Abstract
We investigate a nonlinear wave phenomenon described by the perturbation Rosenau-Hyman equation within the concept of derivative with fractional order. We used the Caputo fractional derivative and we proposed an iteration method in order to find a particular solution of the extended perturbation equation. We proved the stability and the convergence of the suggested method for solving the extended equation without any restriction on and also on the perturbations terms. Using the inner product we proved the uniqueness of the special solution. By choosing randomly the fractional orders and , we presented the numerical solutions.
1. Introduction
Fractional calculus is referring to calculus of integrals and derivatives with any real or complex order and has increased significantly attractiveness throughout the past three decades, owing for the most part to its established relevance in many fields of science and engineering. It does without a doubt make quite a lot of potentially advantageously helpful equipment available for finding the solutions of differential and integral equations and further problems connecting exceptional functions of mathematics physics in addition to their additional rooms and generalizations in one and more variables. No wonder researchers nowadays are interested in modelling several physical problems within the scope of the fractional calculus. This leads us to have a look at the nonlinear scattering waves.
Generalized Korteweg-de Vries equations with nonlinear dispersion can propagate compactly supported solitary waves, referred to as compactons [1–7]. Numerical simulations give you an idea about an original pulse wider than a compacton with small amount of radiation; in addition compactons collide elastically suffering only a phase shift after the collision and generating a small amplitude, zero-mass, compact (see [8–13]). Primarily discovered in the Rosenau-Hyman equation for the modeling of pattern developments in liquid drops, compactons have more than a few functions in physics and science [1], for instance, pattern configuration on liquid plane [14]. The equation is in addition the permanent boundary of the disconnected equations of a nonlinear lattice [1–15] and has been generalized to advanced dimensions [16]. Let us also note that the equation has other kinds of solutions, for instance, the elliptic compactons [1, 17, 18]. At long last, several generalizations of the equations have also been well thought-out in the literature, such as the insertion of time-dependent damping and dispersion [19] or the addition of fifth-order dispersion [20]. In this work we are very much interested in investigating the special solution of the perturbation Rosenau-Hyman equation, which we will generalize as follows:where are small parameters. The above equation will be called the perturbation fractional Rosenau-Hyman equation. Because of the plentiful profits offered by these fractional derivatives, many researchers have been stressing on proposing new definitions of fractional derivatives. We shall present the brief summary of these derivatives in the following section.
2. Some Fractional Calculus Definitions
It is perhaps important to mention that the concept of noninteger order derivative is an older concept, since it is considered to have stanched after an interrogation rose in the year 1695 by Marquis de L’Hopital. We shall present some of these definitions here.
Definition 1 (see [21–26]). The Riemann-Liouville fractional derivative is as follows: according to Riemann-Liouville the fractional derivative of a function says is given as
Definition 2. The Riemann-Liouville fractional integral is as follows: according to Riemann-Liouville, the fractional integral that is considered as antifractional derivative of a function is given as
Definition 3. Caputo fractional derivative is as follows: according to Caputo, the fractional derivative of a continuous and -time differentiable function is given as
Definition 4. The modified Riemann-Liouville fractional derivative of a function is given asThere are other definitions that are not mentioned here. We shall present the derivation of the special solution in the next section. However, in this paper we shall use the Caputo type.
3. Derivation of the Special Solution
One of the great challenges in the field of partial of differential equation is to derive the solution of especially nonlinear equations, not to mention nonlinear equations described with the fractional order derivative, for example, the perturbation fractional Rosenau-Hyman equation that has stronger nonlinearity. No wonder scholars of this area are always trying to find suitable and easier methods to derive at least approximate solution of these difficult equations. In this paper we will make use of a simple iteration method to propose a special solution to the perturbation fractional Rosenau-Hyman equation; the methods is called the new variational iteration method (NVIM) [27], which uses the idea of Lagrange multiplier. We shall first show the methodology to accommodate readers that are not used to this method. Now consider a general partial differential equation with high order () with respect to time; then, in this new VIM, the first step is to apply the Laplace transform on both sides of equation to obtainThe recursive formula of (6) can now be used to put forward the main recursive method connecting the Lagrange multiplier asNow considering as the restricted term, the Lagrange multiplier can be obtained as [27]Now applying the inverse Laplace transform on both sides of (6), we obtain the following iteration:with the first term to beNow following the above methodology, we can obtain the Lagrange multiplier of (1) to beand the iteration formula associate is given byand then
Theorem 5 (convergence analysis). Let us considerand consider the initial and boundary condition for (1); then the new variation iteration method leads to a special solution of (1).
Proof. To achieve this we shall think about the following fractional sub-Hilbert space of the Hilbert space [28] that can be defined as the set of those functions:We harmoniously assume that the differential operators are restricted under the norms. Using the definition of the operator, , we have the following:Our next task is to evaluate with being the inner product. Then, we have the following:To evaluate the above expression, we shall consider case by case: we shall start with the following:With the benefit of the Cauchy-Schwartz inequality, we have the following relationThe most important part in the above inequality is ; let us handle this part; first making use of the continuity properties of the derivative, it is possible for us to find a positive constant such thatThereforeNow if one makes use of the triangular inequality, we transform the above toHowever to proceed with our demonstration, we must assume that are bounded, implying that we can find a positive constant, say , such that . Therefore, Thus, we have the following:We shall now consider the following:Again, using the Cauchy-Schwartz inequality, we have the following relationship:Using the properties of the inner product the above can further be converted toHowever,using the same argument as before, we obtainAlsoNow replacing these relations into (17), we obtain the following relation:Takesuch thatThe next step in this proof will be to evaluateNevertheless, following the discussion presented, we shall havewhere is defined asThis completes the proof.
Our next concern will consist of investigating on the uniqueness of the special solution. We shall assume that there exist two special solutions, say and , satisfying (2); then, using (33), we haveSince both solutions satisfy (2), thus ; then, . Nonetheless, Thus if is the exact solution of (2) we have the following relation, with being a very small positive parameter closer to zero. Then the inequality (37) can be converted toAnd this implied, with the benefits of the norm, that
3.1. Application of the Scheme
We shall present the special solutions for some examples using the scheme presented earlier.
Example 1. Here we assume that the perturbation fractional Rosenau-Hyman equation is such that ; then we have the following equation:Using the presented scheme and also using the fractional integral proposed as antiderivative of the conformable fractional derivative, we obtain the following iteration formula:Now using the algorithm, we reach the following:Using the iteration formula the remaining terms can be calculated; however, if we go to infinity, the special solution for this version can be given aswhich of course satisfies that, for alpha equal to 1, we obtain the exact solution of the Rosenau-Hyman equationWe shall present the numerical results in Figures 1–4 for different value of alpha and . We have depicted the numerical results for different alpha and in Figures 1, 2, 3, and 4; for example, if = 50 and alpha = 0.5, we have Figure 1.
For example if alpha is 0.55 and is 50 we will observe nonlinear wave as represented in Figure 2.
Figures 1–4 as a result of introduction of fractional order derivative help us to understand better the role of nonlinear dispersion in prototype configuration. The solitary wave solutions of these equations have extraordinary assets according to the variation of the order of fractional derivative together with ; in addition the solitary wave collides elastically and they have compact support. The figures also reveal that, when two compactons collide, the interface location is indicated by the beginning of low-amplitude compacton-anticompacton pairs. These equations seem to have only a finite number of local conservation laws [1]. Nonetheless, the behaviour and the stability of these compactons are very similar to those observed in completely integrable systems [1].
4. Conclusion
One of the most difficult tasks in the area of applied mathematics is perhaps to make use of mathematical equations in order to explain adequately the physical phenomenon. Partial differential equations have been intensively used for this purpose in the past decades. However researchers have encountered some limitation by modelling physical phenomenon using the integer derivative order. In the way of trying to extend these limitations, the concept of noninteger order derivative has been introduced. In the contemporary day numerous objective occurrences were clarified with pronounced accomplishment in the light of the perception of noninteger order derivatives. Exclusively, the compensations of fractional calculus and fractional order replicas and their diligences in the field of nonlinear wave motion have beforehand been intensively reexamined throughout the last few epochs with exceptional conclusion. We have therefore investigated within that scope of fractional calculus the nonlinear wave phenomenon that is usually described via the Perturbation Rosenau-Hyman equation. We derived the special solution of the extended equation using the so-called Laplace transform and the Lagrange multiplier. We proved with great success the convergence of this method for solving the extended equation. On the other hand we showed the uniqueness of the special solution using some properties of the inner product within a well-constructed Hilbert space. Some numerical results were depicted from Figures 1–4. The special solution seems to describe the real world problem better than the standard solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research project was supported by a grant from the “Research Center of the Center for Female Scientific and Medical Colleges,” Deanship of Scientific Research, King Saud University. The authors are thankful to Visiting Professor Program at King Saud University for support.