An efficient hybrid computational technique for  solving nonlinear local fractional partial differential equations arising in fractal media

Amit Prakash; Hardish Kaur

doi:10.1515/nleng-2017-0100

Artikel Open Access

An efficient hybrid computational technique for solving nonlinear local fractional partial differential equations arising in fractal media

Amit Prakash und Hardish Kaur

Veröffentlicht/Copyright: 22. Februar 2018

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen

Aus der Zeitschrift Nonlinear Engineering Band 7 Heft 3

Abstract

In present work, nonlinear fractional partial differential equations namely transport equation and Fokker-Planck equation involving local fractional differential operators, are investigated by means of the local fractional homotopy perturbation Sumudu transform method. The proposed method is a coupling of homotopy perturbation method with local fractional Sumudu transform and is used to describe the non-differentiable problems. Numerical simulation results are projected to show the efficiency of the proposed technique.

Keywords: Local fractional Sumudu transform; Homotopy perturbation method; Approximate solution; Nonlinear partial differential equations; Local fractional derivative

1 Introduction

Fractional calculus [1, 2] is playing a vital role in various fields, such as physics, electronics, robotics, bioengineering, anomalous transport and many others including fractal phenomena. Fractional differential equations (FDE) have a wide range of applications in modeling many problems such as the study of aerodynamic forces, electrochemistry, mathematical biology, signal and image processing, etc. The study of FDE have become a topic of interest for the researchers in recent years and several numerical techniques have been developed to find their analytical and approximate solutions as modified homotopy analysis transform method (MHATM) [3], Fractional Adams-Bashforth-Moulton method [4], improved Bernoulli sub-equation function method [5], fractional variational iteration method (FVIM) [6], Adomian decomposition method (ADM) [7] and many more [8,9,10,11,12,13,14,15,16,17,18,19]. The transport equation [20] and Fokker-Planck equation [21] has been studied in past few years by using various analytical techniques. Moreover, Fokker-Planck equation of fractional order has also been investigated by homotopy perturbation method (HPM) [22], iterative Laplace transform method [23] etc. But, the classical fractional order calculus is not valid for differential equations on Cantor sets because of its nonlocal nature. So, local fractional calculus theory has been used to deal with such problems.

Recently, researchers have introduced local fractional derivative and calculus theory [24,25,26,27], which is used to elucidate the non-differentiable function defined on Cantor sets. The fractional differential equations involving local fractional differential operators befit to deal with phenomena that occurs in fractal space and time. The local fractional calculus theory has crucial applications in studying the numerous important models in fractal mathematics, such as heat conduction theory, transport problems in Cantorian space. In the present article, we study the one-dimensional local fractional nonlinear transport equation (LFNTE) and one-dimensional local fractional nonlinear Fokker-Planck equation (LFNFPE) defined on Cantor set given by equations (1) and (2) respectively.

∂αφ(x,t)∂tα−ax,t∂2αφx,t∂x2α−bx,tφx,t2∂αφx,t∂xα=0,(1)

with the given condition φ (x, 0) = f(x), where a and b are real constants, and

∂αφ(x,t)∂tα=ax,φ∂2αφx,t∂x2α−bx,φ∂αφx,t∂xα,(2)

with the given condition φ (x, 0) = f(x), where a(x, φ) and b(x, φ) are parameters.

In [28], X. J. Yang et al. presented the local fractional variational iteration method (LFVIM) which is an extended version of VIM via local fractional operators and has employed this scheme to solve LFNTE and LFNFPE in fractal domain. The homotopy perturbation method (HPM), firstly proposed by Chinese mathematician J. H. He [29] is an analytical approximate solution method for linear and nonlinear differential equations without any discretization, transformation or perturbation parameters. The proposed local fractional homotopy perturbation Sumudu transform method (LFHPSTM) is an elegant compilation of local fractional Sumudu transform and HPM using He’s polynomials. In LFHPSTM only a few iterates yields a reasonably accurate solution.

2 Local fractional order calculus

Definition 2.1

Consider a function ψ(t) ∈ C_μ(σ, ϑ), while |ψ(t) – ψ(t₀)| < ε^μ, 0 < μ ≤ 1, is valid, when ([24,25,26,27]) |t – t₀| < δ, for ε, δ > 0 and ε ∈ R.

Definition 2.2

For 0 < μ < 1 and ψ(t) ∈ C_μ(σ, ϑ), the local fractional derivative of ψ(t) of order μ is defined ([24,25,26,27]) as follows:

dμψ(t0)dtμ=Dtμψt0=Δμ(ψt−ψ(t0))(t−t0)μ,

where

Δμψt−ψt0≅Γ1+μΔ(ψt−ψ(t0)).

Definition 2.3

Let 0 < μ < 1, ψ(t) ∈ C_μ(σ, ϑ) ΔS_j = S_j+1–S_j, ΔS = max {ΔS₁, ΔS₂, ΔS₃, … } and [S_j, S_j+1], j = 0, 1, … N – 1, S₀ = σ, S_N = ϑ, be partitions of the interval (σ, ϑ). The local fractional integral ([24,25,26,27]) of ψ(t) of order μ is defined by

σIϑμψs=1Γ(1+μ)∫σϑψ(s)(ds)μ=1Γ(1+μ)limΔs→0∑j=0N−1ψ(Sj)(ΔSj)μ.(3)

From Eq. (3), we get the following results:

σIϑμψs=0(σ=ϑ)

and σIϑμψs=−ϑIσμψs(σ<ϑ).

Definition 2.4

In fractal space, the well-known Mittag-Leffler function ([24,25,26,27]) is defined as

Eμtμ=∑k=0∞tkμΓ(1+kμ).

Definition 2.5

The local fractional Sumudu transform of ψ(t) of order μ ([24,25,26,27]) is

LFSμψt=Fμu=1Γ(1+μ)∫0∞Eμ−u−μtμψ(t)uμ(dt)μ,0<μ≤1.

The inverse operator is defined in the following way

LFSμ−1Fμu=ψt,0<μ≤1.

Theorem 1

If LFS_μ {ψ(t)} = F_μ(u) and LFS_μ {φ (t)} = G_μ(u), then [30] one has

LFSμψt+φ(t)=Fμu+Gμu.

Theorem 2(a)

If LFS_μ {ψ(t)} = F_μ(u) then [30] one has

LFSμdμψ(t)dtμ=Fμu−F(0)uμ.(4)

And Eq. (4) gives us the following results.

If LFS_μ {ψ(t)} = F_μ(u), we obtain

LFSμdnμψ(t)dtnμ=1unμFμu−∑k=0n−1ukμψkμ(0),(5)

when n = 2, from (5), we obtain

LFSμd2μψ(t)dt2μ=1u2μFμu−ψ0−uμψμ(0).

Theorem 2(b)

If LFS_μ {ψ(t)} = F_μ(u), then we have LFSμ0Itμψt=uμFμu.

3 Description of local fractional homotopy perturbation Sumudu transform method (LFHPSTM)

To illustrate the solution procedure of LFHPSTM, we take a general nonlinear local fractional partial differential equation of the following form:

Lγφχ,t+Nγφχ,t=0,(6)

where L_y and N_y are local fractional linear and nonlinear differential operators, respectively.

Taking the local fractional Sumudu transform on Eq. (6), we obtain

φyχ,z=φχ,0+zyφyχ,0+z2yφ2yχ,0+…+zk−1yφk−1yχ,0−zkyLFSy[Nyφχ,t].(7)

Now, using the inverse of local fractional Sumudu transform on Eq. (7), we have φ(χ, t) = φχ,0+tyΓ(1+y)φyχ,0+t2yΓ(1+2y)φ2yχ,0+⋯+t(k−1)yΓ(1+(k−1)y)φk−1yχ,0

−LFSy−1zkyLFSyNyφχ,t.(8)

Table 1

List of local fractional derivatives and integrals of some functions

dμψ(t)dtμ	0It(μ)ψt	Notations
dμEμtμdtμ=Eμtμ	0It(μ)Eμtμ=Eμtμ−1	Eμtμ=∑k=0∞tkμΓ(1+kμ)
dμdtμt(n+1)μΓ(1+(n+1)μ)=tnμΓ(1+nμ)	0It(μ)tnμΓ(1+nμ)=t(n+1)μΓ(1+(n+1)μ)	t^μ is a cantor function
dμsinμ(tμ)dtμ=cosμ(tμ)	0Itμcosμtμ=sinμ(tμ)	sinμtμ=∑k=0∞(−1)kt(2k+1)μΓ(1+(2k+1)μ)
dμcosμ(tμ)dtμ=−sinμ(tμ)	0Itμsinμtμ=1−cosμtμ	cosμtμ=∑k=0∞(−1)kt2kμΓ(1+2kμ)

Next, we apply the classical HPM

φχ,t=∑n=0∞pnφnχ,t,(9)

and the nonlinear term is decomposed as

Nyφχ,t=∑n=0∞pnHnφ,(10)

for some He’s polynomial H_n(φ) that are given by

Hnφ0,φ1,φ2,..,φn=1n!∂n∂pnN∑i=0∞piφip=0,n=0,1,2,…(11)

Substituting Eq. (9) and Eq. (10) in Eq. (8), we get

∑n=0∞pnφnχ,t=φχ,0+tyΓ(1+y)φyχ,0+t2yΓ(1+2y)φ2yχ,0+⋯+t(k−1)yΓ(1+(k−1)y)φk−1yχ,0−pLFSy−1zkyLFSy∑n=0∞pnHnφ.

Combining the local fractional Sumudu transform and classical HPM using He’s polynomial, the above equation is obtained. Equating the coefficients of alike powers of p, we get the following iterates

p0: φ0(χ,t)=φ(χ,0)+tyΓ(1+y)φy(χ,0)+t2yΓ(1+2y)φ2y(χ,0)+⋯+t(k−1)yΓ(1+(k−1)y)φ(k−1)y(χ,0),p1:φ1(χ,t)=−LFSy−1[zkyLFSy{H0(φ)}],p2: φ2(χ,t)=−LFSy−1[zkyLFSy{H1(φ)} ],⋮

Hence, the non-differentiable solution of Eq. (6) is given as

φχ,t=limp→1limN→∞∑n=0Npnφnχ,t.(12)

4 Numerical examples

In this section, LFNTE and LFNFPE are discussed in fractal domain and their non-differentiable solutions are presented.

Example 4.1

In the first model, we study the local fractional nonlinear transport equation in fractal dimensional space [31, 32]:

∂yφ(χ,t)∂ty−aχ,t∂2yφχ,t∂χ2y−bχ,tφχ,t2∂yφχ,t∂χy=0,(13)

subject to the initial condition

φχ,0=χyΓ(1+y),

where

aχ,t=1andbχ,t=2.(14)

Making use of the differential property of local fractional Sumudu operator on both sides of Eq. (13), we have

φyχ,z=φχ,0+zyLFSy∂2yφχ,t∂χ2y+φ(χ,t)∂yφχ,t∂χy=χyΓ(1+y)+zyLFSy∂2yφχ,t∂χ2y+φ(χ,t)∂yφχ,t∂χy.(15)

Now, applying inverse local fractional Sumudu transform on both sides of Eq. (15), we obtain

φχ,t=χyΓ(1+y)+LFSy−1zyLFSy∂2yφχ,t∂χ2y+φ(χ,t)∂yφχ,t∂χy.(16)

Applying homotopy perturbation method, we obtain

∑n=0∞pnφnχ,t=χyΓ(1+y)+pLFSy−1×zyLFSy∂2y∂χ2y∑n=0∞pnφnχ,t+∑n=0∞pnHnφ,(17)

where H_n(φ) are He’s polynomials representing the nonlinear terms. Thus, He’s polynomials are given by

∑n=0∞pnHnφ=φ(χ,t)∂yφχ,t∂χy.(18)

On evaluating, some components of He’s polynomials are given by

H0(φ)=φ0∂yφ0∂χy,H1φ=φ0∂yφ1∂χy+φ1∂yφ0∂χy,H2φ=φ0∂yφ2∂χy+φ1∂yφ1∂χy+φ2∂yφ0∂χy,H3φ=φ0∂yφ3∂χy+φ1∂yφ2∂χy+φ2∂yφ1∂χy+φ3∂yφ0∂χy,⋮

comparing the coefficients of like powers of p in Eq. (17), we have the following successive approximations

p0:φ0χ,t=χyΓ(1+y),

p1:φ1χ,t=LFSy−1zyLFSy∂2yφ0(χ,t)∂χ2y+H0(φ)=χyty(Γ(1+y))2,

p2:φ2χ,t=LFSy−1zyLFSy∂2yφ1(χ,t)∂χ2y+H1(φ)=2χyt2y(Γ(1+y))2,

p3:φ3χ,t=LFSy−1zyLFSy∂2yφ2(χ,t)∂χ2y+H2(φ)=3χyt3y(Γ(1+y))4+2χyt3y(Γ(1+y))3.

In this way, the remaining components of numerical solution can be obtained. Hence, the non-differentiable approximate solution of Eq. (13) is given as

φχ,t=limp→1limN→∞∑n=0Npnφnχ,t=χyΓ(1+y)+χyty(Γ(1+y))2+2χyt2y(Γ(1+y))2+3χyt3y(Γ(1+y))4+2χyt3y(Γ(1+y))3+….(19)

$Fig. 1 Numerical simulation of first order approximation with non-differentiable terms when y=ln⁡2ln⁡3$\begin{array}{} \displaystyle y=\frac{\ln2}{\ln3} \end{array}$ for Ex. 4.1.$

Fig. 1

Numerical simulation of first order approximation with non-differentiable terms when y=ln⁡2ln⁡3 for Ex. 4.1.

$Fig. 2 Numerical simulation of second order approximation with non-differentiable terms when y=ln⁡2ln⁡3$\begin{array}{} \displaystyle y=\frac{\ln2}{\ln3} \end{array}$ for Ex. 4.1.$

Fig. 2

Numerical simulation of second order approximation with non-differentiable terms when y=ln⁡2ln⁡3 for Ex. 4.1.

$Fig. 3 Numerical simulation of third order approximation with non-differentiable terms when y=ln⁡2ln⁡3$\begin{array}{} \displaystyle y=\frac{\ln2}{\ln3} \end{array}$ for Ex. 4.1.$

Fig. 3

Numerical simulation of third order approximation with non-differentiable terms when y=ln⁡2ln⁡3 for Ex. 4.1.

Example 4.2

In the final model, we study the local fractional nonlinear Fokker-Planck equation in fractal domain as follows:

∂yφ(χ,t)∂ty=aχ,φ∂2yφχ,t∂χ2y−bχ,φ∂yφχ,t∂χy,(20)

with the initial condition given by

φχ,0=Eyχy,(21)

where a (χ, φ) = φ (χ, t) and b (χ, φ) = φ (χ, t).

Making use of differential property of local fractional Sumudu operator on both sides of Eq. (20), we have

φyχ,z=φχ,0+zyLFSyφ(χ,t)∂2yφχ,t∂χ2y−φ(χ,t)∂yφχ,t∂χy=Eyχy+zyLFSyφ(χ,t)∂2yφχ,t∂χ2y−φ(χ,t)∂yφχ,t∂χy.(22)

Now, applying inverse local fractional Sumudu transform on both sides of Eq. (22), we get

φχ,t=Eyχy+LFSy−1zyLFSyφ(χ,t)∂2yφχ,t∂χ2y−φ(χ,t)∂yφχ,t∂χy.(23)

Applying homotopy perturbation method, we have

∑n=0∞pnφnχ,t=Eyχy+pLFSy−1zyLFSy∑n=0∞pnHnφ−∑n=0∞pnHn′φ,(24)

where H_n (φ) and Hn′φ are He’s polynomials representing the nonlinear terms. Thus, He’s polynomials are given by

∑n=0∞pnHnφ=φ(χ,t)∂2yφχ,t∂χ2y,

and

∑n=0∞pnHn′φ=φ(χ,t)∂yφχ,t∂χy.(25)

Some components of He’s polynomials are evaluated as

H0(φ)=φ0∂2yφ0∂χ2y,H0′(φ)=φ0∂yφ0∂χy,

H1φ=φ0∂2yφ1∂χ2y+φ1∂2yφ0∂χ2y,H1′φ=φ0∂yφ1∂χy+φ1∂yφ0∂χy,H2φ=φ0∂2yφ2∂χ2y+φ1∂2yφ1∂χ2y+φ2∂2yφ0∂χ2y,H2′φ=φ0∂yφ2∂χy+φ1∂yφ1∂χy+φ2∂yφ0∂χy,H3φ=φ0∂2yφ3∂χ2y+φ1∂2yφ2∂χ2y+φ2∂2yφ1∂χ2y+φ3∂2yφ0∂χ2y,H3′φ=φ0∂yφ3∂χy+φ1∂yφ2∂χy+φ2∂yφ1∂χy+φ3∂yφ0∂χy,⋮

Equating the coefficients of like powers of p in Eq. (24), we obtain the following iterates of solution

p0:φ0χ,t=Eyχy,p1:φ1χ,t=LFSy−1zyLFSyH0φ−H0′(φ)=0,p2:φ2χ,t=LFSy−1zyLFSyH1φ−H1′(φ)=0,p3:φ3χ,t=LFSy−1zyLFSyH2φ−H2′(φ)=0.

In this way, the remaining components of numerical solution can be obtained. Hence, the closed-form solution of Eq. (20) is given as

φχ,t=limp→1limN→∞∑n=0Npnφnχ,t=Eyχy,

and the corresponding graph is plotted in Figure 4 when y=ln⁡2ln⁡3.

$Fig. 4 Numerical simulation of the solution in closed form with non-differentiable term when y=ln⁡2ln⁡3$\begin{array}{} \displaystyle y=\frac{\ln2}{\ln3} \end{array}$ for Ex. 4.2.$

Fig. 4

Numerical simulation of the solution in closed form with non-differentiable term when y=ln⁡2ln⁡3 for Ex. 4.2.

5 Conclusion

In this paper, LFHPSTM has successfully been employed to obtain the non-differentiable solutions of LFNTE and LFNFPE arising in mathematical physics. The numerical solution for LFNTE is obtained with non-differentiable terms, whereas LFNFPE has the non-differentiable solution in closed-form. The proposed numerical technique is very simple, easily applicable and computationally effective for solving various local fractional differential equations emerging in numerous real-world phenomenon.

Tel. +91-8683953406, E-mail address: amitmath0185@gmail.com

Acknowledgement

The authors are extremely thankful to the reviewer’s for carefully reading the paper and useful comments and suggestions which have helped to improve the paper.

References

[1] I. Podlubny, Fractional differential equations, Academic Press, San Diego (1999).Suche in Google Scholar

[2] A. A. Kilbas, H. M. Srivastva, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematical Studies (Elsevier Publications), 204 (2006).Suche in Google Scholar

[3] S. Kumar, A. Kumar, IK Argyros, A new analysis for the Keller-Segel model of fractional order, Numerical Algorithms, 75(1) (2017) 213-228.10.1007/s11075-016-0202-zSuche in Google Scholar

[4] H. M. Baskonus, H. Bulut, On the numerical solutions of some fractional ordinary differential equations by fractional Adam-Bashforth-Moulton Method, Open Mathematics, 13(1) (2015) 547-556.10.1515/math-2015-0052Suche in Google Scholar

[5] H. M. Baskonus, H.Bulut, Regarding on the prototype solutions for the nonlinear fractional-order biological population model, AIP Conference Proceedings 1738, 290004, 2016.10.1063/1.4952076Suche in Google Scholar

[6] A. Prakash, M. Kumar, Numerical method for fractional dispersive partial differential equations, Communications in Numerical Analysis, 1 (2017) 1-18.10.5899/2017/cna-00266Suche in Google Scholar

[7] Z. Odibat, S. Momani, Numerical solution of Fokker-Planck equation with space and time-fractional derivatives. Physics Letters A: General, Atomic and Solid-State Physics 369 (5-6) (2007)349-358.Suche in Google Scholar

[8] A. Kumar, S. Kumar, A modified analytical approach for fractional discrete KdV equations arising in particle vibrations, Proceed. National Acad. Sci., Sect. A: Phys. Sci., 1-12 (2017).10.1007/s40010-017-0369-2Suche in Google Scholar

[9] J. Singh, D. Kumar, R. Swroop, S. Kumar, An efficient computational approach for time-fractional Rosenau -Hyman equations, Neur. Comput. Appl., 1-8 (2017).10.1007/s00521-017-2909-8Suche in Google Scholar

[10] S. Kumar, A. Kumar, Z. Odibat, A nonlinear fractional model to describe the population dynamics of two interacting species, Math. Meth App. Sci., 40(11) (2017) 4134-4148.10.1002/mma.4293Suche in Google Scholar

[11] S. Kumar, D. Kumar, J. Singh, Fractional modelling arising in unidirectional propagation of long waves in dispersive media, Advances in Nonlinear Analysis, 5(4) (2016) 2013-2033.10.1515/anona-2013-0033Suche in Google Scholar

[12] H. M. Baskonus, T. Mekkaoui, Z. Hammouch, H. Bulut, Active control of a chaotic fractional economic system, Entropy, 17(8) (2015) 5771-5783.10.3390/e17085771Suche in Google Scholar

[13] H. M. Baskonus, Z. Hammouch, T. Mekkaoui, H. Bulut, Chaos in the fractional order logistic delay system: Circuit realization and synchronization, AIP Conference Proceedings 1738, 290005, 2016.10.1063/1.4952077Suche in Google Scholar

[14] M.T. Gencoglu, H. M. Baskonus, H. Bulut, Numerical simulations to the nonlinear model of interpersonal relationships with time fractional derivative, AIP Conference Proceedings 1798, 1-9 (020103), 2017.10.1063/1.4972695Suche in Google Scholar

[15] A. Prakash, Analytical method for space-fractional telegraph equation by homotopy perturbation transform method, Nonlinear Engineering, 5(2) (2016) 123-128.10.1515/nleng-2016-0008Suche in Google Scholar

[16] A. Prakash, M. Kumar, Numerical solution of two-dimensional time fractional order biological population model, Open Physics, 14 (2016) 177–186.10.1515/phys-2016-0021Suche in Google Scholar

[17] A. Prakash, M. Kumar, He’s variational iteration method for the solution of nonlinear Newell-White-head-Segel equation, Journal of Applied Analysis and Computation, 6(3) (2016) 738–748.10.11948/2016048Suche in Google Scholar

[18] A. Prakash, M. Kumar, K. K. Sharma, Numerical method for solving fractional coupled Burgers equations, Applied Mathematics and Computation, 260 (2015) 314–320.10.1016/j.amc.2015.03.037Suche in Google Scholar

[19] A. Prakash, M. Kumar, Numerical method for solving time-fractional multi-dimensional diffusion equations, Int. J. Computing Science and Mathematics, 8(3) (2017) 257-267.10.1504/IJCSM.2017.085725Suche in Google Scholar

[20] X. J. Yang, D. Baleanu, J. H. He, Transport equations in fractal porous media within fractional complex transform method. Proc. Roman, Acad. Ser., A 14(4) (2013) 287-292.Suche in Google Scholar

[21] D. Baleanu, H. M. Srivastva, X. J.Yang, Local fractional variational iteration algorithms for the parabolic Fokker-Planck equation defined on cantor sets, Prog. Fract. Diff. Appl., 1(1) (2015) 1-11.Suche in Google Scholar

[22] Z. Odibat, S. Momani, Numerical solution of Fokker-Planck equation with space and time fractional derivatives, Physics Letters A: General, Atomic and Solid-State Physics, 369 (5-6) (2007) 349-358.Suche in Google Scholar

[23] L. Yan, Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method, Abstract and applied analysis, vol. 2013 (2013), Article ID 465160.10.1155/2013/465160Suche in Google Scholar

[24] X. J. Yang, Advanced local fractional calculus and its applications, World Science Publisher, New York (2012).Suche in Google Scholar

[25] X. J. Yang, Local fractional functional analysis and its applications, Asian Academic Publisher Limiited, Hong Kong (2011).Suche in Google Scholar

[26] X. J. Yang, D. Baleanu, H. M. Srivastva, Local fractional integral transforms and their applications, Academic Press (2015).10.1016/B978-0-12-804002-7.00004-8Suche in Google Scholar

[27] X. J. Yang, D. Baleanu, H. M. Srivastva, Local fractional similarity solution for the diffusion equation defined on Cantor sets, Appl. Math. Lett., 47 (2015) 54-60.10.1016/j.aml.2015.02.024Suche in Google Scholar

[28] X. J. Yang, Y. Zhang, An efficient analytical method for solving local fractional nonlinear PDEs arising in mathematical physics, Applied Mathematical Modelling 40 (2016) 1793-1799.10.1016/j.apm.2015.08.017Suche in Google Scholar

[29] J. H. He, Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng., 178 (1999) 257-262.10.1016/S0045-7825(99)00018-3Suche in Google Scholar

[30] D. Ziane, D. Baleanu, K. Belghaba, M. Hamdi Cherif, Local fractional Sumudu decomposition method for linear partial differential equations with local fractional derivatives, Journal of King Saud University-Science (2017).10.1016/j.jksus.2017.05.002Suche in Google Scholar

[31] K. M. Kolwankar, A. D. Gangal, Local fractional Fokker-Planck equation, Phys. Rev. Lett., 80 (2) (1998) 214-217.10.1103/PhysRevLett.80.214Suche in Google Scholar

[32] X. J. Yang, H. M. Srivastva, J. H. He, D. Baleanu, Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Phys. Lett., A 377 (27) (2013) 1696-1700.10.1016/j.physleta.2013.04.012Suche in Google Scholar

Received: 2017-08-08

Revised: 2017-10-24

Accepted: 2017-12-29

Published Online: 2018-02-22

Published in Print: 2018-09-25

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Artikel in diesem Heft

https://doi.org/10.1515/nleng-2017-0100

Schlagwörter für diesen Artikel

Local fractional Sumudu transform; Homotopy perturbation method; Approximate solution; Nonlinear partial differential equations; Local fractional derivative

Creative Commons

BY-NC-ND 3.0