Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model

Mehmet Yvuz ,,Ndolne Sene ,

a Department of Mathematics and Computer Sciences,Faculty of Science,Necmettin Erbakan University,42090 Konya,Turkey

b Department of Mathematics,College of Engineering,Mathematics and Physical Sciences,University of Exeter,Cornwall TR10,UK

c Laboratoire Lmdan,Département de Mathématiques de la Décision,Université Cheikh Anta Diop de Dakar,Faculté des Sciences Economiques et Gestion,BP 5683 Dakar Fann,Senegal

Abstract Before going further with fractional derivative which is constructed by Rabotnov exponential kernel,there exist many questions that are not addressed.In this paper,we try to recapitulate all the fundamental calculus,which we can obtain with this new fractional operator.The problems in this paper are to determine the solutions of the fractional differential equations where the second members are constant functions,polynomial functions,exponential functions,trigonometric functions,or Mittag-Leffler functions.For all the fractional differential equations,the obtained solutions are represented graphically.The Laplace transform of the fractional derivative with Rabotnov exponential kernel is the primary tool in the investigations.Finally,we give the fundamental solution to the nonlinear time-fractional modified Degasperis-Procesi equation by considering the fractional operator with Rabotnov exponential kernel.

Keywords: Fractional differential equation;Nonlinear dispersive wave model;Rabotnov exponential kernel;Mittag-Leffler function;Laplace transformation.

1.Introduction

Fractional calculus theory and its informative applications are attracting attention all over the world day by day.All of them have appeared since the Leibnitzs’ question.The question was,what is the value of the derivative

New fractional operators that have different features have been defined and have been used extensively to model reallife problems [1-5].The emergence of the new operators in the literature can be considered as a result of the reproduction of new problems that model different types of real-life events.For this reason,approaching to the real-life problems in terms of their fractional order versions has facilitated to model and solve them with a proper method.Therefore,it has been applied to a very wide area of science.

Many responses have addressed this question in the literature of fractional calculus.The mathematician as Riemann,Liouville,and Caputo were the first to bring a response to the Leibniz question.The derivatives as the Riemann-Liouville derivative [6,7]and the Caputo drivative [6,7]born from Leibniz question.These two fractional derivatives are the most used in fractional calculus,their numerical approximations are now well known,their Laplace transforms are proposed for the resolutions of the fractional differential equations.The diversity of the fractional derivatives comes to the fact the first proposed fractional derivative as the Riemann-Liouville derivative presents a singularity and the limiting case does not give the first-order derivative.It’s for these reasons Caputo was the first to reconsider the Riemann-Liouville derivative by proposing another fractional derivative known as the Caputo derivative.After that many other fractional derivatives appear in the literature as the conformable derivative [8,9],the Atangana-Baleanu derivative [10],the Caputo-Fabrizio derivative [11],the fractional derivative with Rabotnov exponential kernel (FDREK) [12]and others [13-21].

The applications of the fractional differential equations in real-world problems continue to preoccupy researchers.But it is proved now in many researches papers,due the memory aspect,the fractional derivatives explain more accurately the physics phenomena as the diffusion processes,as electrical circuits [22-26],the fluids mechanics [1,2,27],viscoelasticity [28],financial models [29-31],and many others domains [32-35,46,47].For the applications of the fractional derivatives without singular kernels in real-world problems,refer to the following research papers [13].Note that this last decade the applications of the fractional operators in the epidemics models receive many investigations,too[48-56].

The fractional differential equations play many roles in fractional calculus.There exist many classes of the fractional differential equations: the diffusion equations,the Cauchy equation,Euler equation,and many others.The fractional differential equations are subject to many investigations in fractional calculus.Notably,the problem concerning to determine the analytical solutions and to propose the numerical schemes.For analytical solutions of some fractional differential equations,refer to the papers [15,36,35],and for the numerical investigations,refer to the papers [5,26,37-39].In this paper,the problem consists of solving the basics fractional differential equations relevant in many issues as the powers series,the domain decomposition series,and others.The main attraction of this investigation is the use of the recent fractional derivative without singular kernel,namely the fractional derivative with Rabotnov exponential kernel [12].The tool used in this paper to solve the fractional differential equations is the Laplace transform.In short,with this paper,we prove the applications of the fractional derivative with Rabotnov exponential kernel [12]in real-world problems.We demonstrate the use of this fractional operator is useful and can be used in modeling physical phenomena and others.

On the other hand,it is notable that there exist a number of multi-peakon and approximate solutions for the standard Degasperis-Procesi (DP) equation.But the characteristics of these peakon solutions are transformed into the bell-shaped solitons in case of the modified DP equation.This equation is relevant and helpful in analyzing the shallow water dynamics and is integrable.There have been lots of important studies which model and give the fundamental analysis of the fractional modified Degasperis-Procesi model or its relevant types [40-45].

In Section 2,we recall the fractional derivative with Rabotnov exponential kernel,and its associated integral and its Laplace transform.In Section 3,we recall some basics of fractional differential equations by solving them using the Laplace transform.Morever,in Section 3,we represent the solutions of the fractional differential equations and their corresponding graphical representations.In Section 4,we give an approximate solution to the time-fractional modified Degasperis-Procesi equation and we finish with Section 5 by giving the concluding remarks for our paper.

2.Fractional calculus tools

Before beginning our investigations,we recall the fundamental notions as the fractional derivatives used in fractional calculus.In this section,we are interested in a new fractional operator propose in [12].That is the fractional derivative with Rabotnov exponential kernel.This new fractional derivative is a fractional derivative with the nonsingular kernel.

Definition 1[12].The fractional derivative with Rabotnov exponential kernel of the functionf:a,+∞ -→ R,of orderαcan be represented in the following form

with the timet＞a,the orderα∈(0,1),λ∈ R+,anda∈R+.Furthermore,the condition described by the Rabotnov exponential function is described by the relationship

where the variablez∈ C and the Gamma function is represented by the functionΓ(...).

The integral form of a fractional derivative is very important notably to solve the fractional differential equations using the numerical method as Adam’s Basforth numerical scheme or the homotopy methods which are strongly relied on the integral.The integral representation of the fractional derivative with Rabotnov with Rabotnov exponential kernel is presented in the following definition [12].

Definition 2[12].The fractional integral with Rabotnov exponential kernel of the functionf:a,+∞ -→ R,of orderαis presented by the relation

wheret＞a,the orderα∈(0,1)and withλ∈ R+.

We recall the fundamental tool of resolution used in our paper.We define the Laplace transform of the fractional derivative with Rabotnov exponential kernel;we will use it to solve the fractional differential equations.We have [12]

whereLdenotes the classical Laplace transform.

An interesting function in fractional calculus that is used to represent all the analytical solutions for the fractional differential equations is the Mittag-Leffler function introduced in the literature by Mittag-Leffler in [58].This function plays an important role and can be considered as a generalization of the classical exponential function.We have the following definition.

Fig.1.Solution of Eq.(12) for various values of α,t and λ.

Definition 3[57,58].We consider the parameterα ＞0,andβ∈ R.The Mittag-Leffler function is represented by the function

where the variablez∈ C and the gamma function is represented byΓ(...).

3.Fundamental calculus with Rabotnov exponential kernel

In this section,we solve the fractional differential equations using the Laplace transform of the fractional derivative with the Rabotnov exponential kernel.We consider a certain number of the fractional differential equations described by the fractional derivative with Rabotnov exponential kernel.We consider in these equations the fundamental functions like the exponential functions,Mittag-Leffler functions,constant functions,and the trigonometric functions.We pose a series of exercises,and we solve them with the Laplace transform of the fractional derivative with the Rabotnov exponential kernel.

The first problem under consideration in this section concerns particularly the derivative of a constant function,we have the following equation

under the initial condition defined by the relationshipx(0)constant.We have the following procedure to get the solution.We use the Laplace transform method;we have the following relations

Inverting Eq.(7),we get the analytical solution of the fractional differential Eq.(6) given by the representation

Fig.2.Analytical solution of Eq.(15) for different values of t and λ when α=0.9.

The fractional differential equation in (6) means that the fractional derivative with the Rabotnov exponential kernel of a constant function yields 0.This result is interesting because the fractional derivative with Rabotnov exponential kernel respects the classical rule of concerning the derivative of a constant function,which is zero (Figs.1-6).

The second problem concerns the fractional differential equation described by the following equation

under the initial condition defined by the relationshipx(0)constant.Our objective here is to determine the function,satisfying Eq.(9).We have the following procedure

Fig.3.Analytical solution of Eq.(18) for different values of t and α when β=2,λ=1.

Inverting Eq.(10),we get the analytical solution of the fractional differential Eq.(9) given by the representation

The result obtained is interesting.In comparison with the classical derivative and when the Caputo fractional derivative is used,the result of Eq.(9) gives us an affine function for the classical derivative andx(t)=x(0)+cλtαfor the Caputo-Liouville derivative.We notice the affine function is obtained with Eq.(9) when the time is long.We can consider that our result is in good agreement with the classical results.

The third problem focuses on an essential basic calculus used when the power series method is used to determine the solution of a class of the fractional differential equations.In this paragraph,we consider the fractional differential equation given by the equation

under the initial condition defined by the relationshipx(0)constant,whereβis a positive constant.We have the following procedure for getting the analytical solution

Inverting Eq.(13),we get the analytical solution of the fractional differential Eq.(12) given by the representation

When we want to solve a fractional differential equation described the fractional derivative with the Rabotnov exponential kernel by using the power series method,the result in Eq.(14) can be used in the application.

In problem fifth,we consider the Mittag-Leffler function with two-parameters.The Mittag-Leffler function is very important in fractional calculus;it will be interesting to see what happens with this function in basic calculus.We consider the following fractional differential equation

under the initial condition defined by the relationshipx(0)constant.We apply the Laplace transform

Applying inverse of the Laplace transform to both sides of Eq.(16),we obtain the following analytical solution of Eq.(15)

Fig.4.Analytical solution of Eq.(24) for different values of t and α when a=3,b=5,λ=5 (left).And the solution with different values of α=1,0.99,0.95,0.9 (bottom to top-right).

Fig.5.Analytical solution of Eq.(30) with respect to the fractional operator with Rabotnov exponential kernel for Rabotnov constant λ=1 (left) and λ=3(right).

Fig.6.Analytical solution of Eq.(33) with respect to the fractional operator with Rabotnov exponential kernel for Rabotnov constant λ=1 (left) and λ=3(right).

We continue with the problem where we use the Mittag-Leffler function,but here we consider the Mittag-Leffler function with one parameter.We have to solve the fractional differential equation represented by the equation defined by

under initial condition defined by the relationshipx(0)constant.We apply to both sides of Eq.(18),the Laplace transform,we get the following relation

Applying the inverse of the Laplace transform to both sides of Eq.(19),we obtain the following analytical solution of Eq.(18)

The main observation of this paragraph is that to obtain the Mittag-Leffler function with one parameter,we calculate the fractional derivative of the Mittag-Leffler function with two parameters.

In another problem,we consider the fractional differential equation including now the Rabotnov fractional exponential function and defined as the following form

under the initial condition defined by the relationshipx(0)constant.Applying the Laplace transform to Eq.(21),we obtain that

Finally,after inverting Eq.(22),the form of the analytical solution is represented by

The problem in Eq.(21) means that there exists a relation between the exponential function in Rabotnov sense and the Mittag-Leffler function.

In this new section,we consider the following fractional differential equation

whereaandbare real constants and the initial condition defined by the relationshipx(0)is constant as well.We repeat the same procedure as in the previous equations.We apply the Laplace transform;we have the following procedure

Inverting Eq.(25),we get the analytical solution of the fractional differential equation represented by Eq.(24),that is

From Eq.(26),we can observe the fractional derivative for the polynomial function can be predicted.We consider the classical exponential function.Many applications in real-world problems use this function.We consider the fractional differential equation defined by the following equation

under the initial condition defined by the relationshipx(0)constant.Applying the Laplace transform to both sides of Eq.(27),we obtain that

Therefore,the analytical solution of the fractional differential equation described by Eq.(27) is obtained after inverting Eq.(28).We have

We finish this section by considering the trigonometric functions.We begin with the fractional differential equation defined by the following trigonometric equation

Applying the Laplace transform to both sides of Eq.(30),we obtain that

Hence,the analytical solution of the fractional DE shown by Eq.(30) is obtained by taking the inverse LT of the last equation.Then we have

wherepFq(a;b;c)is the generalized hypergeometric function introduced by Pochhammer [59].Now we consider the another trigonometric fractional differential equation:

If we apply the Laplace transform to both sides of Eq.(33),we have

Hence,the analytical solution of the fractional DE shown by Eq.(33) is obtained by taking the inverse LT of Eq.(34).Then we have

wherepFq(a;b;c)is the generalized hypergeometric function.

The second part of this paper is to investigate certain fractional differential equations recurrent in the literature.These equations appear in many real-world problems.We begin this section by considering the Cauchy problem defined by

under initial condition defined by the relationshipx(0)constant,andkis a constant.The method of resolution does not change.We use the Laplace transform,that is

Applying the inverse Laplace transform of Eq.(37),we obtain the following analytical solution of the Cauchy problem described by the fractional derivative with Rabotnov exponential kernel

The second problem is to consider the second-order fractional differential equation defined by the following equation

under the initial condition defined by the relationshipx(0)constant,the parametera,bandcare constants.This class of the fractional differential equations has many applications,notably in electrical circuits,as in theRLCcircuit.It has importance also in the spring-mass-damper system.Eq.(39) is not in general trivial.For the procedure of the solution,we use the following decomposition for Eq.(39).That is

combined with the equation defined by

Eqs.(40) -(41) can be represented in their matrix form,and the problem will be to get the eigenvalue of the matrix.We have the following representation

For simplification in the calculations,we use an concrete example,we consider for the rest of the investigationsa=1,b=3 andc=-4.Consider the matrix

We can observe that Eq.(43) can be written as the following form when we takez=(x,y),

Using the eigenvectors,we can decompose the matrixAas the formA=PDP-1where the matrixPandDare obtained with the calculations given by

To solve Eq.(45),we first consider the fractional differential equation described by the fractional derivative with Rabotnov exponential kernel defined by the equation

whereu=Pz.We apply the Laplace transform to both sides of Eq.(46),we have the following relationship

The solution is obtained after the application of the inverse of the Laplace transform for Eq.(47),we get the following solution

and the second solution is represented as the form

The solution of Eq.(39) follows from the change variableu=Pz.Many other applications and basics calculus can be presented,but we limit the most important of them,which are current in real-world problems.In the next section,we try to represent some of the solutions obtained in the previous investigations.In conclusion,the fractional derivative of Rabotnov exponential kernel is useful,and many calculations with this fractional operator are possible.They can be done using,in particular,the Laplace transform of such operator.

4.Application to the nonlinear dispersive wave model

4.1.Procedure of the solution method

In this part of the paper,we give an illustrative application of the fractional operator with Rabotnov exponential kernel.The application examines the nonlinear time-fractional modified Degasperis-Procesi (FMDP) equation arising in the modelling of the propagation of nonlinear dispersive waves [60].Firstly,we present the fundamental methodology which has been used in the application.To investigate this methodology we take into account the following general form of fractional nonlinear PDE:

with initial condition

and the boundary conditions

whereμz,φ,γ0andγ1are known functions.In Eq.(50),we represent the linear part of the equation withL〈.〉,the nonlinear part withN〈.〉 andDα,λ tshows the FDREK.We characterize the recursive steps for solving the suggested FMDP equation.Using the Laplace transform of the mentioned fractional operator given in Eq.(4),we define thefor Eq.(50).Then we can obtain the transformed functions for the FDREK as

Then,applying the perturbation method,we achieve the solution of Eqs.(50) -(51) as

The nonlinear part in Eq.(50) can be computed from and the componentsΛψ(x,t)are given in [61]as

Substituting Eqs.(55) and (56) into Eq.(53),we get the solution components as:

Then,by solving Eq.(58) with respect toϑ,we identify the following corresponding homotopies regarding to the Rabotnov exponential kernel operator:

whenϑ→ 1,we obtain that Eq.(59) shows the approximate solution for the proposed problem,thus the solution is given by

Applying the inverse LT of Eq.(60),we obtain the approximate solution of Eq.(50),

4.2.Solution to the modified Degasperis-Procesi model

In this subsection,we examine the laplace transform by considering nonlinear time-fractional modified Degasperis-Procesi equation arising in the modelling of the propagation of nonlinear dispersive waves which is given [60]:

with the initial condition

Firstly,we consider the problem (62) with its initial condition(63) by using the Laplace transform method coupled with the FDREK.We get the following by applying the Laplace transform to Eq.(62)

At this step,we apply the perturbation to Eq.(64),we have

Then,we apply the inverse Laplace transform to Eq.(65),we get

whereΛψ=3ξxξxx+ξξxxx-4ξ2ξxvalues are the functions that show the nonlinear terms given in Eq.(57) and they are examined by the following way:

Then we have the solution steps by considering the equal powers ofϑin Eq.(66):

Fig.7.Approximate solution of the modified Degasperis-Procesi Eq.(62) with respect to the FDREK for Rabotnov constant λ=1 and fractional parameter α=1.

Fig.8.Analytical solution of the modified Degasperis-Procesi equation (62).

Therefore,the approximate solution of the problem is given by

where it gives the integer-order(α=1,λ=1)solution of the problemFollowing Fig.7 and Fig.8 show the approximate solution that has been obtained by using the homotopy method coupled with the FDREK and the exact solution of the problem,respectively.

5.Conclusion

In this paper,we have discussed an important fractional kernel which is constructed by the Rabotnov exponential function.Also,we have examined some illustrative applications on the fractional differential equations that have important roles in the modelling of real-life phenomena.Moreover,different-types of fractional differential equations described by the fractional derivative includes Rabotnov function,have been considered.And the fundamental results of outstanding certain functions as exponential,trigonometric,Mittag-Leffler and constant functions have been focused.The graphical representations of the obtained solutions after the resolutions of the fractional differential equations have been presented.We have also given the solution of the modified Degasperis-Procesi equation arising in the modelling of the propagation of nonlinear dispersive waves as another implementation of the operator.These obtained results have pointed out that the mentioned fractional operator which has the Rabotnov kernel has advantages to obtain approximate solutions for which the real-life phenomena including the ocean engineering problems have been modeled.

Declaration of Competing Interest

The authors have declared no conflict of interest.

Acknowledgment

M.Yavuz was supported by TUBITAK (The Scientific and Technological Research Council of Turkey).