On the Fractional Nagumo Equation with Nonlinear Diffusion and Convection

Abstract

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.

1. Introduction

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 𝜕^𝛼𝑢𝜕𝑡^𝛼+𝛽𝑢^𝑛𝜕𝑢=𝜕𝜕𝑥𝜕𝑥𝛼𝑢^𝑛𝜕𝑢𝜕𝑥+𝛾𝑢1−𝑢^𝑚𝑢^𝑚−𝛿,0<𝛼≤1,𝑢(𝑥,0)=𝑓(𝑥),𝑢(0,𝑡)=𝑔(𝑡), (1)(2)(3) where 𝛼,𝛽,𝛾, and 𝛿 are constants, and for the sake of simplicity in this paper, we consider 𝑚=𝑛=1=𝛿. 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 𝑓(𝑥),𝑥>0, is said to be in the space 𝐶_𝜇, 𝜇∈ℝ if there exists a real number 𝑝>𝜇, such that 𝑓(𝑥)=𝑥^𝑝ℎ(𝑥), where ℎ(𝑥)∈𝐶[0,∞), and it is said to be in space 𝐶^𝑚_𝜇 if 𝑓^(𝑚)∈𝐶_𝜇, 𝑚∈ℕ.

Definition 2 (partial derivatives of fractional order [31–34]). Assume now that 𝑓(𝑥) is a function of 𝑛 variables 𝑥_𝑖,  𝑖=1,…,𝑛 also of class 𝐶 on 𝐷∈ℝ_𝑛. We define partial derivative of order 𝛼 for 𝑓 with respect to 𝑥_𝑖, the function 𝑎𝜕^𝛼_𝑥1𝑓=Γ(𝑚−𝛼)^𝑥𝑖_𝑎𝑥_𝑖−𝑡^{𝑚−𝛼−1}𝜕^𝑚_𝑥𝑖𝑓𝑥_{𝑗𝑥𝑗=𝑡}𝑑𝑡, (4) where 𝜕^𝑚_𝑥𝑖 is the usual partial derivative of integer order 𝑚.

Definition 3 (see [26–30]). The Riemann-Liouville fractional integral operator of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇, 𝜇≥−1, is defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝐽𝑑𝑡,𝛼>0,𝑥>0,₀𝑓(𝑥)=𝑓(𝑥). (5) Properties of the operator can be found in [31–38]; we mention only the following.

For 𝑓∈𝐶_𝜇𝐽,𝜇≥−1,𝛼,𝛽≥0,𝛾>−1,_𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛼+𝛽𝑓(𝑥),𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛽𝐽^𝛼𝑓(𝑥),𝐽^𝛼𝑥^𝛾=Γ𝛾+1Γ𝑥𝛼+𝛾+1^𝛼+𝛾. (6)

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 𝛼 is a natural number: 𝑤_𝑛+1(𝑥,𝑡)=𝑤_𝑛()+𝑥,𝑡_𝑡0𝜆(−𝜕𝑡,𝜏)^𝑚𝑤(𝑥,𝜏)𝜕𝑡^𝑚+𝐿𝑤_𝑛(𝑤𝑥,𝜏)+𝑁_𝑛(𝑥,𝜏)+𝑘(𝑥,𝜏)𝑑𝜏, (7) where 𝜆(𝑡,𝜏) is the so-called Lagrange multiplier [43] and 𝑤_𝑛(𝑥,𝑡) is the 𝑛-approximate solution. However, this development was not suitable for equations with fractional order derivative [43]. Therefore, in their work, they apply the new development of the VIM proposed in [44] 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 𝑠^𝑚𝑤(𝑥,𝑠)−𝑠^𝑚−1𝑤(𝑥,0)−⋯𝑤^𝑚−1()[𝐿(𝑤((𝑤((.𝑥,0=ℒ𝑥,𝑡))+𝑁𝑥,𝑡))+𝑘𝑥,𝑡)] (8) The recursive formula of (8) can now be used to put forward the main recursive method connecting the Lagrange multiplier as 𝑤_𝑛+1(𝑥,𝑠)=𝑤_𝑛()(𝑠𝑠𝑥,𝑠+𝜆)^𝑚𝑤_𝑛(𝑥,𝑠)−𝑠^𝑚−1𝑤(𝑥,0)−⋯𝑤^𝑚−1()𝐿𝑤𝑥,0−ℒ_𝑛(𝑥,𝑡)+𝑁𝑤_𝑛(𝑥,𝑡)+𝑘(𝑥,𝑡). (9) Now considering ℒ[𝐿(𝑤_𝑛(𝑥,𝑡))+𝑁(𝑤_𝑛(𝑥,𝑡))+𝑘(𝑥,𝑡)], the restricted term; the Lagrange multiplier can be obtained as [43] 𝜆(𝑠)1=−𝑠^𝑚. (10) Now, applying the inverse Laplace transform on both sides of (9), we obtain the following iteration: 𝑤_𝑛+1(𝑥,𝑡)=𝑤_𝑛(𝑥,𝑡)−ℒ⁻¹1𝑠^𝑚𝑠^𝑚𝑤_𝑛(𝑥,𝑠)−𝑠^𝑚−1𝑤(𝑥,0)−⋯𝑤^𝑚−1()𝐿𝑤𝑥,0−ℒ_𝑛()𝑤𝑥,𝑡+𝑁_𝑛(𝑥,𝑡)+𝑘(𝑥,𝑡). (11)

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: 𝑢(𝑥,0)=𝑥²,𝜕^𝛼𝑢𝜕𝑡^𝛼+𝛽𝑢𝜕𝑢=𝜕𝜕𝑥𝜕𝑥𝑎𝑢𝜕𝑢(),𝜕𝑥+𝛾𝑢1−𝑢)(𝑢−10<𝛼≤1. (12) Using the steps involved in the HDM we arrive at the following: 𝑢()()+1𝑥,𝑡=𝑢𝑥,0Γ[1−𝛼]^𝑡₀(𝑡−𝜏)^1−𝛼−𝛽𝑢𝜕𝑢+𝜕𝜕𝑥𝜕𝑥𝑎𝑢𝜕𝑢(𝜕𝑥+𝛾𝑢1−𝑢)(𝑢−1)𝑑𝜏. (13) Now, assume the solution of the above equation can be expressed in series form as follows: 𝑢(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡). (14) Replacing this in (13) and after comparing the term of the same power of 𝑝, we obtain the following recursive formulas: 𝑢₀()(),𝑢𝑥,𝑡=𝑢𝑥,𝑡₁()=1𝑥,𝑡Γ[𝛼]^𝑡₀(𝑡−𝜏)^𝛼−1−𝛽𝑢₀𝜕𝑢₀+𝜕𝜕𝑥𝜕𝑥𝑎𝑢₀𝜕𝑢₀𝜕𝑥+𝛾𝑢₀1−𝑢₀𝑢₀𝑢−1𝑑𝜏,₂()=1𝑥,𝑡Γ[𝛼]^𝑡₀(𝑡−𝜏)^𝛼−1−𝛽𝐻¹₂+𝑎𝐻²₂+𝑎𝐻³₂+𝛾𝐻⁴₂+𝛾𝐻⁵₂−𝑢₁𝑑𝜏. (15) Here, 𝐻¹₂=𝑢₀(𝑥,𝑡)𝜕_𝑥𝑢₁(𝑥,𝑡)+𝑢₁(𝑥,𝑡)𝜕_𝑥𝑢₀(),𝐻𝑥,𝑡₂₂=2𝜕_𝑥𝑢₀(𝑥,𝑡)𝜕_𝑥𝑢₁(),𝐻𝑥,𝑡₃₂=𝑢₀(𝑥,𝑡)𝜕_𝑥𝑥𝑢₁(𝑥,𝑡)+𝑢₁(𝑥,𝑡)𝜕_𝑥𝑥𝑢₀(),𝐻𝑥,𝑡₄₂=𝑢²₀(𝑥,𝑡)𝑢₁(𝑥,𝑡)+𝑢³₁+𝑢²₁(𝑥,𝑡)𝑢₀(),𝐻𝑥,𝑡₅₂=2𝑢₀(𝑥,𝑡)𝑢₁(𝑥,𝑡). (16) The general recursive formula for 𝑝≥3 is given as 𝑢_𝑛()=1𝑥,𝑡Γ[𝛼]^𝑡₀(𝑡−𝜏)^𝛼−1−𝛽𝐻¹_𝑛+𝑎𝐻²_𝑛+𝑎𝐻³_𝑛+𝛾𝐻⁴_𝑛+𝛾𝐻⁵_𝑛−𝑢_𝑛−1𝑑𝜏, (17) with 𝐻¹_𝑛(𝑥,𝑡)=^𝑛−1_𝑗=0𝑢_𝑗(𝑥,𝑡)𝜕_𝑥𝑢_𝑛−𝑗(),𝐻𝑥,𝑡_2𝑛(𝑥,𝑡)=^𝑛−1_𝑗=0𝜕_𝑥𝑢_𝑗(𝑥,𝑡)𝜕_𝑥𝑢_𝑛−𝑗(),𝐻𝑥,𝑡_3𝑛(𝑥,𝑡)=^𝑛−1_𝑗=0𝑢_𝑗(𝑥,𝑡)𝜕_𝑥𝑥𝑢_𝑛−𝑗(),𝐻𝑥,𝑡_4𝑛(𝑥,𝑡)=^𝑛−1𝑗_𝑗=0𝑘=0𝑢_𝑗(𝑥,𝑡)𝑢_𝑗−𝑘(𝑥,𝑡)𝑢_{𝑛−𝑗−1}(),𝐻𝑥,𝑡_5𝑛(𝑥,𝑡)=^𝑛−1_𝑗=0𝑢_𝑗(𝑥,𝑡)𝑢_𝑛−𝑗(𝑥,𝑡), (18) so that, integrating the above set of integral equations, we obtain the following: 𝑢₀(𝑥,𝑡)=𝑥²,𝑢₁()𝑡𝑥,𝑡=−^𝛼𝑥²−6𝑎+2𝑥𝛽+−1+𝑥²²𝛾Γ(),𝑢1+𝛼₂(𝑥,𝑡)=𝑡^2𝛼𝑥²48𝑎²+𝛾−2𝑎3+24𝑥𝛽+4𝛾−22𝑥²𝛾+𝑥10𝑥𝛽²+𝑥𝛾−2+𝑥²−2𝛾+3𝑥²𝛾−𝑥⁶𝛾+2𝛽1+2−8𝑥²+3𝑥⁴𝛾×1Γ()+𝑡1+2𝛼_𝛼𝑥⁴𝛾−6𝑎+2𝑥𝛽+−1+𝑥²²𝛾²Γ(1+2𝛼)Γ²()Γ()−𝑡1+𝛼1+3𝛼_2𝛼−6𝑎+2𝑥𝛽+−1+𝑥²²𝛾³Γ(1+3𝛼)Γ³()Γ().1+𝛼1+4𝛼 (19) 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: 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+⋯+⋯. (20)

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 𝜆(𝑠)1=−𝑠^𝛼 (21) and the recursive formula is given as 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−ℒ⁻¹1𝑠^𝛼ℒ−𝛽𝑢_𝑛𝜕𝑢_𝑛+𝜕𝜕𝑥𝜕𝑥𝑎𝑢_𝑛𝜕𝑢_𝑛𝜕𝑥+𝛾𝑢_𝑛1−𝑢_𝑛𝑢_𝑛−1 (22) with the initial term 𝑢₀(𝑥,𝑡)=𝑥², (23) so that using the iteration formulas we obtain 𝑢₁(𝑥,𝑡)=𝑥²−𝑡^𝛼Γ()×1+𝛼6𝑎𝑥²−2𝑥³𝛽+𝑥²1−𝑥²𝑥²−1𝛾,𝑢₂()=𝑥𝑥,𝑡₂𝑡^3𝛼−144𝑎³+12𝑎²×18𝑥𝛽+4𝛾−22𝑥²𝛾+21𝑥⁴𝛾−4𝑎12𝑥𝛽𝛾−54𝑥³𝛽𝛾+48𝑥⁵𝛽𝛾+𝛾²+43𝑥⁴𝛾²−48𝑥⁶𝛾²+18𝑥⁸𝛾²+2𝑥²12𝛽²−7𝛾²+𝑥2𝛽𝛾²+52𝑥⁴𝛽𝛾²−52𝑥⁶𝛽𝛾²+18𝑥⁸𝛽𝛾²+26𝑥⁷𝛾³−14𝑥⁹𝛾³+3𝑥¹¹𝛾³+4𝑥²3𝛽³−5𝛽𝛾²+6𝑥⁵5𝛽²𝛾−4𝛾³+2𝑥5𝛽²𝛾−𝛾³+𝑥³−36𝛽²𝛾+11𝛾³×Γ(1+𝛼)Γ²(1+2𝛼)Γ(1+4𝛼)+𝑡^2𝛼180𝑎²+6𝑥𝛽𝛾−20𝑥³𝛽𝛾+14𝑥⁵𝛽𝛾+𝛾²+12𝑥⁴𝛾²−10𝑥⁶𝛾²+3𝑥⁸𝛾²−4𝑎23𝑥𝛽+9𝛾−25𝑥²𝛾+16𝑥⁴𝛾+2𝑥²5𝛽²−3𝛾²×Γ(1+3𝛼)Γ³()Γ()()Γ()×𝑡1+𝛼1+4𝛼+Γ1+2𝛼1+3𝛼_4𝛼𝑥⁴𝛾×−6𝑎+2𝑥𝛽+𝛾−2𝑥²𝛾+𝑥⁴𝛾³()−×Γ1+3𝛼2𝑡^𝛼16𝑎−2𝑥𝛽−𝛾+2𝑥²𝛾−𝑥⁴𝛾−Γ(1+𝛼)Γ(1+𝛼)×Γ³()Γ()×Γ1+𝛼1+4𝛼₃()Γ()()Γ()1+𝛼1+2𝛼×Γ1+3𝛼1+4𝛼⁻¹. (24)(25) 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 𝑢(𝑥,𝑡)=𝑢₂(𝑥,𝑡). (26) 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 𝛼=1.5, 𝛾=1, and 𝑎=4. 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 𝛼=0.9, Figure 2 shows the approximate solution via the NDVIM. Figure 3 Show the approximate solution obtained via the HDM for 𝛼=0.25 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.

Table 1 

Comparison of numerical values for the approximate solutions via HDM and NDVIM.

Figure 1 

Approximate solution of the time-fractional nonlinear Nagumo equation via the HDM for the value of alpha equal to 0.9.

Figure 2 

Approximate solution of the time-fractional nonlinear Nagumo equation via the NDVIM for the value of alpha equal to 0.9.

Figure 3 

Approximate solution of the time-fractional nonlinear Nagumo equation via the HDM for the value of alpha equal to 0.25.

Figure 4 

We present in Table 1 the numerical values of the approximate solutions obtained via both methods 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.

6. Conclusion

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.

Acknowledgment

Abdon Atangana would like to thank Claude Leon foundation for their financial support.