Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: A new iterative algorithm

Abstract

In this paper, explicit and approximate solutions of the nonlinear fractional KdV–Burgers equation with time–space-fractional derivatives are presented and discussed. The solutions of our equation are calculated in the form of rabidly convergent series with easily computable components. The utilized method is a numerical technique based on the generalized Taylor series formula which constructs an analytical solution in the form of a convergent series. Five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method. Graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution. The results reveal that the method is very effective and simple in determination of solution of the fractional KdV–Burgers equation.

Introduction

The Korteweg–de Vries–Burgers (KdV–Burgers) equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a nonlinear partial differential equation (PDE) whose solution can be exactly and precisely specified. The mathematical theory behind the KdV–Burgers equation is rich and interesting, and, in the broad sense, is a topic of active mathematical and physical research [1], [2], [3], [4], [5], [6]. Due to this, the present paper is going to present an analytical method, so-called residual power series (RPS) [7], [8], [9], [10], [11], so that, it enables us to predict and calculate the solution which admits nonlinear KdV–Burgers equation of fractional order.

In the last century notable contributions have been made to both the theory and applications of the fractional differential equations (DEs). These equations are increasingly used to model problems in research areas as diverse as dynamical systems, mechanical systems, control theory, heat transfer, chaos synchronization, mixed convection flows, anomalous diffusive, unification of diffusion, wave propagation phenomenon, image processing, and entropy theory [10], [11], [12], [13], [14], [15], [16], [17], [18]. The most important advantage of using fractional DEs in these and other applications is their nonlocal property. It is well known that the integer order differential operators and the integer order integral operators are local but on the other aspect as well, the fractional order differential operators and the fractional order integral operators are nonlocal. This means that the next state of a system depends not only upon its current state but also upon all of its historical states. In fact, this is main reason why differential operators of fractional order provide an excellent instrument for description of memory and hereditary properties of various mathematical, physical, and engineering processes. The reader is asked to refer to [19], [20], [21], [22], [23] in order to know more details about the fractional DEs, including their history and kinds, their existence and uniqueness of solutions, their applications and methods of solutions, etc.

Numerical techniques are widely used by scientists and engineers to solve their problems. A major advantage for numerical techniques is that a numerical answer can be obtained even when a problem has no analytical solution. In some cases, PDEs of fractional order can be solved analytically, where finding their solutions in the general case is hard and limited to the linear one. Anyhow, in most real-life applications, it is too complicated to obtain the exact solutions to PDEs of fractional order in terms of composite elementary functions in a simple manner, so an efficient, reliable numerical algorithm for the solutions of such equations is required; it is little wonder that with the development of fast, efficient digital computers, the role of numerical methods in mathematics, physics, and engineering problem solving has increased dramatically in recent years.

The RPS method was developed by the second author [7] as an efficient method for determining values of coefficients of the power series solution for the first- and the second-order fuzzy DEs. It has been successfully applied in the numerical solution of the generalized Lane–Emden equation, which is a highly nonlinear singular DE [8], in the numerical solution of higher-order regular DEs [9], in the solution of composite and noncomposite fractional DEs [10], and in predicting and representing the multiplicity of solutions to boundary value problems of fractional order [11]. The RPS method is effective and easy to construct power series solution for strongly linear and nonlinear equations without linearization, perturbation, or discretization [7], [8], [9], [10], [11]. Different from the classical power series method, the RPS method does not need to compare the coefficients of the corresponding terms and a recursion relation is not required. This method computes the coefficients of the power series by a chain of algebraic equations of one or more variables. In fact, the RPS method is an alternative procedure for obtaining analytic solution for PDEs of fractional order. By using residual error concept, we get a series solution, in practice a truncated series solution. Moreover, the obtained solution and all its fractional derivatives are applicable for each arbitrary point and each multi-dimensional variables in the given domain. On the other aspect as well, the RPS method does not require any converting while switching from the low-order to the higher-order and from simple linearity to complex nonlinearity; as a result the method can be applied directly to the given problem by choosing an appropriate initial guess approximation.

The aim of this paper is to directly extend the application of the RPS method in the sense of the Caputo's fractional derivative to consider and construct an approximate solution of the nonlinear fractional KdV–Burgers equation with time–space-fractional derivatives of the following form:

$$\frac{{\partial }^{\alpha }u(x,t)}{\partial t^{\alpha }}+\epsilon u^r(x,t)\frac{{\partial }^{\beta }u(x,t)}{\partial x^{\beta }}+\eta \frac{{\partial }^{2}u(x,t)}{\partial x^2}+\vartheta \frac{{\partial }^{3}u(x,t)}{\partial x^3}=0,0<\alpha ,\beta \le 1,x,t>0,$$

subject to the initial condition

$$u(x,0)=f(x),$$

where $\epsilon ,\eta ,\vartheta$ are given constants, $r=0,1,2$, and $\alpha ,\beta$ are parameters describing the order of fractional time-derivative and fractional space-derivative, respectively. The function $u(x,t)$ is assumed to be a causal function of time and space, which means that $u(x,t)$ is vanishing for $t<0$ and $x<0$ and this function is assumed to be analytic on $t>0$. Also, the function $f(x)$ is assumed to be analytic on $x>0$. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. As special cases, when $(\alpha ,\beta )=(1,1)$ the fractional KdV–Burgers equation (1.1) reduces to the classical KdV–Burgers equation, while on the other hand, when $\beta =1$ and $0<\alpha \le 1$, it reduces to the time-fractional equation and when $\alpha =1$ and $0<\beta \le 1$, it reduces to the space-fractional equation.

Today, nonlinear fractional PDEs are widely used to describe many important phenomena and dynamic processes in physics, engineering, electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science [19], [20], [21], [22], [23]. For better understanding the phenomena that a given nonlinear fractional PDEs describes, the solutions of DEs of fractional order is much involved. Anyhow, to customizable more, numerous authors have discussed the approximate and exact solutions to the nonlinear KdV–Burgers equation of fractional order using some of the well-known methods. The reader is asked to refer to [24], [25], [26], [27], [28], [29], [30], [31], [32] in order to know more details about these methods, including their types, their motivation for use, their characteristics, and their applications.

The outline of the paper is as follows. In the next section, we utilize some necessary definitions and results from fractional calculus theory. In Section 3, basic idea of the RPS method is presented in order to construct and predict the solution of the fractional KdV–Burgers equation (1.1) subject to initial condition (1.2). In Section 4, five fractional KdV–Burgers models are performed in order to illustrate the capability, the potentiality, and the simplicity of the proposed method. Finally, conclusions are presented in Section 5.