Research Article | Open Access
Numerical Solutions for the Time and Space Fractional Nonlinear Partial Differential Equations
We implement relatively analytical techniques, the homotopy perturbation method, and variational iteration method to find the approximate solutions for time and space fractional Benjamin-Bona Mahony equation. The fractional derivatives are described in the Caputo sense. These methods are used in applied mathematics to obtain the analytic approximate solutions for the nonlinear Bejamin-Bona Mahoney (BBM) partial fractional differential equation. We compare between the approximate solutions obtained by these methods. Also, we present the figures to compare between the approximate solutions. Also, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved -expansion function method to find exact solutions of nonlinear fractional BBM equation.
In recent years, there has been a great deal of interest in fractional differential equations. First there were almost no practical applications of fractional calculus, and it was considered by many as an abstract area containing only mathematical manipulations of little or no use. Nearly 30 years ago, the paradigm began to shift from pure mathematical formulations to applications in various fields. During the last decade, fractional calculus has been applied to almost every field of science, engineering, and mathematics. Several fields of application of fractional differentiation and fractional integration are already well established, and some others have just started. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, and astrophysics [1–11]. There has been some attempt to solve linear problems with multiple fractional derivatives (the so-called multiterm equations) [1, 12]. Not much work has been done for nonlinear problems, and only a few numerical schemes have been proposed to solve nonlinear fractional differential equations. More recently, applications have included classes of nonlinear equation with multiorder fractional derivative, and this motivates us to develop a numerical scheme for their solutions . Numerical and analytical methods have included Adomian decomposition method (ADM) [14–17], variational iteration method (VIM) [18–20], homotopy perturbation method [21–24], homotopy analysis method [25–27], and the fractional complex transformation [28, 29] to get some special exact solutions for nonlinear partial fractional differential equation.
Recently, Zhao  investigated the nature of protein dynamics and thermodynamics. Olszewski  studied the uncertainty relation between intervals of energy and time derived for the electromagnetic radiation of a harmonic oscillator. El-Naggar et al.  investigated the initial stress, magnetic field, voids, and rotation effects on plane waves in generalized thermoelasticity. Khrennikov  discussed “Einstein’s Dream” Quantum Mechanics as Theory of Classical Random Fields.
The main objective of this paper is to use two different methods such as homotopy perturbation method and variational iteration method for calculating the analytic approximate solutions of the Bejamin-Bona Mahoney equation. We compare between the approximate solutions which obtained from these methods. We make the figures to compare the approximate solutions. Also, we find some exact solutions to time and space fractional Bejamin-Bona Mahoney equation by using the improved -expansion function method.
2. Preliminaries and Notations
We give some basic definitions and properties of the fractional calculus theory which are used further in this paper [1, 2]. For the finite derivative [a,b], we define the following fractional integral and derivatives.
Definition 1. If , the set of all integrable functions, and then, the Riemann-Liouville fractional integral of order , denoted by , is defined by
Definition 2. For , the Caputo fractional derivative of order , denoted by , is defined by
where is such that and .
If is an integer, then this derivative takes the ordinary derivative
Finally the Caputo fractional derivative on the whole space is defined by the following.
Definition 3. For , the Caputo fractional derivative of order on the whole space, denoted by , is defined by
In recent years, in order to investigate the local behavior of fractional models, several local versions of fractional derivatives have been proposed, that is, the Kolwankar-Gangal local fractional derivative , Chen’s fractal derivative , Cresson’s derivative , and Jumarie’s modified Riemann-Liouville derivative [38, 39]. Jumarie’s derivative is defined as where , denotes a continuous (but not necessarily first-order-differentiable) function. We can obtain the following properties.
Property 3. Some useful formulas include
Function should be differentiable with respect to , and is fractional differentiable in (8). The above results are employed in the following sections. The modified Riemann-Liouville derivative has been successfully applied in the probability calculus , the fractional Laplace problems , the fractional variational approach with several variables , the fractional variational iteration method , the fractional variational approach with natural boundary conditions , and the fractional Lie group method .
3. Basic Idea of the Homotopy Perturbation Method (HPM)
In this section, we illustrate some basic concepts of the homotopy perturbation method for the following nonlinear partial fractional differential equation [21–24]: subject to the initial and boundary conditions where is a linear operator, while is a nonlinear operator, is a known analytical function, and denotes the fractional derivative in the Caputo sense. The solution is assumed to be a causal function of time, that is, vanishing for . Also, is the th derivative of , are the specified initial conditions, and is a boundary operator. Using He’s homotopy perturbation technique , we can construct the following simple homotopy: or
Applying the homotopy perturbation method, we use the homotopy parameter to expand the solution into the following form:
For nonlinear problems, let us set . Substituting (16) into (13) or (14) and equating the terms with identical powers of , we can obtain a series of equations of the form and so on, where the functions satisfy the following equations:
Applying the operator on both sides of (17) and considering the initial and boundary conditions, the terms of the series solution can be given by
On setting , we get an accurate approximate solution in the following form:
4. Basic Idea of Variational Iteration Method (VIM)
The basic concepts of the variational iteration method can be expressed as follows. Consider the differential equation of the form where is a linear operator, is a nonlinear operator, and is the inhomogeneous term. According to VIM, a correction functional for (23) can be constructed as follows: where is a general Lagrange multiplier, which can be identified optimally via the variational theory, the subscript denotes the th approximations, and is considered as restricted variation; that is, . The successive approximations , , of the solution can be obtained after finding the Lagrange multiplier and by using the selective function which is usually selected from initial conditions. Consequently, the solution is obtained as
5. Approximate Solutions for the Time and Space Fractional BBM Equation
In this section, we use the homotopy perturbation method and variational iteration method to calculate the approximate solutions for the time and space fractional Bejamin-Bona Mahoney equation. Also, we compare the approximate solutions obtained from these methods.
5.1. The HPM for the Time and Space Fractional Nonlinear BBM Equation
In this subsection, we use the HPM to construct the approximate solution of the time and space fractional nonlinear BBM equation in the form where , , and are constants.
With the initial condition,
By the homotopy perturbation technique, we construct a homotopy which satisfies
According to the homotopy perturbation method, we can first use the embedding parameter as a small parameter and assume that the solution of (28) can be written as a power series in as follows:
Substituting (29) into (28) and arranging the coefficients of the powers of , after some calculation we obtain and so on. We substitute the initial condition (27) into the system (30), then the approximate solutions of (26) take the following form: and so on. Thus the approximate solution of (26) is given by where
5.2. The VIM for the Time-Space Fractional Nonlinear BBM Equation
In this subsection, we use the VIM to construct the approximate solution of the time and space fractional nonlinear BBM equation (26).
The correction functional for (26) can be approximately expressed as follows: where is a general Lagrange multiplier, is considered as restricted variations, and is considered as a restricted variation. Making the above correction functional stationary and noticing that , we obtain Therefore, it can be identified as , and the following variational iteration formula can be obtained:
By the iteration formula (36), we can obtain the other components as and so on, and in the same manner, the rest of components of the iteration formula (36) can be obtained using Mathematica package where
Now we make the comparison between the approximate solutions obtained by the variation iteration method and the approximate solution obtained by the homotopy perturbation method as shown in Figures 1, 2, 3, 4, and 5.
Remark 4. From Figures 2–5, the approximate solution obtained by the homotopy perturbation method is different from the approximate solutions obtained by variational iteration when and are not positive integers.
6. The Improved -Expansion Function Method
Consider the following nonlinear partial fractional differential equation: where is an unknown function and is a polynomial of and its partial fractional derivatives, in which the highest order derivatives and the nonlinear terms are involved.
In the following, we give the main steps of the improved -expansion method.
Step 1. Li and He  proposed a fractional complex transformation to convert fractional differential equations into ordinary differential equations (ODE), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. The traveling wave variable
where , , , and are nonzero arbitrary constants, permits us to reduce (40) to an ODE of in the form
If possible, we should integrate (43) term by term one or more times.
Step 3. The positive integer can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in (43). More precisely, we define the degree of as , which gives rise to the degrees of the other expressions as follows:
Therefore, we can obtain the value of in (44).
Step 4. Substituting (44) into (43), using (45), collecting all terms with the same order of together and equating each coefficient of the resulting polynomial to zero, we obtain a set of algebraic equations for , , , , , , and .
Step 5. Since the general solutions to (45) are well known, then substituting , , , , , , , and the general solutions of (45) into (44), we get more traveling wave solutions of the nonlinear partial fractional derivatives (40).
In this section, we use the improved -expansion method to construct the exact solutions for some nonlinear partial fractional differential equations, via the time-space fractional nonlinear BBM equation.
We can see that the fractional complex transformation where and are constants, permits us to reduce (26) to an ordinary differential equation. After that, we integrate to obtain where is the integration constant.
Considering the homogeneous balance between the highest order derivative and the nonlinear term in (48), we deduce that . Thus, we get where , , , , and are arbitrary constants to be determined later. Substituting (49) into (48), collecting all the terms of powers of , and setting each coefficient to zero, we get a system of algebraic equations. With the aid of Maple or Mathematica, we can solve this system of algebraic equations to obtain the following sets of solutions.
Family 1. If , then we have the hyperbolic solution
Family 2. If , then we have the trigonometric solution
Family 3. If , then we have where
Family 1. If , then we have the hyperbolic solution
Family 2. If , then we have the trigonometric solution
Family 3. If , then we have where
In this paper, we use the homotopy perturbation method and variational iteration method to obtain the analytic approximate solutions for linear and nonlinear partial fractional differential equations. Section 5 shows the comparative study between the factional variational itearation methods and the homotopy perturbation method. The two methods are powerful and effective tools for the solutions of fractional partial differential equations, and both give two different approximate solutions of higher accuracy and closed form solutions if existing. Also, we use the improved -expansion function method to calculate the exact solutions for the time and space fractional nonlinear BBM equation. This method is reliable and simple and gives many new exact solutions for the nonlinear fractional differential equations. When the parameters take certain special values, the solitary waves are derived from the traveling waves. This method is very efficient and powerful in finding the exact solutions for the nonlinear fractional differential equations.
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