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Open Physics

formerly Central European Journal of Physics

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Volume 14, Issue 1

Issues

Volume 13 (2015)

Solitary and compacton solutions of fractional KdV-like equations

Bo Tang
  • Corresponding author
  • School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang, Hubei, 441053, P.R. China; School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P.R. China
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  • De Gruyter OnlineGoogle Scholar
/ Yingzhe Fan / Jianping Zhao / Xuemin Wang
  • Department of Mechanical Engineering, University of Texas at Dallas, Richardson, United States America
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Published Online: 2016-10-04 | DOI: https://doi.org/10.1515/phys-2016-0038

Abstract

In this paper, based on Jumarie’s modified Riemann-Liouville derivative, we apply the fractional variational iteration method using He’s polynomials to obtain solitary and compacton solutions of fractional KdV-like equations. The results show that the proposed method provides a very effective and reliable tool for solving fractional KdV-like equations, and the method can also be extended to many other fractional partial differential equations.

Keywords: Fractional variational iteration method; Homotopy perturbation method; Modified Riemann-Liouville derivative; Fractional partial differential equation

PACS: 02.30.Jr; 02.60.-x; 04.20.Fy; 04.20.Jb

1 Introduction

In recent years, theory and numerical analysis of fractional partial differential equations (FPDEs) have received considerable interest due to their numerous applications in the areas of physics, biology, fluid and continuum mechanics, and engineering [111]. For example, in [10], Devendra Kumar, Jagdev Singh and Sunil Kumar used the homotopy perturbation transform method to study the nonlinear fractional Zakharov-Kuznetsov equation arising in ion-acoustic waves; Devendra Kumar, Jagdev Singh and Sushila used the homotopy analysis transform method to study the fractional biological population model in [11]. For better understanding of the complicated nonlinear physical phenomena, the solution of the fractional differential equation is much involved. In the past, various methods have been proposed to obtain solutions of FPDEs, such as homotopy perturbation method [1214], homotopy perturbation Sumudu transform method [15, 16], Adomian decomposition method [17, 18], homotopy analysis method [19], fractional variational iteration method [20, 21], finite difference method [22], fractional sub-ODE method [2326], and so on. Based on these methods, many fractional differential equations have been investigated.

Recently, Jumarie proposed a new definition of the fractional derivative which is a simple alternative definition to the Riemann-Liouville derivative. His definition has the advantages of both the standard Riemann-Liouville and the Caputo fractional derivatives, because the Jumarie derivative of a constant is equal to zero, an arbitrary continuous function needs not to be differentiable, and more importantly it removes singularity at the origin for all functions, for example, the exponentials functions and Mittag-Leffler functions.

In this paper, we will apply the the fractional variational iteration method using He’s polynomials (FVIMHP) [2729] and Jumarie’s modified Riemann-Liouville derivative to get solitary and compacton solutions of the following KdV-like equations

  • The time fractional K(2, 2) equation:

    Dtαu+a(u2)x+b(u2)xxx=0,0<α1.

  • (2+1)-dimensional time fractional Z-K equation:

    Dtαu+a(u2)x+b(u2)xxx+k(u2)yyx=0,0<α1.

  • (3+1)-dimensional time fractional Z-K equation:

    Dtαu+a(u2)x+b(u2)xxx+k(u2)yyx+r(u2)zzx=0,0<α1.

  • (3+1)-dimensional time fractional K-P equation: [Dtαu+a2(u2)x+b(u(u)xx)x]x+uyy+uzz=0,0<α1.

The rest of this paper is organized as follows: In Section 2, we introduce some basic definitions of Jumarie’s modified Riemann-Liouville derivative and give the main steps of the method here. In Section 3, we construct solitary and compacton solutions of fractional KdV-like equations by the proposed method. Some conclusions are given in Section 4.

2 Description of Modified Riemann-Liouville derivative and FVIMHP method

In this section, we first introduce some basic definitions of the fractional calculus theory.

The Riemann-Liouville fractional integral [30] is defined as

Ixαf(x)=1Γ(a)0x(xξ)α1f(ξ)dξ,α>0.(1)

The Jumarie’s modified Riemann-Liouville derivative [31-34] is defined as

Dxαf(x)=1Γ(α)ddx0x(xξ)α1(f(ξ)f(0))dξ,α<0,1Γ(1α)ddx0x(xξ)α(f(ξ)f(0))dξ,0<α<1,(f(n)(x))(αn),nα<n+1,n1,(2)

The integral with respect to (dx)α is defined by Jumarie [31] as follows: 0xf(ξ)(dξ)α=α0x(xξ)α1f(ξ)dξ,0<α1.(3)

We present the essential steps of the FVIMHP method as follows: Consider the following initial value problem

Dtru(x,t)+L(u(x,t))+N(u(x,t))=f(x,t)(4)

where Dtr is the Jumarie’s modified Riemann-Liouville derivative, L is the linear operator, N is the nonlinear operator.

step 1. By using the fractional variational iteration method (FVIM), we can get the iteration formula as follows: um+1(x,t)=um(x,t)+Itγλ(τ)Dtrum(x,τ)+L(um(x,τ))+N(u~m(x,τ))f(x,τ),(5)

where λ is the Lagrangian multiplier, the subscript m denotes the m-th order approximation, and ũm is considered as a restricted variation.

step 2. By means of the homotopy perturbation method (HPM), Eq.(5) becomes the following form: m=0pmum(x,t)=u0(x,t)+pm=1pmum(x,t)+Itγλ(τ)tm=0pmDtrum(x,τ)+m=0pmL(um(x,τ))+m=0pmN(u~m(x,τ))f(x,τ),(6)

where p ∈ [0, 1] is an imbedding parameter, and u0 is an initial approximation of Eq. (4).

step 3. Comparing the coefficients of the same order of p on both sides of Eq. (6), we can obtain um (m = 0, 1, 2, …). According to the HPM, the solution of Eq. (4) can be expressed as: u=m=0um.(7)

3 Application of the proposed method

In this section, we apply the FVIMHP method to obtain solitary and compacton solutions of the time fractional KdV-like equations.

3.1 Time fractional K(2, 2) equation

We consider the following time fractional K(2, 2) equation: Dtαu+a(u2)x+b(u2)xxx=0,0<α1,(8)

with the initial condition

u(x,0)=4c3acosh2(14abx).(9)

According to the FVIMHP method given in Section 2, we can get the iteration formula of Eq. (8) in the following form: m=0pmum=u0+pm=1pmum1Γ(1+α)0tm=0pmDταu+am=0pmumright2x+bm=0pmum2xxx(dτ)α.(10)

By the above iteration formula and initial approximation u0(x,t)=u(x,0)=4c3acosh2(14abx)=2c3a[cosh(12abx)+1], we can obtain

p1:u1(x,y,t)=2c3ac2absinh12abxtαΓ(1+α),p2:u2(x,y,t)=2c3ac2ab2p3:u3(x,y,t)=2c3ac2ab3sinh12abxt3αΓ(1+3α),p4:u4(x,y,t)=2c3ac2ab4cosh12abxt4αΓ(1+4α),

So we have the solitary solution of Eq. (8): u(x,t)=2c3acosh12abx1+c2ab2t2αΓ(1+2α)+c2ab4t4αΓ(1+4α)+sinh12abxc2abtαΓ(1+α)+c2ab3t3αΓ(1+3α)++1=2c3acosh12abxcosh12abctα,αsinh12abxsinh12abctα,α+1,(11)

where the functions sinh(z, α) and cosh(z, α) are defined as follows: sinh(z,α)=Eα(z)Eα(z)2,cosh(z,α)=Eα(z)+Eα(z)2.

Here Eα(z)=k=0zkΓ(1+kα)(α>0) is the Mittag-Leffler function.

We show some properties of the approximate solution (11) obtained by the proposed method in Fig.1. The plot (a) shows that approximate solution (11) is in good agreement with the exact solution. The plot (b) shows the fifth-order approximate solution (11) when α = 0.89. The plot (c) shows the approximate solution (11) for different values of α.

(a) the absolute error between the fifth-order approximate solution (11) and exact solution when α = 1 (b) the fifth-order approximate solution (11) when α = 0.89 (c) the fifth-order approximate solution (11) for different values of α when t = 0.5; c = 1; a = –1; b = 10
Figure 1

(a) the absolute error between the fifth-order approximate solution (11) and exact solution when α = 1 (b) the fifth-order approximate solution (11) when α = 0.89 (c) the fifth-order approximate solution (11) for different values of α when t = 0.5; c = 1; a = –1; b = 10

If we select the initial approximation u(x,0)=4c3asinh2(14abx), we can get the solitary solution as follows: u(x,t)=2c3acosh12abxcosh12abctα,αsinh12abxsinh12abctα,α1.(12)

In order to search compacton solutions of Eq. (8), we select the following initial approximations

u(x,0)=4c3asin2(14abx),|x|<πμ,0,

and

u(x,0)=4c3acos2(14abx),|x|<π2μ,0.

Using the FVIMHP in the same manner, we could get the compacton solutions as follows: u(x,t)=2c3a[1cos(12abx)cos(12abctα,α)sin(12abx)sin(12abctα,α)],|xctα|<πμ,0,

and

u(x,t)=2c3a[1+cos(12abx)cos(12abctα,α)+sin(12abx)sin(12abctα,α)],|xctα|<π2μ,0,

where μ=14ab.

3.2 (2+1)-dimensional time fractional Z-K equation

Consider the following (2+1)-dimensional time fractional Z-K equation: Dtαu+a(u2)x+b(u2)xxx+k(u2)yyx=0,0<α1.(13)

with the initial condition

u(x,y,0)=4c3acosh2(14ab+k(x+y)).(14)

According to the method given in Section 2, its iteration formula can be constructed in the following form: m=0pmum=u0+pm=1pmum1Γ(1+α)0tm=0pmDταum+am=0pmum2x+bm=0pmum2xxx+km=0pmum2yyx(dτ)α.(15)

By initial approximation u0(x,y,t)=u(x,y,0)=4c3acosh2(14ab+k(x+y)), we can obtain

p1:u1(x,y,t)=2c3ac2ab+ksinh12ab+k(x+y)tαΓ(1+α),p2:u2(x,y,t)=2c3ac2ab+k2cosh12ab+k(x+y)t2αΓ(1+2α),p3:u3(x,y,t)=2c3ac2ab+k3sinh12ab+k(x+y)t3αΓ(1+3α),p4:u4(x,y,t)=2c3ac2ab+k4cosh12ab+k(x+y)t4αΓ(1+4α),

Consequently, we have the following solitary solution of Eq.(13)

u(x,y,t)=2c3acosh12ab+k(x+y)cosh12ab+kctα,αsinh12ab+k(x+y)sinh12ab+kctα,α+1,(16)

We show some properties of the approximate solution (16) obtained by the proposed method in Fig. 2. The plot (a) shows that the approximate solution (16) is in good agreement with the exact solution when α = y = 1. The plot (b) shows the fifth-order approximate solution (16) when x = 2, α = 0.75. The plot (c) shows the approximate solution (16) for different values of α.

(a) the absolute error between the fifth-order approximate solution (16) and exact solution when α = y = 1 (b) the fifth-order approximate solution (16) when x = 2, α = 0.75 (c) the fifth-order approximate solution (16) for different values of α when t = 0.5, y = 2, c =1; a = –1; b = k = 1
Figure 2

(a) the absolute error between the fifth-order approximate solution (16) and exact solution when α = y = 1 (b) the fifth-order approximate solution (16) when x = 2, α = 0.75 (c) the fifth-order approximate solution (16) for different values of α when t = 0.5, y = 2, c =1; a = –1; b = k = 1

If we choose u(x,y,0)=4c3asinh2(14ab+k(x+y)), then we can obtain the solitary solutions of Eq.(13) as follows: u(x,y,t)=2c3acosh12ab+k(x+y)cosh12ab+kctα,αsinh12ab+k(x+y)sinh12ab+kctα,α1.

In order to construct compacton solutions of Eq.(13), we select initial approximations as follows: u(x,y,0)=4c3asin2(14ab+k(x+y)),|x+y|<πμ,0,

and

u(x,y,0)=4c3acos2(14ab+k(x+y)),|x+y|<π2μ,0.

By using the FVIMHP method, we can get the following compacton solutions of Eq.(13)

u(x,y,t)=2c3a[1cos(12ab+k(x+y))cos(12ab+kctα,α)sin(12ab+k(x+y))sin(12ab+kctα,α)],|x+yctα|<πμ,0.

and

u(x,y,t)=2c3a[1+cos(12ab+k(x+y))cos(12ab+kctα,α)+sin(12ab+k(x+y))sin(12ab+kctα,α)],|x+yctα|<π2μ,0,

where μ=14ab+k.

3.3 (3+1)-dimensional time fractional Z-K equation

We consider the following (2+1)-dimensional time fractional Z-K equation: Dtαu+a(u2)x+b(u2)xxx+k(u2)yyx+r(u2)zzx=0,0<α1.(17)

with the initial condition

u(x,y,z,0)=4c3acosh214ab+k+r(x+y+z).(18)

According to the FVIMHP method, its iteration formula can be constructed in the following form: m=0pmum=u0+pm=1pmum1Γ(1+α)0tm=0pmDταum+am=0pmum2x+bm=0pmum2xxx+km=0pmum2yyx+rm=0pmum2zzx(dτ)α.(19)

By initial approximation u0(x,y,z,t)=u(x,y,z,0)=4c3acosh2(14ab+k(x+y)), we can obtain

p1:u1(x,y,t)=2c3ac2ab+k+rsinh12ab+k+r(x+y+z)tαΓ(1+α),p2:u2(x,y,t)=2c3ac2ab+k+r2cosh12ab+k+r(x+y+z)t2αΓ(1+2α),p3:u3(x,y,t)=2c3ac2ab+k+r3sinh12ab+k+r(x+y+z)t3αΓ(1+3α),p4:u4(x,y,t)=2c3ac2ab+k+r4cosh12ab+k+r(x+y+z)t4αΓ(1+4α),

Consequently, we have the following solitary solution of Eq. (17)

u(x,y,z,t)=2c3acosh12ab+k+r(x+y+z)cosh12ab+k+rctα,αsinh12ab+k+r(x+y+z)sinh12ab+k+rctα,α+1.(20)

We show some properties of the approximate solution (20) obtained by the proposed method in Fig. 3. The plot (a) shows that our approximate solution (20) is in good agreement with the exact solution. The plot (b) shows the fifth-order approximate solution expressed by (20) when α, x, y are fixed. The plot (c) shows the approximate solution (20) for different values of α when t, y, z are fixed.

(a) the absolute error between the fifth-order approximate solution (20) and exact solution when α = 1, y = 0.1, z = 0.2 (b) the fifth-order approximate solution (20) when x = 2, y = 1, α = 0.94 (c) the fifth-order approximate solution (20) for different values of α when t = 0.6, y = 1, z = 5, c = 5 a = –0.01; b = 10; k = 20; r = 30
Figure 3

(a) the absolute error between the fifth-order approximate solution (20) and exact solution when α = 1, y = 0.1, z = 0.2 (b) the fifth-order approximate solution (20) when x = 2, y = 1, α = 0.94 (c) the fifth-order approximate solution (20) for different values of α when t = 0.6, y = 1, z = 5, c = 5 a = –0.01; b = 10; k = 20; r = 30

If we choose u(x,y,z,0)=4c3asinh2(14ab+k+r(x+y+z)), we can get the solitary solution of Eq. (17) as follows: u(x,y,z,t)=2c3acosh12ab+k+r(x+y+z)cosh12ab+k+rctα,αsinh12ab+k+r(x+y+z)sinh12ab+k+rctα,α1.

In order to construct compacton solutions of Eq. (17), we select initial approximations as follows: u(x,y,z,0)=4c3asin2(14ab+k+r(x+y+z)),|x+y+z|<πμ,0,

and

u(x,y,z,0)=4c3acos2(14ab+k+r(x+y+z)),|x+y+z|<π2μ,0.

By using the FVIMHP method, we can get the compacton solutions of Eq. (17) as follows

u(x,y,z,t)=2c3a[1cos(12ab+k+r(x+y+z))cos(12ab+k+rctα,α)sin(12ab+k+r(x+y+z))sin(12ab+k+rctα,α)],|x+y+zctα|<πμ,0.

and

u(x,y,z,t)=2c3a[1+cos(12ab+k+r(x+y+z))cos(12ab+k+rctα,α)+sin(12ab+k+r(x+y+z))sin(12ab+k+rctα,α)],|x+y+zctα|<π2μ,0,

where μ=14ab+k+r.

3.4 (3+1)-dimensional time fractional K-P equation

We consider the following (3+1)-dimensional time fractional K-P equation: Dtαu+a2(u2)x+b(u(u)xx)xx+uyy+uzz=0,0<α1.(21)

with the initial condition

u(x,y,z,0)=4(c2)acosh212a2b(x+y+z).(22)

According to the FVIMHP method, its iteration formula can be constructed in the following form: m=0pmum=u0+pm=1pmum1Γ(1+α)0tm=0pmDtαum+a2m=0pmum2x+bm=0pmumm=0pmumxx)xx+m=0pm(umyy+umzz)(dτ)α,(23)

By initial approximation u0(x,y,z,t)=u(x,y,z,0)=4(c2)acosh2(12a2b(x+y+z)), we can obtain

p1:u1(x,y,t)=2(c2)ac2a2bsinha2b(x+y+z)tαΓ(1+α),p2:u2(x,y,t)=2(c2)ac2a2b2cosha2b(x+y+z)t2αΓ(1+2α),p3:u3(x,y,t)=2(c2)ac2a2b3sinha2b(x+y+z)t3αΓ(1+3α),p4:u4(x,y,t)=2(c2)ac2a2b4cosha2b(x+y+z)t4αΓ(1+4α),

Consequently, we have the following solitary solution of Eq. (21)

u(x,y,z,t)=2(c2)acosha2b(x+y+z)cosha2bctα,αsinha2b(x+y+z)sinha2bctα,α+1.(24)

We show some properties of the approximate solution (24) obtained by the proposed method in Fig. 4. The plot (a) shows the absolute error between the approximate solution (24) and exact solution. The plot (b) shows the approximate solution expressed by (24) when α is fixed. And the plot (c) shows the approximate solution (24) for different values of α when time is fixed.

(a) the absolute error between the fifth-order approximate solution (24) and exact solution when α = 1, y = 0.1, z = 0.2 (b) the fifth-order approximate solution (24) when x = 1, y = 0, α = 0.82 (c) the fifth-order approximate solution (24) for different values of α when t = 0.6, x = 1, y = 1, c = 1; a = –1; b = 1
Figure 4

(a) the absolute error between the fifth-order approximate solution (24) and exact solution when α = 1, y = 0.1, z = 0.2 (b) the fifth-order approximate solution (24) when x = 1, y = 0, α = 0.82 (c) the fifth-order approximate solution (24) for different values of α when t = 0.6, x = 1, y = 1, c = 1; a = –1; b = 1

If we choose u(x,y,z,0)=4(c2)asinh2(12a2b(x+y+z)), we can get the solitary solution of Eq. (22) as follows: u(x,y,z,t)=2(c2)acosha2b(x+y+z)cosha2bctα,αsinha2b(x+y+z)sinha2bctα,α1.

In order to construct compacton solutions of Eq.(22), we select initial approximations as follows: u(x,y,z,0)=4(c2)asin2(12a2b(x+y+z)),|x+y+z|<πμ,0,

and

u(x,y,z,0)=4(c2)acos2(12a2b(x+y+z)),|x+y+z|<π2μ,0.

By using the FVIMHP method, we can get the following compacton solutions of Eq. (22)

u(x,y,z,t)=2(c2)a[1cos(a2b(x+y+z))cos(12a2bctα,α)sin(a2b(x+y+z))sin(a2bctα,α)],|x+y+zctα|<πμ,0,

and

u(x,y,z,t)=2(c2)a[1+cos(a2b(x+y+z))cos(a2bctα,α)+sin(a2b(x+y+z))sin(a2bctα,α)],|x+y+zctα|<π2μ,0,

where μ=12a2b.

4 Conclusion

In this paper, we apply the FVIMHP method to obtain the solitary and compacton solutions of fractional KdV-like equations. The numerical results given in Section 3 demonstrate the good accuracy of the proposed method. The results show that the FVIMHP method is direct, effective, and can be useful in dealing with many other fractional partial differential equations. Our results confirm that the method takes all the advantages of the variational iteration method, homotopy perturbation method, and Jumarie’s modified Riemann-Liouville derivative. The comparison made with the exact solutions, enables us to see the accuracy of the FVIMHP method clearly. It is worth mentioning that the FVIMHP method is capable of reducing the volume of the computational work. In our future studies, we will solve many other nonlinear fractional partial differential equations by this method.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (11526088).

References

  • [1]

    Meerschaert M.M., Zhang Y., Baeumerc B, Particle tracking for fractional diffusion with two time scales, Comput. Math. Appl., 2010, 59, 1078–1086. Google Scholar

  • [2]

    Meerschaaert M., Benson D., Scheffler H. P., and Baeumer B., Stochastic solution of space time fractional diffusion equations, Phys. Rev. E, 2002, 65, 1103–1106.Google Scholar

  • [3]

    Baleanu D., Defterli O., Agrawal O.P., A central difference numerical scheme for fractional optimal control problems, J. Vib. Control, 2009, 15, 583–597.Google Scholar

  • [4]

    Özis T., Aǧıseven D., He’s homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients, Phys. Lett. A, 2008, 372(38), 5944–5950.Google Scholar

  • [5]

    Hilfer R., Applications of Fractional Calculus in Physics, World Scientific, New Jersey, 2000.Google Scholar

  • [6]

    Tenreiro Machado J.A., Analysis and design of fractional-order digital control systems, Syst. Aanl. Model. Simul., 1997, 27, 107–122.Google Scholar

  • [7]

    Momani S. and Odibat Z., A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula, J. Comput. Appl. Math., 2008, 220(1), 85–95.Google Scholar

  • [8]

    West B.J., Bolognab M., Grigolini P., Physics of Fractal Operators, Springer, New York, 2003.Google Scholar

  • [9]

    Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar

  • [10]

    Kumar D., Singh J. and Kumar S., Numerical Computation of Nonlinear Fractional Zakharov-Kuznetsov Equation arising in Ion-Acoustic Waves, J. Egyptian Math. Soc., 2014, 22(3), 373–378. Google Scholar

  • [11]

    Kumar D., Singh J. and Sushila, Application of homotopy analysis transform method to fractional biological population model, Romanian Reports in Physics, 2013, 65(1), 63–75. Google Scholar

  • [12]

    He J.H., Wu X.H., Variational iteration method: New development and applications, Comput. Math. Appl., 2007, 54, 881–894. Google Scholar

  • [13]

    He J.H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 1999, 178(3-4), 257–262.Google Scholar

  • [14]

    Rajeev and Kushwaha M. S., Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation, Appl. Math. Model., 2013, 37(5), 3589–3599. Google Scholar

  • [15]

    Singh J., Kumar D., and Adem Kılıçman, Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations, Abstr. Appl. Anal., 2014, Article ID 535793, 12 pages. Google Scholar

  • [16]

    Sushila, Singh J., Shishodia Y.S., A New Reliable Approach for Two-Dimensional and Axisymmetric Unsteady Flows between Parallel Plates, Zeitschrift fr Naturforschung A, 2013, 68a, 629–634. Google Scholar

  • [17]

    Daftardar-Gejji V. and Bhalekar S., Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method, Appl. Math. Comput., 2008, 202 (1), 113–120. Google Scholar

  • [18]

    Hu Y., Luo Y., Lu Z., Analytical solution of the linear fractional differential equation by Adomian decomposition method, J. Comput. Appl. Math., 2008, 215, 220–229. Google Scholar

  • [19]

    Ganjiani M., Solution of nonlinear fractional differential equations using homotopy analysis method, Appl. Math. Model., 2010, 34, 1634–1641. Google Scholar

  • [20]

    Wu G., Lee E.W.M., Fractional variational iteration method and its application, Phys. Lett. A, 2010, 374, 2506–2509. Google Scholar

  • [21]

    Khan Y., Faraz N., Yildirim A., and Wu Q. B., Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science, Comput. Math. Appl., 2011, 62, 2273–2278. Google Scholar

  • [22]

    Cui M., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 2009, 228, 7792–7804. Google Scholar

  • [23]

    Zhang S., Zhang H., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 2011, 375(7), 1069–1073. Google Scholar

  • [24]

    Guo S., Mei L., Li Y., Sun Y., The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Phys. Lett. A, 2012, 376(4), 407–411.Google Scholar

  • [25]

    Tang B., He Y., Wei L., and Zhang X., A generalized fractional sub-equation method for fractional differential equations with variable coefficients, Phys. Lett. A, 2012, 376(38-39), 2588–2590.Google Scholar

  • [26]

    Zhao J. P., Tang B., Kumar S., and Hou Y. R., The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations, Math. Probl. Eng., 2012, Article ID 367802, 8 pages. Google Scholar

  • [27]

    Guo S., Mei L., The fractional variational iteration method using He’s polynomials, Phys. Lett. A, 2011, 375(3), 309–311. Google Scholar

  • [28]

    Guo S., Mei L., Fang Y., Qiu Z., Compacton and solitary pattern solutions for nonlinear dispersive KdV-type equations involving Jumarie’s fractional derivative, Phys. Lett. A, 2012, 376(3), 158–164.Google Scholar

  • [29]

    Tang B., Wang X., Wei L., and Zhang X., Exact solutions of fractional heat-like and wave-like equations with variable coefficients, Internat. J. Numer. Methods Heat Fluid Flow, 2014, 24, 455–467. Google Scholar

  • [30]

    Sayevand K., Golbabai A., Yildirim A., Analysis of differential equations of fractional order, Appl. Math. Model., 2012, 36(9), 4356–4364. Google Scholar

  • [31]

    Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl., 2006, 51(9-10), 1367–1376. Google Scholar

  • [32]

    Jumarie G., Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution, J. Appl. Math. Comput., 2007, 24, 31–48.Google Scholar

  • [33]

    Jumarie G., Cauchy’s integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order, Appl. Math. Lett., 2010, 23, 1444–1450.Google Scholar

  • [34]

    Jumarie G., New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Math. Comput. Modelling, 2006, 44(3-4), 231–254. Google Scholar

About the article

Received: 2015-06-13

Accepted: 2016-07-02

Published Online: 2016-10-04

Published in Print: 2016-01-01


Citation Information: Open Physics, Volume 14, Issue 1, Pages 328–336, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2016-0038.

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© 2016 B. Tang et al. published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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