New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method

  • 1 Department of Computer Engineering, Faculty of Engineering, Ardahan, Turkey
  • 2 Department of Mathematics, Faculty of Science and Art, Denizli, Turkey
  • 3 Department of Mathematics, Faculty of Science and Art, Hatay, Turkey
Hülya Durur
  • Corresponding author
  • Department of Computer Engineering, Faculty of Engineering, Ardahan University, Ardahan, Turkey
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, Ali Kurt and Orkun Tasbozan

Abstract

This paper proposes obtaining the new wave solutions of time fractional sixth order nonlinear Equation (KdV6) using sub-equation method where the fractional derivatives are considered in conformable sense. Conformable derivative is an understandable and applicable type of fractional derivative that satisfies almost all the basic properties of Newtonian classical derivative such as Leibniz rule, chain rule and etc. Also conformable derivative has some superiority over other popular fractional derivatives such as Caputo and Riemann-Liouville. In this paper all the computations are carried out by computer software called Mathematica.

1 Introduction

In the last years, the number of studies on fractional partial differential equations have increased since they can be used in many fields such as physics, engineering, biology and chemistry [1, 2, 3]. Most of these studies focused on obtaining the exact solutions of fractional partial differential equations [13, 14]. But, some of the fractional derivative definitions such as Riemann-Liouville and Caputo do not have capabilities to achieve the exact solutions. Because they do not satisfy some main principles of classical integer order derivative. It is not possible to solve some fractional derivatives by using these definitions. For example

  • Riemann-Liouville derivative definition does not satisfy Dαc = 0 where c is real constant and α is not a natural number. (This property satisfies for Caputo derivative definition)
  • Riemann-Liouville and Caputo derivatives do not satisfy the derivative of the product of two functions.
    Dα(fg)=gDα(f)+fDα(g).
  • Caputo and Riemann-Liouville derivatives do not satisfy the derivative of the quotient of two functions.
    Dα(fg)=gDα(f)fDα(g)g2.
  • Caputo and Riemann-Liouville derivatives do not satisfy the well-known chain rule.
    Dα(fοg)=f(α)(g(t))g(α)(t).
  • Caputo and Riemann-Liouville derivatives do not satisfy Dα Dβ (f) = Dα+β (f) usually.
  • Caputo derivative definition accepts that the function f is differentiable.

Although the concept of fractional derivative appeared in the middle of the 17th century, in recent years, interest in this subject has increased. The reason is that physical systems can be expressed clearer by fractional derivative. When the literature is examined, we see that a lot of studies have been carried out on fractional derivatives. Different definitions of fractional derivative have been made since 1730s. Because of the limitations of the popular fractional derivative definitions, the scientists worked to find a new definition for fractional derivative which can satisfy all the main principles. Recently, by Khalil et al. introduced the conformable fractional derivative which is a simple, understandable and efficient fractional derivative definition [4] .

Definition 1

For all t > 0 and α ∈ (0, 1), an α-th order “conformable fractional derivative” of a function is defined by [4] as

Tα(f)(t)=limɛf(t+ɛt1α)f(t)ɛ
for f : [0, ∞) → R.

Definition 2

If f is α differentiable in some (0, a), a > 0 and limt0+f(α)(t) exist then define f(α)(0)=limt0+fα(t).. For a function f starting from a ≥ 0, the conformable fractional integral is defined such as following:

Iaα(f)(t)=atf(x)dαx=atf(x)x1αdx
where α ∈ (0, 1] and the integral is the Riemann improper integral.

In the following theorem [1.1], the properties of this new definition are given.

Theorem 1.1

Let α ∈ (0, 1], t > 0 and f, g be α− differentiable functions. Then

  1. Tα(cf + dg) = cTα(f) + dTα(g), for all a, b ∈.
  2. Tα(tp) = ptp−α for all p.
  3. Tα(λ) = 0 for all constant functions f(t) = λ.
  4. Tα(fg) = fTα(g) + gTα(f).
  5. Tα(fg)=gTα(g)fTα(f).g2
  6. If f is differentiable function, then Tγ(f)(t)=t1αdfdt.

The conformable fractional derivative has been used to provide new solutions for existing differential equations by many scientists. For example, Ilie et al. [5] studied general solutions of Riccati and Bernoulli fractional differential equations with conformable fractional derivative. Ta¸sbozan and Kurt [12] obtained new travelling wave solutions of time-space fractional Liouville and Sine-Gordon equations using conformable fractional derivative definition. Many references can be seen in the literature [15, 16, 17] that used conformable fractional derivative to build the mathematical model of a natural event.

In this study authors aimed to find the new exact solutions of conformable time fractional (1+1) and (2+1) dimensional KdV6 equations [10, 11] with the aid of sub equation method.

2 The Sub-Equation Method

In this section, we mention a biref description of fractional sub-equation method [8] . We assume the nonlinear fractional partial differential equation

P(u,Dtαu,Dxu,Dt2αu,Dt2u,)=0
where all the fractional derivatives are in conformable form. u(x, t) is unknown function and Dt means n times conformable fractional derivative u(x, t). The sub-equation method will be explained step by step as follows:
  1. Step 1:Using the wave transformation [7], we get the following equalities.
    u(x,t)=U(ξ),ξ=kx+wtαα
    where k, w are constants to be examined later and. Equation (2.1) can be rewritten in the form of the following ODE by using chain rule [6] :
    G(U,U,U,)=0
    where prime indicates the known derivative with respect to ξ.
  2. Step 2. Assume that Equation (2.3) has one solution in the following form
    U(ξ)=i=0Naiϕi(ξ),aN0,
    where ai (0 ≤ iN) are constants to be determined. N represents a positive integer which is going to found using balancing procedure [9] in Eq. (2.3) and φ(ξ) satisfies the ordinary differential equation below
    ϕ(ξ)=σ+(ϕ(ξ))2
    where σ is a constant. For the Eq. (2.5) , some special solutions are given in the following formulas.
    ϕ(ξ)={σtanh(σξ),σ<0σcoth(σξ),σ<0σtan(σξ),σ>0σcot(σξ),σ>01ξ+ϖ,ϖisacons.,σ=0
  3. Step 3. Eqs. (2.4) and (2.5) are substituted into Eq. (2.3) and the coefficients of φi(ξ) are set to zero. This procedure gives a nonlinear algebraic system in ai (i = 0, 1,...,N).
  4. Step 4.Finally, solving the obtained non-linear algebraic equations system gives us the values of unknown constants. Substituting obtained constants from the nonlinear algebraic system and by help of the formulas (2.6) the solutions of Eq. (2.5) into Eq. (2.4) . This provide the exact solutions for Eq. (2.1) .

3 Implementation of the Sub-Equation Method

3.1 Solution of (1+1) Dimensional Time Fractional KdV6 Equation

Firstly, we consider (1 + 1) KdV6 equation

Dx6u+20DxuDx4u+40Dx2uDx3u+120Dxu2Dx2u+Dx3Dtαu+8DxuDxDtαu+4DtαuDx2u=0
Employing the chain rule [6] and wave transform [7] in Eq. (3.1) and integrating once yield following differential equation
k6u(v)+k3wu+6k2w(u)2+20k5uu+10k5(u)2+40k4(u)3=0
where prime denotes integer order derivative of function u(ξ) with respect to variable ξ. Assuming the solution of Equation (3.2) is denoted by the following series
u(ξ)=i=0Naiϕi(ξ),aN0
where φ(ξ) is the exact solutions of Riccati differential equation (2.5). With aid balancing procedure [9] , we get N = 1. Substituting all the obtained values in Eq. (3.2) led to an equation with respect to φ(ξ). Equating all the coefficients of φi(ξ) to zero we obtain an equation system. After solving the equation system by using Mathematica, we get,
a1=k,w=4k3σ.
Due to the this solution set the new wave solutions of Eq. (3.1) can be obtained as
u1(x,t)=a0+kσtanh(σ(kx+4k3tασα)),u2(x,t)=a0+kσcoth(σ(kx+4k3tασα)),u3(x,t)=a0kσtan(σ(kx+4k3tασα)),u4(x,t)=a0+kσcot(σ(kx+4k3tασα)).

3.2 Solution of (2+1) Dimensional KdV6 Equation

Consider the time fractional (2+1) KdV6 equation as follows:

Dx(Dx6u+20DxuDx4u+40Dx2uDx3u+120Dxu2Dx2u+Dx3Dtαu+8DxuDxDtαu+4DtαuDx2u)+Dy3u=0
then applying chain rule [6] and with the aid of the conformable wave transform [7] ξ=kx+wtαα+ly,, u ( x,y,t )= u(ξ), and integrating once we have the following ODE
k6u(v)+k3wu+6k2w(u)2+20k5uu+10k5(u)2+40k4(u)3+l3u=0
Using balancing principle [9] we have N = 1. Subrogating all the obtained results in Eq. (3.2) we have an equation system. Solving the system yields
w=20k3σ,a1=k,l=222/331/3k7/3σ2/3
Using the solution set above, we acquire the exact solutions for time fractional (2+1) dimensional KdV6 Equation
u1(x,y,t)=a0+kσtanh(σ(kx222/331/3k7/3yσ2/320k3tασα)),u2(x,y,t)=a0+kσcoth(σ(kx222/331/3k7/3yσ2/320k3tασα)),u3(x,y,t)=a0kσtan(σ(kx222/331/3k7/3yσ2/320k3tασα)),u4(x,y,t)=a0+kσcot(σ(kx222/331/3k7/3yσ2/320k3tασα)).

4 Conclusions

In study the sub equation method is implemented to get the new traveling wave solutions of time fractional (1+1) and (2+1) dimensional KdV6 equation successfully. The obtained results indicate that the sub equation method is an efficient, reliable and applicable technique for obtaining the exact solutions of fractional derivatives in conformable sense.

References

  • [1]

    R. Geroch, “Partial Differential Equations of Physics,” in General Relativity, Routledge, pp.19–60, 2017.

  • [2]

    E. E. Holmes, M. A. Lewis, J. E. Banks, and R. R. Veit, “Partial differential equations in ecology: Spatial Interactions and Population Dynamics,” Ecology, vol. 75, no. 1, pp. 17–29, 1994.

  • [3]

    M. Epstein, “Partial Differential Equations in Engineering,” in Mathematical Engineering, Springer, Cham, pp. 25–47, 2017.

  • [4]

    R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” J. Comput. Appl. Math., vol. 264, pp.65–70, 2014.

  • [5]

    M. Ilie, J. Biazar, and Z. Ayati, “General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative,” Int. J. Appl. Math. Res., vol. 6, no. 2, pp.49–51, 2017.

  • [6]

    T. Abdeljawad, “On conformable fractional calculus,” J. Comput. Appl. Math., vol. 279, pp. 57–66, 2015.

  • [7]

    Cenesiz, Y., Kurt, A., “New fractional complex transform for conformable fractional partial differential equations,” Journal of Applied Mathematics, Statistics and Informatics, 12, 41–47, 2016.

  • [8]

    Zhang, Sheng, and Hong-Qing Zhang., “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, 375.7, 1069–1073, 2011.

  • [9]

    Malfiet, W., “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics 60, 650–654, 1992.

  • [10]

    Wazwaz AM. “Multiple soliton solutions for a (2+1)-dimensional integrable KdV6 equation,” Commun Nonlinear Sci Numer Simulat 15, 1466–1472, 2010.

  • [11]

    Wazwaz AM., “The integrable KdV6 equations: multiple soliton solutions and multiple singular soliton solutions,” Appl Math Comput, 204:963–72, 2008.

  • [12]

    Taşbozan O., Kurt, A., “New Travelling Wave Solutions for Time-Space Fractional Liouville and Sine-Gordon Equations,” Journal of the Institute of Science and Technology, 8(4): 295–303, 2018.

  • [13]

    Nawaz, B., Rizvi, S. T. R., Ali, K., Younis, M., “Optical soliton for perturbed nonlinear fractional Schrödinger equation by extended trial function method,” Optical and Quantum Electronics, 50(5): 204, 2018.

  • [14]

    Rizvi, S. T. R., Ali, K., Bashir, S., Younis, M., Ashraf, R., Ahmad, M. O., “Exact soliton of (2+ 1)-dimensional fractional Schrödinger equation,” Superlattices and Microstructures, 107, 234–239, 2017.

  • [15]

    Uddin, M. H., Khan, M., Akbar, M. A., Haque, M., Analytical wave solutions of the space time fractional modified regularized long wave equation involving the conformable fractional derivative. Karbala International Journal of Modern Science, 5(1): 7, 2019.

  • [16]

    Senol, M., Tasbozan, O., Kurt, A., Numerical Solutions of Fractional Burgers’ Type Equations with Conformable Derivative. Chinese Journal of Physics, 58, 75–84, 2019.

  • [17]

    Karayer, H., Demirhan, D., Buyukkilic, F., Solutions of local fractional sine-Gordon equations. Waves in Random and Complex Media, 29(2), 227–235, 2019.

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  • [1]

    R. Geroch, “Partial Differential Equations of Physics,” in General Relativity, Routledge, pp.19–60, 2017.

  • [2]

    E. E. Holmes, M. A. Lewis, J. E. Banks, and R. R. Veit, “Partial differential equations in ecology: Spatial Interactions and Population Dynamics,” Ecology, vol. 75, no. 1, pp. 17–29, 1994.

  • [3]

    M. Epstein, “Partial Differential Equations in Engineering,” in Mathematical Engineering, Springer, Cham, pp. 25–47, 2017.

  • [4]

    R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” J. Comput. Appl. Math., vol. 264, pp.65–70, 2014.

  • [5]

    M. Ilie, J. Biazar, and Z. Ayati, “General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative,” Int. J. Appl. Math. Res., vol. 6, no. 2, pp.49–51, 2017.

  • [6]

    T. Abdeljawad, “On conformable fractional calculus,” J. Comput. Appl. Math., vol. 279, pp. 57–66, 2015.

  • [7]

    Cenesiz, Y., Kurt, A., “New fractional complex transform for conformable fractional partial differential equations,” Journal of Applied Mathematics, Statistics and Informatics, 12, 41–47, 2016.

  • [8]

    Zhang, Sheng, and Hong-Qing Zhang., “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, 375.7, 1069–1073, 2011.

  • [9]

    Malfiet, W., “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics 60, 650–654, 1992.

  • [10]

    Wazwaz AM. “Multiple soliton solutions for a (2+1)-dimensional integrable KdV6 equation,” Commun Nonlinear Sci Numer Simulat 15, 1466–1472, 2010.

  • [11]

    Wazwaz AM., “The integrable KdV6 equations: multiple soliton solutions and multiple singular soliton solutions,” Appl Math Comput, 204:963–72, 2008.

  • [12]

    Taşbozan O., Kurt, A., “New Travelling Wave Solutions for Time-Space Fractional Liouville and Sine-Gordon Equations,” Journal of the Institute of Science and Technology, 8(4): 295–303, 2018.

  • [13]

    Nawaz, B., Rizvi, S. T. R., Ali, K., Younis, M., “Optical soliton for perturbed nonlinear fractional Schrödinger equation by extended trial function method,” Optical and Quantum Electronics, 50(5): 204, 2018.

  • [14]

    Rizvi, S. T. R., Ali, K., Bashir, S., Younis, M., Ashraf, R., Ahmad, M. O., “Exact soliton of (2+ 1)-dimensional fractional Schrödinger equation,” Superlattices and Microstructures, 107, 234–239, 2017.

  • [15]

    Uddin, M. H., Khan, M., Akbar, M. A., Haque, M., Analytical wave solutions of the space time fractional modified regularized long wave equation involving the conformable fractional derivative. Karbala International Journal of Modern Science, 5(1): 7, 2019.

  • [16]

    Senol, M., Tasbozan, O., Kurt, A., Numerical Solutions of Fractional Burgers’ Type Equations with Conformable Derivative. Chinese Journal of Physics, 58, 75–84, 2019.

  • [17]

    Karayer, H., Demirhan, D., Buyukkilic, F., Solutions of local fractional sine-Gordon equations. Waves in Random and Complex Media, 29(2), 227–235, 2019.

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