Solving the fractional BBM–Burgers equation using the homotopy analysis method

doi:10.1016/j.chaos.2007.09.042

Chaos, Solitons & Fractals

Volume 40, Issue 4, 30 May 2009, Pages 1616-1622

https://doi.org/10.1016/j.chaos.2007.09.042 Get rights and content

Abstract

Based on the homotopy analysis method, a scheme is developed to obtain approximation solution of a fractional BBM–Burgers equation with initial condition, which is introduced by replacing some integer-order space derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the traditional homotopy analysis method for differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional differential equations. The solutions of our model equation are calculated in the form of convergent series with easily computable components.

Introduction

In the past decades, both mathematicians and physicists have devoted considerable effort to the study of explicit solutions to nonlinear integer-order differential equation. Many powerful methods have been presented [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34]. Among them, the homotopy analysis method (HAM) [22], [23], [24], [25], [26], [27] provides an effective procedure for explicit and numerical solutions of a wide and general class of differential systems representing real physical problems. Based on homotopy of topology, the validity of the HAM is independent of whether or not there exists small parameters in the considered equation. Therefore, the HAM can overcome the foregoing restrictions and limitations of perturbation techniques so that it provides us with a possibility to analyze strongly nonlinear problems. This method has been successfully applied to solve many types of nonlinear problems by others [28], [29], [30], [31], [32], [33], [34]. However, the application of HAM only circumscribes integer-order differential equation. Here, we introduce HAM to nonlinear fractional differential equation.

In recent years, considerable interest in fractional differential equation has been stimulated due to their numerous applications in the areas of physics and engineering [35]. Many important phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are well described by fractional differential equation [36], [37], [38]. The solution of fractional differential equation is much involved. In general, there exists no method that yields an exact solution for fractional differential equation. Only approximation solution can be derived using linearization or perturbation method.

The aim of this paper is to directly extend the HAM to consider the numerical solution of the fractional BBM–Burgers equation with space-fractional derivatives of the form $u_{t} + D_{x}^{α} u + {uu}_{x} - {au}_{xx} - {bu}_{xxt} = 0, t > 0, 0 < α ⩽ 1,$ where a, b are constants and α is parameter describing the order of the fractional space-derivatives. The function u(x, t) is assumed to be a causal function of space, i.e. vanishing for t < 0 and x < 0. The fractional derivatives are considered in the Caputo sense [39]. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. In the case of α = 1, Eq. (1.1) reduces to the classical nonlinear BBM–Burgers equation. More important, above procedure is just an algebraic algorithm and can be easily applied in the symbolic computation system Maple.

Although there are a lot of studies for the classical BBM–Burgers equation and some profound results have been established, it seems that detailed studies of the nonlinear fractional differential equation are only beginning. According to our knowledge, the paper represents the first available numerical solution of the fractional BBM–Burgers equation with space-fractional derivatives.

The paper has been organized as follows. In Section 2, a brief review of the theory of fractional calculus will be given to fix notation and provide a convenient reference. In Section 3, we give analysis of the HAM. In Section 4, we extend the application of the HAM to construct numerical solution for the fractional BBM–Burgers equation. Conclusions are presented in Section 5.

Section snippets

Preliminaries and notations

In this section, let us recall essentials of fractional calculus first. The fractional calculus is a name for the theory of integrals and derivatives of arbitrary order, which unifies and generalizes the notions of integer-order differentiation and n-fold integration. There are many books [35], [36], [37], [38] that develop fractional calculus and various definitions of fractional integration and differentiation, such as Grünwald–Letnikov’s definition, Riemann–Liouville’s definition, Caputo’s

Analysis of the homotopy analysis method

In the paper, we apply the homotopy analysis method [22], [23], [24], [25], [26], [27] to the discussed problem. We extend Liao’ basic ideas to the fractional differential equation.

Let us consider the fractional differential equation $FD (u (x, t)) = 0,$ where $FD$ is a fractional differential operator, x and t denote independent variables, u(x, t) is an unknown function. For simplicity, we ignore all boundary or initial conditions, which can be treated in the same way.

Based on the constructed zero-order

Application

To demonstrate the effectiveness of the method, we consider Eq. (1.1) with the initial condition $u (x, 0) = ρ x^{2}, ρ \in C$

We choose the linear integer-order operator $L [U (x, t; q)] = \frac{\partial U (x, t; q)}{\partial t} .$

Furthermore, Eq. (1.1) suggests to define the nonlinear fractional differential operator $NFD [U (x, t; q)] = U_{t} (x, t; q) + D_{x}^{α} U (x, t; q) + U (x, t; q) U_{x} (x, t; q) - {aU}_{xx} (x, t; q) - {bU}_{xxt} (x, t; q) .$

Using above definition, we construct the zeroth-order deformation equation $(1 - q) L [U (x, t; q) - u_{0} (x, t)] = qh NFD [U (x, t; q)] .$

Obviously, when q = 0 and q = 1, $U (x, t; 0) = u_{0} (x$

Conclusions

Here, we solve fractional BBM–Burgers equation with space-fractional derivatives. We can also apply the HAM to solve fractional BBM–Burgers equation with time-fractional derivatives, $D_{t}^{α} u + u_{x} + {uu}_{x} - {au}_{xx} - {bu}_{xxt} = 0, t > 0, 0 < α ⩽ 1,$

At the same time, we choose linear integer-order operator $L [U (x, t; q)] = U_{x} (x, t; q),$ U_xx(x, t; q), or U_xxt(x, t; q). If we choose linear fractional operator, there are difficulties in solving process. How to conquer the difficulties by modifying the method need us study further.

The

Acknowledgement

The work is partially supported by the National Key Basic Research Project of China under the Grant No. 2004CB318000.

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Citation Excerpt :
This technique has been applied to solve linear and non-linear fractional partial differential equations [25,26]. For instance, the fractional KdV–Burgers–Kuramoto equation [27], the non-linear fractional Riccati equation [28], fractional initial value problems (fIVPs) [29], the Benjamin–Bona–Mahony–Burgers equation [30], the fractional Klein–Gordon equation [31,32], this method has also been used to solve linear homogeneous one and two-dimensional fractional heat-like partial differential equations subject to the Neumann boundary conditions [33], linear and non-linear homogeneous fractional diffusion-wave equations [34] or linear and non-linear systems of FDEs [35]. Other interesting applications of HAM can be found in [36–40].
In this paper, we present a new definition of fractional-order derivative with a smooth kernel based on the Caputo–Fabrizio fractional-order operator which takes into account some problems related with the conventional Caputo–Fabrizio factional-order derivative definition. The Modified-Caputo–Fabrizio fractional-order derivative here introduced presents some advantages when some approximated analytical methods are applied to solve non-linear fractional differential equations. We consider two approximated analytical methods to find analytical solutions for this novel operator; the homotopy analysis method (HAM) and the multi step homotopy analysis method (MHAM). The results obtained suggest that the introduction of the Modified-Caputo–Fabrizio fractional-order derivative can be applied in the future to many different scenarios in fractional dynamics.

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Solving the fractional BBM–Burgers equation using the homotopy analysis method

Abstract

Introduction

Section snippets

Preliminaries and notations

Analysis of the homotopy analysis method

Application

Conclusions

Acknowledgement

Phys Lett A

Chaos Solitons & Fractals

Phys Lett A

Phys Lett A

Chaos Solitons & Fractals

Chaos Solitons & Fractals

Appl Math Comput

Chaos Solitons & Fractals

Chaos Solitons & Fractals

Appl Math Comput

Int J Non-linear Mech

Appl Math Comput

Int J Heat Mass Transfer

Appl Math Comput

Int J Heat Mass Transfer

Phys Lett A

Phys Lett A

Solitons, nonlinear evolution equations and inverse scattering

J Phys Soc Jpn