Numerical approximation of Newell-Whitehead-Segel equation of fractional order

1 Introduction

In natural phenomena, non-equilibrium systems are usually shown in many extended states: uniform, oscillatory, chaotic and pattern states. Many stripes patterns such as ripples in sand, stripes of seashells arise in a variety of spatially extended systems which can be modeled by a set of equations called amplitude equations. One of the most well known amplitude equation is the Newell-Whitehead-Segel equation which describes the appearance of the stripe pattern in two dimensional systems. Moreover, this type of equations describe the dynamical behavior near the bifurcation point of the Rayleigh-Benard convection of binary fluid mixtures [1].

The Rayleigh-Benard convection is a type of natural convection arising in a plane, horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bernard cells. When the heating is ample high, convective motion of the fluid developed spontaneously then the hot fluid moves upward, and the cold fluid moves downward. Rayleigh-Benard convection is the one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. The convection patterns are the most carefully examined example of self-organizing nonlinear systems [2]. Buoyancy and gravity are responsible for the appearance of convection cells. The initial movement is the upwelling of warmer liquid from the heat bottom layer [3].

There are two types of patterns that are notice normally. First is the roll pattern in which the fluid stream lines form cylinders.

These cylinders may be bent and they may form spirals or target-like patterns. Second pattern is the hexagonal one in which the liquid flow is divided into honeycomb cells. For some fluids, the motion is downward in the centre of each cell and upward on the border between the cells; for other fluids, the motion is in the opposite direction. The same patterns, stripes and hexagons appear in completely different physical systems and on different spatial scales. For example, stripe patterns are observed in human fingerprints, on Zebra’s skin and in the visual cortex. Hexagonal patterns result from the propagation of laser beams through a nonlinear medium and in systems with comical reaction and diffusion species [4].

The Newell-Whitehead-Segel equation is written as

where a,b and k are real numbers with k > 0, q is a positive integer, u(x, t) is a function of the spatial variable x and the temporal variable t, with x ∈ R and t ≥ 0. The function u(x, t) may be thought of as the distribution of temperature in an infinitely thin and long rod or as the flow velocity of a fluid in an infinitely long pipe with small diameter. In Eq. (1) the first term on the left hand side ∂u∂t express the variations of u(x, t) with time at a fixed location, the first term on the right hand side ∂2u∂x2, represents the variations of u(x, t) with spatial variable x at a specific time and au-bu^q is the source term. In recent years various methods have been used to handle Newell-Whitehead-Segel equation such as Adomian decomposition method [5], differential transform method [6], homotopy perturbation method [7], finite difference scheme [8], cubic B-spline collocation method [9], etc.

In this article, we consider the fractional model of Newell-Whitehead-Segel Eq. (1)of the form

where α is a parameter which describe the order of the fractional derivative. The fractional derivative is taken in the Caputo sense. If we take α = 1, the fractional Newell-Whitehead-Segel Eq. (2) reduces to the classical Newell-Whitehead-Segel Eq. (1). The most important the advantage of making use of fractional models of differential equations in these and other physical problems is there non-local property. We know that the integer order derivative is the local operator but the fraction derivative is non local in nature. It shows that the next state of physical system depends not only upon its current state but also upon all of its historical states. Hence the fraction models are more realistic and it is one reason why fraction derivative have become more and more popular to model problems in fluid mechanics, acoustics, biology, electromagnetism, diffusion, signal processing, and many other physical processes [10–17]. Very recently, Caputo and Fabrizio [18] introduced a new fractional derivative without singular kernel. Further more, the properties and applications of this new derivative is studied by Losada and Nieto [19], Atangana [20] and Atangana and Badr [21]. In recent years various methodologies have been proposed to solve non linear fractional differential equations such as Adomian decomposition method [22], differential transform method [23], homotopy perturbation method [24], variation iteration method [25], etc. But these methods have their own limitations. The homotopy analysis method (HAM) was introduced and developed by chines researcher Liao [26–28] to handle nonlinear problems. The HAM have also been applied to solve various physical problems [29–31]. The HAM is also combined with Laplace transform method [32–35] and Sumudu transform method [36] to produce highly effective techniques to derive the solution of non linear differential equations. In the present paper, we apply a hybrid approach based on the homotopy analysis method and Sumudu transform algorithm. This technique is called homotopy analysis Sumudu transform method (HASTM). The advantage of HASTM is its capability of combining two powerful techniques to derive exact and approximate analytical solutions of nonlinear problems. Its give the solution in terms of convergent series with easily computable components in a direct way which does not require linearization, perturbation, or restrictive assumptions.

2 Basic Definitions

In this section, we present the following basic definitions of fractional calculus and Sumudu transform.

3 Application of HASTM to fractional Newell-Whitehead-Segel equation

In this section, we demonstrate the efficiency of HASTMto solve the following Newell-Whitehead-Segel equations.

Example 1.

Consider the linear fractional Newell-Whitehead-Segel equation

with the initial condition

Applying the Sumudu transform on both the sides of Eq. (9), we have

Using the differentiation property of Sumudu transform, we have

The nonlinear operator is

and thus

where

The m^th−order deformation equation is given by

Applying the inverse sumudu transform, we get

Solving above equation (17), for m = 1, 2, 3, ..., we get

U₀(x, t)=e^x

In the similar way, we can drive remaining components of the series solution. If we set ℏ = −1, the HASTM solution is given by

Taking α = 1, we get the solution of classical Newell-Whitehead-Segel equation in the form

which converge very fastly to the to the exact solution

which is the same solution as obtained by ADM and DTM [6] and HPM [7]. The numerical results obtained by using HASTM and exact solution are depicted through Fig. 1–3. It can be observed that the HASTM solution is approximately same at α = 1 for different values of x and t. It is also seen that as the value of t increases, U decreases, and x increases, U also increases. It is to be noted that only eight terms of the series solution are considered for Fig. 1–3. Hence, the accuracy of HASTM can be enhanced dramatically by increasing the number of iterations.

Fig 1.

The behavior of the exact solution U(x,t) w.r.t.x and t, when α = 1 for Eqs. (9)–(10)

Fig 2.

The behavior of the approximate solution U(x,t) w.r.t.x and t, when α = 1 for Eqs. (9)–(10)

Fig 3.

Error E₈(U) = |Exact − Approximate| for Eqs. (9)–(10)

Example 2. Next, consider the non linear fractional Newell-Whitehead-Segel equation

with the initial condition

Taking the Sumudu transform on both the sides of Eq. (22), we have

The non linear operator is

and thus

The m^th - order deformation equation is given by

Applying the inverse Sumudu transform, we get

Solving above equation (28), for m =1, 2, 3, ..., we get

U₀(x, t)=λ

In the similar way, we can drive remaining components of the series solution. If we set ℏ = −1, the HASTM solution is given by

Taking α= 1, we get the solution of classical nonlinear Newell-Whitehead-Segel equation in the form

which converge very fastly to the to the exact solution

which is the same solution as obtained by ADM and DTM [6] and HPM [7]. The numerical results obtained by using HASTM and Exact solution are presented through Fig. 4 at α = 1 and λ = 1 for different values of t. The Fig. 4 shows that the results obtained with the help of HASTM are approximately same to exact solution. It is also to be noticed that the when we increase the value of t, then U decreases.It is also worth mentioning that only three iterations are used to compute the results for Fig. 4. Hence, if we increase the number of iterations then, efficiency and accuracy can be dramatically enhanced.

Figure 4.

Plots of U(x, t) vs. t at α = 1 and λ = 1 exact solution and HASTM for Eqs. (22)–(23)

4 Conclusions

In this article, a fractional model of Newell-Whitehead-Segel equation is studied by using HASTM and symbolic computation. The numerical results obtained by using the proposed algorithm are in a very good agreement with the exact solution and the results obtained with the help of homotopy perturbation method, Adomian’s decomposition method and differential transform method. It is worth mentioning that the results derived by homotopy perturbation method, Adomian’s decomposition method and differential transform method are particular case of the solution obtained by HASTM. HASTM gives us a very easy way to adjust and control the region of convergence of the series solution by selecting proper value of ћ. The results reveal that HASTM is a very efficient and computationally attractive approach to investigate nonlinear fractional systems of physical significance.