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Advances in Mathematical Physics
Volume 2016, Article ID 7304659, 7 pages
http://dx.doi.org/10.1155/2016/7304659
Research Article

Similarity Solutions for Multiterm Time-Fractional Diffusion Equation

1Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt
2Mathematics & Engineering Physics Department, Faculty of Engineering, Modern University for Technology and Information, Cairo 11585, Egypt

Received 12 January 2016; Revised 19 March 2016; Accepted 29 March 2016

Academic Editor: Xiao-Jun Yang

Copyright © 2016 A. Elsaid 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

Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on the obtained results, we propose a definition for a multiterm error function with generalized coefficients.

1. Introduction

Fractional differential equations (FDEs) appear in modeling many problems in the fields of science and engineering [13]. These problems are modeled using different types of fractional derivative operators which include Riemann-Liouville definition [1], Caputo definition [4], Riesz definition [5], Riesz-Feller definition [6], and the modified Riemann-Liouville definition recently proposed by Jumarie [7].

To obtain analytic solutions to fractional partial differential equations (FPDEs), two methods have been basically used: the first method is the application of both Laplace and Fourier transforms and the second method is the separation of variables technique [1]. But recently several semianalytic methods have been also utilized to present series solution to FPDEs such as Adomian decomposition method [8, 9], homotopy analysis method [10, 11], homotopy perturbation method [12, 13], variational iteration method [14, 15], and fractional differential transformation method [16, 17].

Fractional diffusion models are formulated using fractional derivative operators to replace regular derivatives. Different forms of fractional diffusion equations have been widely researched. For example, a time-fractional diffusion equation has been explicitly introduced in physics by Nigmatullin [18] to describe diffusion in special types of porous media which exhibit a fractal geometry. Giona et al. [19] also presented a time-fractional diffusion equation that describes relaxation phenomena in complex viscoelastic materials. Wyss [20] considered the time-fractional diffusion and wave equations and obtained the solution in closed form in terms of Fox functions. Gorenflo et al. [21] used the similarity method and the Laplace transform method to obtain the scale invariant solution of the time-fractional diffusion-wave equation in terms of the Wright function. Recently, fractional diffusion equations have been modeled and studied via local fractional derivatives as in the work of Liu et al. [22] and Zhao et al. [23].

Fractional diffusion equations have been also utilized to model anomalous diffusion where a particle plume spreads faster than predicted by classical models and may exhibit significant asymmetry. The anomalous diffusion processes differ from regular diffusion in that the dispersion of particles proceeds faster (superdiffusion) or slower (subdiffision) than for the regular case. Anomalous diffusion is one of the most popular phenomena in the theory of random walks and transport processes [2426]. Also anomalous diffusion is a vast and exciting topic, in particular for “crowded” biological systems, where the transport of molecules plays a central functional role [27]. Lenzi et al. investigate the solutions for a fractional diffusion equation subjected to boundary conditions which can be connected to adsorption-desorption processes; the analytical solutions were obtained using the Green function approach [28]. Prehl et al. in [29] deal with diffusion equation with a focus on space-fractional diffusion equation.

The large number of applications that are modeled via fractional diffusion equations motivated mathematicians to develop and apply several techniques to obtain analytic and approximate solutions to this class of FPDEs. Some examples of this work include studying the existence and uniqueness of solution [30], presenting the fundamental solution to some types of fractional diffusion equations [31, 32], applying Green’s function approach [33], obtaining approximate solution via pseudospectral scheme [34], solution via radial basis functions [35], stochastic solution via Monte Carlo simulation [36], approximate solution based on the shifted Legendre-tau technique [37], solution by integral transform method [38], numerical solution by finite difference scheme [39], approximate analytical solution by homotopy analysis method [40], approximate analytical solution by optimal homotopy analysis method [41], approximate analytical solution via Adomian method [42], approximate analytical solution by variational iteration method [43], and approximate analytical solution via generalized differential transform method [44].

Multiterm FPDEs provide more generalizations to fractional order models. Thus many authors have dealt with them either analytically or numerically. For example, the authors of [45] developed a direct solution technique for solving the linear multiterm FDEs with constant coefficients using a spectral tau method. In [46], an effective finite element method is presented for the multiterm time-space Riesz fractional advection-diffusion equations. A numerical method is proposed in [47] to reduce the multiterm FDEs to a system of algebraic equations based on Bernstein polynomials basis. Also in [48], the authors present a numerical method for solving the multiterm time-fractional wave-diffusion equations. A study on the nonhomogeneous generalized multiterm fractional heat propagation and fractional diffusion-convection equation in three-dimensional space is presented in [49].

Though symmetry methods have been applied to different types of linear and nonlinear partial differential equations, the research on using these methods for solving FPDEs is still in the initial stage. The work reported on the Riemann-Liouville fractional derivative includes [50] where scaling transformations are derived for the time-fractional heat equation to reduce it to an ordinary FDE but with Erdelyi-Kober fractional differential operator. Also, similarity solutions for the time-fractional nonlinear conduction equations are presented in [51] to reduce the considered problems to ordinary FDEs that are solved by analytic and numerical techniques. The fractional derivative proposed by Jumarie is studied in [52] where the Lie group method is applied to a space-time-fractional diffusion equation. Choksi and Timol [53] derived similarity solution for fractional diffusion-wave equation using Lie group transformation and Erdelyi-Kober fractional differential operator. Through the invariants of the group of scaling transformations, Duan et al. [54] derived the integrodifferential equation for the similarity variable; then, by virtue of Mellin transform, the fundamental solution of the FDE was expressed in terms of Fox functions. Finally, El Kinani and Ouhadan [55] used Lie symmetry analysis to reduce the number of independent variables of time-fractional partial differential equations; then they employed symmetry properties to construct some exact solutions.

In this work, we use similarity methods to solve multiterm time-fractional diffusion equation with fractional derivatives defined in Caputo sense. The FPDE is reduced to an ordinary FDE which is solved by the power series method. This paper is arranged as follows. The definition and properties of Caputo fractional derivative are listed in Section 2. In Section 3, we illustrate how the similarity method technique is employed to transform a multiterm FPDE into a multiterm FDE. In Section 4, power series method is applied to solve the obtained FDE. The conclusion of this work is summarized in Section 5.

2. Fractional Calculus

Definition 1. A real function , , is said to be in the space , , if there exists a real number , such that , where , and it is said to be in the space if , .

Definition 2. The Riemann-Liouville fractional integral operator of order of a function , is defined as

Definition 3. The fractional derivative in Caputo sense of , , is defined as [1]

One basic property of Caputo fractional derivative is

For more details on Caputo fractional derivative definition and properties see [1, 56, 57].

3. Similarity Method Solution

In this section, we illustrate the technique for using similarity methods in solving multiterm FPDEs with Caputo fractional derivative.

Problem 4. Consider the following problem:

where are constants, with an initial condition . Here denotes the partial fractional derivative of order of with respect to the time variable in the Caputo sense. To solve (4), first we perform its scaling transformation using similarity methods; see [51, 58]. Consider the new independent and dependent variables denoted by , , and defined in the following way:where is called the scaling parameter and , , and are arbitrary constants to be determined such that (4) remains invariant under this transformation. From Caputo definition for , one may easily verify thatwhere . Also, for , we have

Hence, by substituting (6) and (7) into (4), we get

From (8), it is clear that, by setting and , then (4) is invariant under transformation (5). The characteristic equation associated with transformation (5) is given by

At , this shows that can be expressed aswhere

By using formula (10), we havewhere .

Again, by using formula (10), we have

Substituting (11) and (12) into (4), the resulting FDE is given bywith the initial condition .

4. Power Series Solution

In this section, we develop the series solution of (13). Let

Using (3), we have

Substituting (15) and (16) into (13) and comparing the coefficients of identical powers on both sides, we get

For the initial condition , we get . Hence, takes the formBased on these results, we propose a definition for multiterm error function with generalized coefficients and discuss its convergence.

Definition 5. A multiterm error function with generalized coefficients is defined in the formwhere

By this definition, the solution of the ordinary FDE obtained in (13) is given bywhere and are arbitrary constants.

To check the radius of convergence of the multiterm error function with generalized coefficients (19) and (20), we evaluate the following limit:

Hence, the series solution converges for all

We consider the case in (18). Figures 1, 2, and 3 illustrate the effect of changing the orders of fractional derivatives , , and on the behavior of Taylor series representation of the solution function at different values of the constants , .

Figure 1: The effect of changing on the solution at , , , , and .
Figure 2: The effect of changing on the solution at , , , , and .
Figure 3: The effect of changing on the solution at , , , , and .

In the case where and , (13) is reduced towhich has a solution of the formwhere and are arbitrary constants and is the error function with generalized coefficients defined in the formwhere

Hence for the condition , equal zero.

Evidently, when we substitute into (24), the obtained series is the Taylor series of the classical integer-order error function . This coincides with the fact that in this case (23) becomes the ordinary differential equation of the form which has the general solution , with to satisfy the initial condition .

Figures 4 and 5 illustrate that, as approaches one, the graph of the solution coincides with the graph of the classical error function Also, when approaches zero in (23), in this case the obtained series is the Taylor series of the classical trigonometric sine function . This coincides with the fact that in this case (23) becomes the ordinary differential equation which has the general solution of the form , with to satisfy the initial condition .

Figure 4: The solution in the case at different values of
Figure 5: The solution at and different values of , .

5. Conclusion

The similarity method is used to solve multiterm FPDEs where the fractional derivatives are given in Caputo sense. We considered time-fractional diffusion equation and illustrated that the similarity method successfully transforms this equation with two independent variables into an ordinary FDE in the same fractional derivative sense. Also it is noticed that the obtained FDE inherits the multiterm nature from the original FPDE.

The ordinary FDE is solved using power series expansion. We proved that the obtained series solution has an infinite radius of convergence. Thus, not only the solution obtained satisfies the equation, but also it can provide the exact solution at any point. So, any other semianalytic solution to this problem should be compared to this one.

Based on this series solution, a new definition is proposed for what we call the multiterm error function with generalized coefficients. This function is considered as a generalization of the classical error function and it coincides with it when the orders of the fractional derivatives approach one and the sum of the coefficients of the multiterm derivatives equals , whereas as the orders approach zero, the function undergoes oscillation that tend to the trigonometric sine function.

The parameters involved in the definition of multiterm error function with generalized coefficients enrich this function. The figures presented in the considered case study illustrate that as the values of these parameters vary, the function exhibits different types of behaviors that can describe a wider range of applied models.

Competing Interests

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

References

  1. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  2. M. Dalir and M. Bashour, “Applications of fractional calculus,” Applied Mathematical Sciences, vol. 4, pp. 1021–1032, 2010. View at Google Scholar
  3. R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Danbury, Conn, USA, 2006.
  4. M. Caputo, “Linear models of dissipation whose Q is almost frequency independent—II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967. View at Publisher · View at Google Scholar
  5. S. G. Samko, A. A. Kilbas, and O. L. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, NY, USA, 1993.
  6. M. Ciesielski and J. Leszczynski, “Numerical solutions to boundary value problem for anomalous diffusion equation with Riesz-Feller fractional operator,” Journal of Theoretical and Applied Mechanics, vol. 44, no. 2, pp. 393–403, 2006. View at Google Scholar
  7. G. Jumarie, “Tables of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non- differentiable functions,” Applied Mathematics Letters, vol. 22, pp. 378–385, 2009. View at Google Scholar
  8. D. B. Dhaigude and G. A. Birajdar, “Numerical solution of system of fractional partial differential equations by discrete Adomian decomposition method,” Journal of Fractional Calculus and Applications, vol. 3, no. 12, pp. 1–11, 2012. View at Publisher · View at Google Scholar
  9. A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters A, vol. 359, no. 3, pp. 175–182, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. A. Elsaid, “Homotopy analysis method for solving a class of fractional partial differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3655–3664, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. H. Jafari, A. Golbabai, S. Seifi, and K. Sayevand, “Homotopy analysis method for solving multi-term linear and nonlinear diffusion-wave equations of fractional order,” Computers &Mathematics with Applications, vol. 59, no. 3, pp. 1337–1344, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. A. M. El-Sayed, A. Elsaid, I. L. El-Kalla, and D. Hammad, “A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8329–8340, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,” Physics Letters A, vol. 365, no. 5-6, pp. 345–350, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. V. Turut and N. Güzel, “On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximations,” European Journal of Pure and Applied Mathematics, vol. 6, no. 2, pp. 147–171, 2013. View at Google Scholar
  15. A. Elsaid, “The variational iteration method for solving Riesz fractional partial differential equations,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1940–1947, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. A. Secer, M. A. Akinlar, and A. Cevikel, “Efficient solutions of systems of fractional PDEs by the differential transform method,” Advances in Difference Equations, vol. 2012, article 188, 7 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. R. R. Nigmatullin, “The realization of the generalized transfer equation in a medium with fractal geometry,” Physica Status Solidi B, vol. 133, no. 1, pp. 425–430, 1986. View at Publisher · View at Google Scholar
  19. M. Giona, S. Cerbelli, and H. E. Roman, “Fractional diffusion equation and relaxation in complex viscoelastic materials,” Physica A: Statistical Mechanics and Its Applications, vol. 191, no. 1–4, pp. 449–453, 1992. View at Publisher · View at Google Scholar · View at Scopus
  20. W. Wyss, “The fractional diffusion equation,” Journal of Mathematical Physics, vol. 27, no. 11, pp. 2782–2785, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. R. Gorenflo, Y. Luchko, and F. Mainardi, “Wright functions as scale-invariant solutions of the diffusion-wave equation,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 175–191, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. F. Liu, Z. Li, S. Zhang, and H. Liu, “He’s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy,” Thermal Science, vol. 19, no. 4, pp. 1155–1159, 2015. View at Publisher · View at Google Scholar
  23. D. Zhao, X. J. Yang, and H. M. Srivastava, “On the fractal heat transfer problems with local fractional calculus,” Thermal Science, vol. 19, no. 5, pp. 1867–1871, 2015. View at Publisher · View at Google Scholar
  24. F. Huang and F. Liu, “The time fractional diffusion equation and the advection dispersion equation,” The ANZIAM Journal, vol. 46, no. 3, pp. 317–330, 2005. View at Publisher · View at Google Scholar
  25. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  26. R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, and P. Paradisi, “Fractional diffusion: probability distributions and random walk models,” Physica A: Statistical Mechanics and its Applications, vol. 305, no. 1-2, pp. 106–112, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. G. Kneller, “Anomalous diffusion in biomolecular systems from the perspective of non-equilibrium statistical physics,” Acta Physica Polonica B, vol. 46, no. 6, pp. 1167–1199, 2015. View at Publisher · View at Google Scholar
  28. E. Lenzi, M. dos Santos, M. Lenzi, D. Vieira, and L. da Silva, “Solutions for a fractional diffusion equation: anomalous diffusion and adsorption–desorption processes,” Journal of King Saud University—Science, vol. 28, no. 1, pp. 3–6, 2016. View at Publisher · View at Google Scholar
  29. J. Prehl, C. Essex, and K. H. Hoffmann, “Tsallis relative entropy and anomalous diffusion,” Entropy, vol. 14, no. 4, pp. 701–716, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  30. J. Kemppainen, “Existence and uniqueness of the solution for a time-fractional diffusion equation,” Fractional Calculus and Applied Analysis, vol. 14, no. 3, pp. 411–417, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001. View at Google Scholar · View at MathSciNet
  32. S. Shen, F. Liu, and V. Anh, “Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order,” Journal of Applied Mathematics and Computing, vol. 28, no. 1-2, pp. 147–164, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. E. Hernandez-Martinez, F. Valdés-Parada, J. Alvarez-Ramirez, H. Puebla, and E. Morales-Zarate, “A Green's function approach for the numerical solution of a class of fractional reaction–diffusion equations,” Mathematics and Computers in Simulation, vol. 121, pp. 133–145, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  34. S. Esmaeili and R. Garrappa, “A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation,” International Journal of Computer Mathematics, vol. 92, no. 5, pp. 980–994, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. A. Mohebbi, M. Abbaszadeh, and M. Dehghan, “Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions (RBF) meshless method,” Engineering Analysis with Boundary Elements, vol. 38, pp. 72–82, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. D. Fulger, E. Scalas, and G. Germano, “Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation,” Physical Review E, vol. 77, no. 2, Article ID 021122, 2008. View at Publisher · View at Google Scholar · View at Scopus
  37. A. Saadatmandi and M. Dehghan, “A tau approach for solution of the space fractional diffusion equation,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1135–1142, 2011. View at Publisher · View at Google Scholar · View at Scopus
  38. V. B. L. Chaurasia, “Solution of a time-space fractional diffusion equation by integral transform method,” Tamsui Oxford Journal of Information and Mathematical Sciences, vol. 28, no. 2, pp. 153–164, 2012. View at Google Scholar
  39. V. D. Beibalaev and M. R. Shabanova, “A finite-difference scheme for solution of a fractional heat diffusion-wave equation without initial conditions,” Thermal Science, vol. 19, no. 2, pp. 531–536, 2015. View at Publisher · View at Google Scholar · View at Scopus
  40. P. K. Gupta, “An approximate analytical solution of nonlinear fractional diffusion equation by homotopy analysis method,” International Journal of Physical Sciences, vol. 6, no. 34, pp. 7721–7728, 2011. View at Google Scholar · View at Scopus
  41. S. Das, K. Vishal, and P. K. Gupta, “Approximate analytical solution of diffusion equation with fractional time derivative using optimal homotopy analysis method,” Surveys in Mathematics and Its Applications, vol. 8, pp. 35–49, 2013. View at Google Scholar · View at MathSciNet
  42. S. Saha Ray and R. K. Bera, “Analytical solution of a fractional diffusion equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 329–336, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. S. Das, “Analytical solution of a fractional diffusion equation by variational iteration method,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 483–487, 2009. View at Publisher · View at Google Scholar · View at Scopus
  44. A. Çetinkaya and O. Kiymaz, “The solution of the time-fractional diffusion equation by the generalized differential transform method,” Mathematical and Computer Modelling, vol. 57, no. 9-10, pp. 2349–2354, 2013. View at Publisher · View at Google Scholar · View at Scopus
  45. E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations,” Applied Mathematical Modelling, vol. 35, no. 12, pp. 5662–5672, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  46. J. Zhao, J. Xiao, and Y. Xu, “A finite element method for the multiterm time-space Riesz fractional advection-diffusion equations in finite domain,” Abstract and Applied Analysis, vol. 2013, Article ID 868035, 15 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  47. D. Rostamy, M. Alipour, H. Jafari, and D. Baleanu, “Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis,” Romanian Reports in Physics, vol. 65, no. 2, pp. 334–349, 2013. View at Google Scholar · View at Scopus
  48. F. Liu, M. M. Meerschaert, R. J. McGough, P. Zhuang, and Q. Liu, “Numerical methods for solving the multi-term time-fractional wave-diffusion equation,” Fractional Calculus and Applied Analysis, vol. 16, no. 1, pp. 9–25, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  49. A. Aghili and M. R. Masomi, “Nonhomogeneous generalized multi-term fractional heat propagation and fractional diffusion-convection equation in three-dimensional space,” New Trends in Mathematical Sciences, vol. 2, no. 2, pp. 106–116, 2014. View at Google Scholar
  50. E. Buckwar and Y. Luchko, “Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations,” Journal of Mathematical Analysis and Applications, vol. 227, no. 1, pp. 81–97, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  51. V. D. Djordjevic and T. M. Ttanackovic, “Similarity solution to nonlinear heat conduction and Burgers/Korteweg-de Vries fractional equation,” Journal of Computational and Applied Mathematics, vol. 222, no. 2, pp. 701–714, 2008. View at Publisher · View at Google Scholar
  52. G. C. Wu, “A Fractional Lie group method for anomalous diffusion equations,” Communications in Fractional Calculus, vol. 1, pp. 27–31, 2010. View at Google Scholar
  53. H. Choksi and M. G. Timol, “Similarity solution for partial differential equation of fractional order,” International Journal of Engineering and Innovative Technology (IJEIT), vol. 4, no. 2, pp. 64–67, 2014. View at Google Scholar
  54. J.-S. Duan, A.-P. Guo, and W.-Z. Yun, “Similarity solution for fractional diffusion equation,” Abstract and Applied Analysis, vol. 2014, Article ID 548126, 5 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  55. E. H. El Kinani and A. Ouhadan, “Lie symmetry analysis of some time fractional partial differential equations,” International Journal of Modern Physics: Conference Series, vol. 38, Article ID 1560075, 8 pages, 2015. View at Publisher · View at Google Scholar
  56. M. M. Khader, N. H. Sweilam, A. M. S. Mahdy, and N. K. A. Moniem, “Numerical simulation for the fractional SIRC model and influenza A,” Applied Mathematics and Information Sciences, vol. 8, no. 3, pp. 1029–1036, 2014. View at Publisher · View at Google Scholar · View at Scopus
  57. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  58. R. Kandasamy and I. Muhaimin, “Scaling transformation for the effect of temperature-dependent fluid viscosity with thermophoresis particle deposition on MHD-Free convective heat and mass transfer over a porous stretching surface,” Transport in Porous Media, vol. 84, no. 2, pp. 549–568, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus