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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina


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Volume 14, Issue 1

Issues

Volume 13 (2015)

Regarding on the exact solutions for the nonlinear fractional differential equations

Melike Kaplan
  • Corresponding author
  • Eskisehir Osmangazi University, Art-Science Faculty, Department of Mathematics-Computer, Eskisehir-Turkey
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  • Other articles by this author:
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/ Murat Koparan / Ahmet Bekir
Published Online: 2016-12-19 | DOI: https://doi.org/10.1515/phys-2016-0056

Abstract

In this work, we have considered the modified simple equation (MSE) method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW) and the modified equal width (mEW) equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

Keywords: Nonlinear fractional differential equation; the modified Riemann-Liouville derivative; the modified simple equation method; exact solution

PACS: 02.70.Wz; 04.20.Jb; 02.30.Jr; 94.05.Fg

1 Introduction

Fractional calculus, was introduced by Leibniz in 1695, as a generalization of ordinary calculus [1]. Physicists, engineers and mathematicians have always taken intense interest in nonlinear problems related to various scientific applications, such as fluid dynamics, plasma physics, high-energy physics, geochemistry, chemical physics, viscoelastic materials, elastic media, optical fibers, signal processing, polymers, chemical kinematics and biomathematics [24]. The most considerable advantage of utilizing nonlinear fractional differential equations (NFDEs) is their non-local property, which means that the next status of a system depends both upon its current status and all of its historical status. For instance, by using the fractional differential equations, the fluid-dynamic traffic model can eliminate the deficiency arising from the assumptions of continuum traffic flow [5]. Considering their importance, the exploration of exact solutions of NFDEs became a crucial case and matter of interest for researchers in recent years. As a result, numerous influential methods have been proposed. Some of these include as the tanh-sech method [6], the (G′/G)-expansion method [7], the (G′/G,1/G)-expansion method [8], the first integral method [9], the exponential function method [10, 11], the sub equation method [12], the trial equation method [13], ansatz method [14], the modified Kudryashov method [15], variational iteration method [16] and others [1719].

There are different approaches to the generalization of the notion of differentiation to fractional orders such as Caputo, Grünwald-Letnikow, Riemann-Liouville and the Generalized Functions approach [20, 21]. Among these approaches, the modified Riemann-Liouville derivative is mostly used by mathematicians.

2 The Modified Simple Equation Method

We consider a NFDE with independent variables x and t given by: F(u,Dtαu,Dxαu,DtαDtαu,DtαDxαu,DxαDxαu,)=0,0<α<1,(1)

where u(x,t) is an unknown function, F is a polynomial in u and various partial fractional derivatives in which the highest order derivatives and nonlinear terms are involved.

To find the exact solution of Eq. (1), we introduce a fractional complex transformationn as follows: u(x,t)=u(ξ),ξ=kxαΓ(1+α)ctαΓ(1+α).(2)

Applying Eq. (2), Eq. (1) can be converted to a nonlinear ordinary differential equation (ODE). When integrated as many times as possible, the integration constant can be selected as zero.

The exact solution of the reduced equation can be represented by a polynomial in ζξζξ as follows; uξ=mn=0anζξζξn,(3)

where an,(n = 0,1,2,…, m),(am≠ 0) are unknown constants and ζ(ξ) is an unknown function that needs to be found.

Here the positive integer m (the balancing number) can be calculated by linking the homogeneous balance principle with the highest order derivative term and the highest order nonlinear term which appears in the reduced equation.

By substituting Eq. (3) into the reduced equation, we obtain a polynomial of ζ−j(ξ) which also includes the derivatives ofζ(ξ). By equating all the coefficients of ζ−j(ξ) to zero (j ≥ 0), we obtain an algebraic equation system which can be solved for an (n = 0,1,2,…, m), c andζ(ξ). Finally substitution of an, c and ζ(ξ) into Eq. (3) provides the specification of the exact solution of Eq. (1)

In the other expansion methods, such as the tanh-function method, the exp-function method, the (GG) -expansion method, the auxiliary function method, the exact solutions are expressed in terms of some pre-defined functions. However some fresh solutions can be found with the usage of the MSE method, since φ is not a pre-defined function nor is it a solution of any pre-defined equation. This is the main advantage of the proposed method [22, 23].

3 Applications

In the current section, the exact solutions of the space-time fractional EW equation and the space-time fractional modified EW equation NPDEs are obtained by using the MSE method.

3.1 The space-time fractional EW equation

We first consider the space-time fractional EW equation in the form: Dtαu(x,t)+εDxαu2(x,t)δDxxt3αu(x,t)=0,(4)

where ε and δ are positive parameters. This equation is used to model nonlinear dispersive waves [24]. Employing the transformation (2), Eq. (4) reduced to an ODE as follows: cu+εk(u2)+δck2u=0.(5)

Integrating (5)once and taking the constant of the integration as zero, it becomes: cu+εku2+δck2u=0(6)

Balancing the highest order derivative term u″ with the highest order nonlinear term u2, the balancing number is determined to be m = 2. Then assume the exact solution of the Eq. (6)as: uξ=a0+a1ζξζξ+a2ζξζξ2.(7)

We obtain an algebraic equation system which follows by substituting Eq. (7) into Eq. (6) and equating all the coefficients of ζ−j(ξ),(j = 0,1,…,4) to zero: ζ0ξ:εka02ca0=0,ζ1ξ:ck2a1δζ+2εka0a1ζca1ζ=0,ζ2ξ:2ck2δa2ζζ+2εka0a2(ζ)2+εka12(ζ)2+2ck2δa2(ζ)23ck2δa1ζζca2(ζ)2=0,ζ3ξ:2εka1(ζ)3a210ck2δa2(ζ)2ζ+2ck2δa1(ζ)3=0,ζ4ξ:εka22(ζ)4+6ck2δa2(ζ)4=0.(8)

From the first and last equations of the system above we obtain for a0 and a2: a0=cεk,a2=6ckδε.(9)

Thereafter by substituting Eq. (9) into the remaining equations of the Eq. system (8), we find ζ1ξ:ck2δa1ζ+ca1ζ=0,ζ2ξ:12c2k3δ2ζζε6kc2δ(ζ)2ε+εka12(ζ)212c2k3δ2(ζ)2ε3ck2δa1ζζ=0,ζ3ξ:10ck2δa1(ζ)3+60c2k3δ2(ζ)2ζε=0.(10)

It follows from the last equation of the system above that, a1=6ckδζεζ.(11)

By substituting Eq. (11) into the remaing equations of the Eq.system(10), an ODE system results. Next, the solution of this system for ζ(ξ) to show that: ζ(ξ)=C1+C2e±ξδk,(12)

where C1 and C2 are arbitrary constants. Finally, we substitute Eq. (12) into Eq. (7) and the following exact solutions of the space-time fractional EW equation resulted: u(ξ)=cεk16C2coshξδk±sinhξδkC1+C2coshξδk±sinhξδk+6C2coshξδk±sinhξδk2C1+C2coshξδk±sinhξδk2.

Here ξ=kxαΓ(1+α)ctαΓ(1+α). The figures of the obtained solutions are generated by setting special values for the parameters as follows:

The exact solution for the EW equation with α = 0.5 and α = 1 respectively when δ = 2, c = 0.5, ε =0.5, C1 = 1, C2 = −1 and k = –3.
Figure 1

The exact solution for the EW equation with α = 0.5 and α = 1 respectively when δ = 2, c = 0.5, ε =0.5, C1 = 1, C2 = −1 and k = –3.

3.2 The space-time fractional modified EW equation

The space-time fractional modified equal width equation (mEW) can be written in the following form [24]: Dtαu(x,t)+εDxαu3(x,t)δDxxt3αu(x,t)=0.(13)

Here ε and δ are positive parameters. We perform the transformation (2) and reduce Eq. (13) to an ODE: cu+εk(u3)+δck2u=0.(14)

Negleting the constant of integration, we integrate Eq. (14) once with respect to ξ and thus obtain: cu+εku3+δck2u=0.(15)

Balancing the highest order derivative terms and nonlinear terms in Eq. (15), the balancing number as m = 1. According to MSE method, the exact solution turns into uξ=a0+a1ζξζξ.(16)

Here after the following substitutions are made: ζ0ξ:εka03ca0=0,ζ1ξ:δck2a1ζca1ζ+3εka02a1ζ=0,ζ2ξ:3δck2a1ζζ+3εka0a12(ζ)2=0,ζ3ξ:2δck2a1+εka13(ζ)3=0.(17)

From the solution of the first and the last equation of the above system, we find the values of a0 and a1 as follows a0=±cεk,a1=±2δcke

By substituting these valuse into the second equation of Eq. system (17), an ordinary differential equation system is obtained. From the solutions of this system ζξ=C1+C2e±2δξk,

where C1 and C2 are arbitrary constants. Subsequently, we have the following exact solutions of the space-time fractional modified equal width equation uξ=cC1+C2cosh2δξk+sinh2δξkεkC1+C2cosh2δξk+sinh2δξk,

The exact solution for the modified EW equation with α = 0.5 and α = 1 respectively when δ = 2, c = 0.5, ε = 0.5, C1 = 1, C2 = −1 and k = −3.
Figure 2

The exact solution for the modified EW equation with α = 0.5 and α = 1 respectively when δ = 2, c = 0.5, ε = 0.5, C1 = 1, C2 = −1 and k = −3.

where ξ=kxαΓ(1+α)ctαΓ(1+α). We represent graphically the obtained exact solutions of the modified EW equation by setting special values for the parameters as follows:

4 Conclusions

In the current paper, the MSE method has been successfully employed to obtain exact solutions of the space-time fractional EW equation and the modified EW equation. The generated solutions serve to illustrate several new features of waves and can be more useful in theoretical and numerical studies of the considered equation. Moreover, the solutions which are obtained in this work are different from the ones which are given in [24]. To the best of our knowledge, symbolic computation systems (such as MATLAB, MAPLE and MATHEMATICA) played a critical role in the computations. The modified simple equation method is a reliable and effective method. Also it can be applied to many other NFDEs and NFDE systems appearing in mathematical physics and nonlinear sciences.

Acknowledgements

This work was partly supported by Anadolu University Scientific Research Projects (Grant No: 1602E048).

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About the article

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Received: 2016-08-02

Accepted: 2016-11-03

Published Online: 2016-12-19

Published in Print: 2016-01-01


Citation Information: Open Physics, Volume 14, Issue 1, Pages 478–482, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2016-0056.

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