the main recursive method connecting the Lagrange multiplier as Now considering , the restricted term; the Lagrange multiplier can be obtained as  Now, applying the inverse Laplace transform on both sides of (9), we obtain the following iteration:
4. Application to the Fractional Nagumo Equation
In this section, we present the application of the homotopy decomposition method and the new development of the so-called variational iteration method to the nonlinear fractional Nagumo equation. We shall start with the HDM.
4.1. Application of the HDM
Let us consider the fractional nonlinear Nagumo Equation with the following initial condition: Using the steps involved in the HDM we arrive at the following: Now, assume the solution of the above equation can be expressed in series form as follows: Replacing this in (13) and after comparing the term of the same power of , we obtain the following recursive formulas: Here, The general recursive formula for is given as with so that, integrating the above set of integral equations, we obtain the following: Here, we have computed only three terms in the series solution. However, using the recursive formula, we can compute the remaining terms, and the approximate solution is given as follows:
4.2. New Development of the Variational Iteration Method
In this subsection, we test the efficiency of the new development of the variation iteration method by solving the nonlinear fractional equation (1). Therefore, following the methodology of NDVIM, we are at the following.
The Lagrange multiplier is and the recursive formula is given as with the initial term so that using the iteration formulas we obtain Using the recursive formula, the remaining term can be obtained but here, due to the length of this term, we computed only three terms and the approximate solution case given as In the following section, we compare the approximate solution via HDM and NDVIM.
5. Numerical Results
We devote this section to the comparison of the numerical solutions obtained via the HDM and the NDVIM for different values of the fractional order derivative. In this case, we chose , , and . The following figures show the numerical solution of the time fractional nonlinear Nagumo equation. Figure 1 show, the approximate solution obtained via the HDM for , Figure 2 shows the approximate solution via the NDVIM. Figure 3 Show the approximate solution obtained via the HDM for and Figure 4 shows the approximate solution via the NDVIM. Table 1 shows the comparison of the numerical values of the solution obtained via the HDM and the NDVIM, respectively, for different values of alpha.
Both methods used the idea of iteration; the initial components are obtained as the Taylor series of the exact solution. On one hand, the new development of variational iteration method makes use of the Laplace transform, the Lagrange multiplier, and finally the inverse Laplace transform. On the other hand, the HDM uses just a simple integral and the perturbation technique. Both techniques are simple to implement and are very accurate.
The Nagumo equation is a very complex equation, for which the exact solution does not exist. The Nagumo equation was extended to the concept of fractional order derivative. The resulting equation was further analyzed within the framework of the homotopy decomposition method and the new development of variational iteration method. Both methods use a simple idea of integral transform. The numerical results are presented to test the efficiency and the accuracy of both methods. From their iteration formulas, one can conclude that these two methods are simple to be used and are powerful weapons to handle fractional nonlinear equation type.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Abdon Atangana would like to thank Claude Leon foundation for their financial support.
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