Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 364853, 9 pages
http://dx.doi.org/10.1155/2015/364853

Research Article

Differential Transform Method with Complex Transforms to Some Nonlinear Fractional Problems in Mathematical Physics

Syed Tauseef Mohyud-Din,¹ Farah Jabeen Awan,¹ Jamshad Ahmad,¹ and Saleh M. Hassan2,3

¹Faculty of Sciences, HITEC University, Taxila Cantonment 44000, Pakistan
²Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
³Department of Mathematics, College of Science, Ain Shams University, Abbassia, Cairo 11566, Egypt

Received 28 May 2015; Revised 31 August 2015; Accepted 28 September 2015

Academic Editor: Fazal M. Mahomed

Copyright © 2015 Syed Tauseef Mohyud-Din et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.

1. Introduction

Nonlinear partial differential equations (NLPDEs) are mathematical models that are used to describe complex phenomena and dynamic processes arising in the world around us. The NLPDEs appear in many applications of science and engineering such as fluid dynamics, plasma physics, hydrodynamics, solid state physics, optical fibers, and acoustics, as well as other disciplines. Recently, lot of attention is paid to finding appropriate solutions of NLPDEs. In the similar context, various techniques including Adomian’s decomposition method (ADM) [1], Variational Iteration (VIM) [2], Homotopy Perturbation (HPM) [3], Homotopy Analysis (HAM) [4], F-Expansion [5], Exp-function [6], sine-cosine [7], differential transform method (DTM) [8–11], and reduced differential transform [9, 12–15] have been applied on wide range of linear and nonlinear problems of diversified physical nature. Inspired and motivated by ongoing research in this area, we apply reduced differential transform method (RDTM) [12–19] coupled with a complex transform to solve three important mathematical models [20–27], namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions. It is worth mentioning that derivatives are defined in Jumarie’s sense which is relatively a new approach and is easier to handle; however, other approaches like Caputo and Riemann Liouville may also be utilized. It is an established fact that models under discussion [20–23] are of extreme importance and hence appear frequently in various physical phenomena including nonlinear dispersive waves, shallow water waves, ion acoustic plasma waves, Lax pairs of a special form, four-reduction of KP hierarchy, Frenkel-Kontorova dislocation model; see [20–23] and the references therein. It is observed that the proposed technique is extremely simple and user friendly and has shown very useful results. It is to be highlighted that the suggested modified version may be extended to some other important nonlinear problems which have been solved by some other reliable methods; see [28–33]. It is to be highlighted that Ganji et al. have solved wide range of mathematical models [28–33] by making an appropriate use of some recently developed schemes and hence giving a new avenue of research.

2. Jumarie’s Fractional Derivative

Some useful formulas and results of Jumarie’s fractional derivative were summarized [24]

3. Fractional Complex Transform Method (FCTM)

The fractional complex transform was first proposed in [25] and is defined as

where , , , and are unknown constants: , , , and .

4. Reduced Differential Transform Method (RDTM)

To illustrate the basic idea of the DTM, The differential transform of th derivative of a function , which is analytic and differentiated continuously in the domain of interest, is defined as

The differential inverse transform of is defined as follows:

Equation (8) is known as the Taylor series expansion of , around . Combining (7) and (8),

when ; the above equation reduces to

and (2) reduces to

Some properties of the reduced differential transform method are as follows.

(1)If the original function is , then the transformed function is

(2)If , then .(3)If , then .(4)If , then .(5)If , then .(6)If , then .(7)If , then .(8)If , then .

5. Numerical Applications

In this section, we apply the proposed fractional complex transform method coupled with reduced differential transform method to solve three important mathematical models. Numerical results are highly encouraging. For details about such equations, readers are referred to study [22, 23].

5.1. Fractional Kaup-Kupershmidt (FKK) Equation

It is an established fact that fractional Kaup-Kupershmidt (FKK) equation plays a major role in the study of nonlinear dispersive waves. Moreover, it describes a large number of important physical phenomena, such as shallow water waves and ion acoustic plasma waves.

Consider the nonlinear KK equation [22, 23],

with the initial condition

where is an arbitrary constant.

Applying the transformation [25], we get the following partial differential equation:

Applying the differential transform to (15) and (14), we obtain the following recursive formula:

Using the initial condition, we have

Now, substituting (17) into (16), and by straightforward iterative steps, yields

and so on.

The series solution is given by

The inverse transformation will yield

This solution is convergent to the exact solution

In Figures 1(a) and 1(b), we have presented approximate solution at and exact solutions.

(a)

(b)

Figure 1: (a) Approximate solution. (b) Exact solution.

5.2. Generalized Fractional Drinfeld-Sokolov (GFDS) Equations [20, 21]

This system was introduced independently by Drinfeld and Sokolov [20, 21]. This coupled system was given as one of the numerous examples of nonlinear equations possessing Lax pairs of a special form. Also, the coupled system was found as a special case of the four-reduction of the KP hierarchy; see [20, 21] and the references therein.

We consider the system of generalized fractional Drinfeld-Sokolov (GFDS) equations [20, 21]

with the initial conditions

where is a constant.

Applying the transformation [25], we get the following partial differential equations:

Applying the differential transform to (24) and (23), we obtain the following recursive formula:

Using the initial condition, we have

Now, substituting (26) into (25) when (), and by straightforward iterative steps, yields

and so on.

The series solution is given by

The inverse transformation will yields

This solution is convergent to the exact solution

In Figures 2(a)–2(d), we have presented approximate and at and exact solutions.

(a)

(b)

(c)

(d)

Figure 2: (a) Approximate solution. (b) Exact solution. (c) Approximate solution. (d) Exact solution.

5.3. System of Coupled Fractional Sine-Gordon Equations [26, 27]

Coupled Sine-Gordon equations were introduced by Ray et al. [26, 27]. The coupled Sine-Gordon equations generalize the Frenkel-Kontorova dislocation model; see [26, 27] and the references therein.

We now consider a system of coupled Sine-Gordon equations [26, 27]:

with the initial conditions

where is the ratio of the acoustic velocities of the components and .

Applying the transformation [25] to (31), we get the following partial differential equations:

Applying the differential transform to (33) and (32), we obtain the following recursive formula:

Using the initial condition, we have

Now, substituting (35) into (34), and by straightforward iterative steps, yields

and so on.

The series solution is given by

The inverse transformation will yield

This solution is convergent to Adomian’s decomposition method solution [26, 27].

In Figures 3(a) and 3(b) we have presented approximate solutions and at .

(a)

(b)

Figure 3: (a) Approximate solution . (b) Approximate solution .

6. Conclusion

In this research, we present new applications of the fractional complex transform method with coupling reduced differential transform method (RDTM) by handling three nonlinear physical fractional dynamical models. This coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique, which does not require linearization, discretization or perturbation, gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool for solving nonlinear equations and hence may be extended to other nonlinear problems also.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The project was funded by the National Plan for Science, Technology and Innovation (MAARIFA), King Abdul Aziz City for Science & Technology, Kingdom of Saudi Arabia, Award no. 15-MAT4688-02.

References

G. Adomian, “A new approach to nonlinear partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 102, no. 2, pp. 420–434, 1984.

S. Abbasbandy, “Numerical solution of non-linear Klein–Gordon equations by variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 70, no. 7, pp. 876–881, 2007.

M. A. Noor and S. T. Mohyud-Din, “Homotopy perturbation method for solving thomas-fermi equation using pade approximants,” International Journal of Nonlinear Science, vol. 8, no. 1, pp. 27–31, 2009.

S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003.

J.-L. Zhang, M.-L. Wang, Y.-M. Wang, and Z.-D. Fang, “The improved F-expansion method and its applications,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 350, no. 1-2, pp. 103–109, 2006.

S. T. Mohyud-Din, M. A. Noor, K. . Noor, and M. M. Hosseini, “Variational iteration method for re-formulated partial differential equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 2, pp. 87–92, 2010.

A.-M. Wazwaz, “A sine–cosine method for handling nonlinear wave equations,” Mathematical and Computer Modelling, vol. 40, no. 5-6, pp. 499–508, 2004.

J. K. Zhou, Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.

J. Ahmad and S. T. Mohyud-Din, “An efficient algorithm for some highly nonlinear fractional PDEs in mathematical physics,” PLoS ONE, vol. 9, no. 12, Article ID e109127, 2014.

A. Arikoglu and I. Ozkol, “Solution of fractional differential equations by using differential transform method,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1473–1481, 2007.

A. Kurnaz and G. Oturanç, “The differential transform approximation for the system of ordinary differential equations,” International Journal of Computer Mathematics, vol. 82, no. 6, pp. 709–719, 2005.

A. Saravanan and N. Magesh, “A comparison between the reduced differential transform method and the Adomian decomposition method for the Newell-Whitehead-Segel equation,” Journal of the Egyptian Mathematical Society, vol. 21, no. 3, pp. 259–265, 2013.

R. Abazari and M. Abazari, “Numerical study of Burgers-Huxley equations via reduced differential transform method,” Computational & Applied Mathematics, vol. 32, no. 1, pp. 1–17, 2013.

B. Bis and M. Bayram, “Approximate solutions for some nonlinear evolutions equations by using the reduced differential transformmethod,” International Journal of Applied Mathematical Research, vol. 1, no. 3, pp. 288–302, 2012.

R. Abazari and B. Soltanalizadeh, “Reduced differential transform method and its application on Kawahara equations,” Thai Journal of Mathematics, vol. 11, no. 1, pp. 199–216, 2013.

M. A. Abdou and A. A. Soliman, “Numerical simulations of nonlinear evolution equations in mathematical physics,” International Journal of Nonlinear Science, vol. 12, no. 2, pp. 131–139, 2011.

M. A. Abdou, “Approximate solutions of system of PDEEs arising in physics,” International Journal of Nonlinear Science, vol. 12, no. 3, pp. 305–312, 2011.

P. K. Gupta, “Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2829–2842, 2011.

R. Abazari and M. Abazari, “Numerical simulation of generalized Hirota–Satsuma coupled KdV equation by RDTM and comparison with DTM,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 619–629, 2012.

Y. Ugurlu and D. Kaya, “Exact and numerical solutions of generalized Drinfeld-Sokolov equations,” Physics Letters A, vol. 372, no. 16, pp. 2867–2873, 2008.

G. A. Afrouzi, J. Vahidi, and M. Saeidy, “Numerical solutions of generalized Drinfeld-Sokolov equations using the homotopy analysis method,” International Journal of Nonlinear Science, vol. 9, no. 2, pp. 165–170, 2010.

A. Mohebbi, “Numerical solution of nonlinear Kaup-Kupershmit equation, KdV-KdV and hirota-satsuma systems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 13, no. 7-8, pp. 479–486, 2012.

K. A. Gepreel, S. Omran, and S. K. Elagan, “The traveling wave solutions for some nonlinear PDEs in mathematical physics,” Applied Mathematics, vol. 2, no. 3, pp. 343–347, 2011.

G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.

Z.-B. Li and J.-H. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 970–973, 2010.

S. S. Ray, “A numerical solution of the coupled sine-Gordon equation using the modified decomposition method,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1046–1054, 2006.

A. Sadighi, D. D. Ganji, and B. Ganjavi, “Traveling wave solutions of the sine-gordon and the coupled sine-gordon equations using the homotopy-perturbation method,” Scientia Iranica Transaction B: Mechanical Engineering, vol. 16, no. 2, pp. 189–195, 2009.

M. Safari, D. D. Ganji, and M. Moslemi, “Application of He's variational iteration method and Adomian's decomposition method to the fractional KdV-Burgers-Kuramoto equation,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2091–2097, 2009.

Z. Z. Ganji, D. D. Ganji, and Y. Rostamiyan, “Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique,” Applied Mathematical Modelling, vol. 33, no. 7, pp. 3107–3113, 2009.

S. E. Ghasemi, A. Zolfagharian, and D. D. Ganji, “Study on motion of rigid rod on a circular surface using MHPM,” Propulsion and Power Research, vol. 3, no. 3, pp. 159–164, 2014.

M. Hatami and D. D. Ganji, “Thermal and flow analysis of microchannel heat sink (MCHS) cooled by Cu-water nanofluid using porous media approach and least square method,” Energy Conversion and Management, vol. 78, pp. 347–358, 2014.

D. D. Ganji, A. Sadighi, and I. Khatami, “Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences,” Physics Letters A, vol. 372, no. 24, pp. 4399–4406, 2008.

M. Rafei, D. D. Ganji, H. Daniali, and H. Pashaei, “The variational iteration method for nonlinear oscillators with discontinuities,” Journal of Sound and Vibration, vol. 305, no. 4-5, pp. 614–620, 2007.