# Application of Homotopy Perturbation Transform Method to Linear and Non-linear Space-time Fractional Reaction-diffusion Equations

Volume 5, Issue 1, pp 40-52
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### Authors

Jagdev Singh - Department of Mathematics, Jagan Nath University, Jaipur, Rajasthan, India. Devendra Kumar - Department of Mathematics, Jagan Nath Gupta Inst. of Engg. & Tech., Jaipur, Rajasthan, India. Sushila - Department of Physics, Jagan Nath University, Jaipur, Rajasthan, India Sumit Gupta - Department of Mathematics, Jagan Nath Gupta Inst. of Engg. & Tech., Jaipur, Rajasthan, India.

### Abstract

In this paper, we obtain the analytical solutions of linear and non-linear space-time fractional reaction-diffusion equations on a finite domain by the application of homotopy perturbation transform method (HPTM). The HPTM is a combined form of the Laplace transform method with the homotopy perturbation method. Some examples are also given. Numerical results show that the HPTM is easy to implement and accurate when applied to linear and non-linear space-time fractional reaction-diffusion equations.

### Share and Cite

##### ISRP Style

Jagdev Singh, Devendra Kumar, Sushila, Sumit Gupta, Application of Homotopy Perturbation Transform Method to Linear and Non-linear Space-time Fractional Reaction-diffusion Equations , Journal of Mathematics and Computer Science, 5 (2012), no. 1, 40-52

##### AMA Style

Singh Jagdev, Kumar Devendra, Sushila, Gupta Sumit, Application of Homotopy Perturbation Transform Method to Linear and Non-linear Space-time Fractional Reaction-diffusion Equations . J Math Comput SCI-JM. (2012); 5(1):40-52

##### Chicago/Turabian Style

Singh, Jagdev, Kumar, Devendra, Sushila,, Gupta, Sumit. "Application of Homotopy Perturbation Transform Method to Linear and Non-linear Space-time Fractional Reaction-diffusion Equations ." Journal of Mathematics and Computer Science, 5, no. 1 (2012): 40-52

### Keywords

• Homotopy perturbation transform method
• Laplace transform
• fractional reaction-diffusion equation
• Caputo time-fractional derivative
• Caputo space-fractional derivative.

•  35K57
•  35B20
•  65H20

### References

• [1] B. Baeumer, M. Kovacs, M. Meerschaert, Numerical solution for fractional reaction- diffusion equation, Computers and Mathematics with Applications, 55 (2008), 2212–2226.

• [2] M. Bar, N. Gottschalk, M. Eiswirth, G. Ertl, Spiral waves in a surface reaction: model calculations , Journal of Chemical Physics, 100 (1994), 1202–1214.

• [3] D. A. Benson, S. Wheatcraft, M. M. Meerschaert, Application of a fractional advection dispersion equation, Water Resource Research, 36 (2000), 1403–1412.

• [4] M. A. Burke, P. K. Miani, J. D. Murray, Suicide substrate reaction-diffusion equations varying the source, IMA Journal Mathematical Applications in Medicine and Biology, 10 (1993), 97–114.

• [5] M. Caputo, Elasticitá e Dissipazione, Zani-chelli, Bologna (1969)

• [6] L. Cong-Xin, C. Xiao-Fang, W. Peng-Ye, W. Wei-Chi , Effects of CRAC channel on spatiotemporal $Ca^{+2}$ patterns in T cells, Chinese Physics Letters, 27 (2010), 028701.

• [7] S. Das, Solution of Fractional Vibration Equation by the Variational Iteration Method and Modified Decomposition Method, International Journal of Nonlinear Science and Numerical Simulation, 9 (2008), 361–366.

• [8] R. Erban, S. J. Chapman, Stochastic modeling of reaction diffusion processes: algorithms for bimolecular reactions, Physical Biology, 6 (2009), 046001.

• [9] Z. Z. Ganji, D. D Ganji, H. Jafari, Application of the homotopy perturbation Method to coupled system of partial differential equations with time fractional derivatives, Topological Methods in Nonlinear Analysis, 31 (2008), 341–348.

• [10] A. Ghorbani, J. Saberi-Nadjafi, He's homotopy perturbation method for calculating adomain polynomials, International Journal of Nonlinear Sciences and Numerical Simulation, 8 (2007), 229–232.

• [11] A. Ghorbani, Beyond Adomain's polynomials: He's polynomials, Chaos Solitons, Fractals, 39 (2009), 1486–1492.

• [12] R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional calculus and continuous time finance, III: The diffusion limit. Mathematical finance. Konstanz, 2000. Trends Math Birkhuser Basel, (2001), 171–180.

• [13] V. Grafiychuk, B. Datsko, S. V. Meleshko, Mathematical modeling of pattern formation in sub- and superdiffusive reaction diffusion systems, arxiv: nlin.AO/06110005 v3, (2006),

• [14] V. Grafiychuk, B. Datsko, S. V. Meleshko, Nonlinear oscillations and stability domains in fractional reaction-diffusion systems , arxiv: nlin.PS/0702013 v1, (2007),

• [15] M. J. Grimson, G. C. Barker, A continuum model for the growth of bacterial colonies on a surface, Journal of Physics A: Mathematical General, 26 (1993), 5645–5654.

• [16] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng., 167 (1998), 57–68.

• [17] J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178 (1999), 257–262.

• [18] B. I. Henry, S. L. Wearne, Fractional reaction-diffusion, Physica A, 276 (2000), 448–455.

• [19] M. Inc, He’s homotopy perturbation method for Solving Korteweg-de Veries Burgers Equation with Initial Condition, Numerical Methods for Partial Differential Equations, 26(5) (2009), 1224–1235.

• [20] H. Jafari, S. Momani, Solving fractional diffusion and wave equations by modified homotopy perturbation method, Physics Letters A, 370 (2007), 388–396.

• [21] N. A. Khan, N. Khan, A. Ara, M. Jamil, Approximate analytical solutions of fractional reaction-diffusion equations, Journal of King Saud University (Science), 24(2) (2012), 111–118.

• [22] Y. Khan, Q. Wu, Homotopy perturbation transform method for nonlinear equations using He's polynomials, Computer and Mathematics with Applications, 61(6) (2011), 1963–1967.

• [23] M. Khan, M. A. Gondal, S. Kumar, A new analytical approach to solve exponential stretching sheet problem in fluid mechanics by variational iterative Padé method, The Journal of Mathematics and Computer Science, 3 (2) (2011), 135–144.

• [24] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

• [25] F. Mainardi, M. Raberto, R. Gorenflo, R. Scalas, Fractional calculus and continuous-time finance. II: The waiting-time distribution, Physica A, 287 (2000), 468–481.

• [26] M. Matinfar, M. Ghanbari, Solution of systems of integral-differential equations by variational iteration method, The Journal of Mathematics and Computer Science, 1(1) (2010), 46–57.

• [27] K. S. Miller, B. Ross , An introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993)

• [28] S. T. Mohyud-Din, M. A. Noor, K. I. Noor , Travelling wave solutions of seventh- order generalized KdV equations using He’s polynomials, Int. J. Nonlin. Sci. Num. Sim., 10 (1) (2009), 227–233.

• [29] S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A, 365 (2007), 345–350.

• [30] S. Momani, Z. Odibat, I. Hasshim, Algorithms for non-linear fractional partial differential equations: A selection of numerical methods, Topological Methods in Nonlinear Analysis, 31 (2008), 211–226.

• [31] S. C. Muller, T. Plesser, B. Hess, Two-dimensional spectrophotometry of spiral wave propagation in the Belousov–Zhabotinskii reaction. I: Experiments and digital data representation, Physica D, 24 (1987), 71–78.

• [32] J. D. Murray, Lectures on Non-Linear Differential Equation Models in Biology, Clarenden, Oxford (1977)

• [33] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Science and Numerical Simulation, 7 (2006), 27–34.

• [34] Z. Odibat, S. Momani, Application of variational iteration and homotopy perturbation methods to fractional evolution equations, Topological Methods in Nonlinear Analysis, 31 (2008), 227–234.

• [35] M. I. A. Othman, A. M. S. Mahdy, R. M. Farouk, Numerical solution of 12th order boundary value problems by using homotopy perturbation method, The Journal of Mathematics and Computer Science, 1(1) (2010), 14–27.

• [36] M. I. A. Othman, A. M. S. Mahdy, Differential transformation method and variation iteration method for Cauchy reaction-diffusion problems, The Journal of Mathematics and Computer Science, 1(2) (2010), 61–75.

• [37] M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high frequency financial data: an empirical study, Physica A, 314 (2002), 749–755.

• [38] K. Seki, M. Wojcik, Tachiya, Fractional reaction- diffusion equation , Journal of Chemical Physics, 119 (2003), 2165–2174.

• [39] B. M. Slepchenko, J. C. Schaff, Y. S. Choi, Numerical approach to fast reactions in reaction-diffusion systems: application to buffered calcium waves in bistable models, Journal of Computational Physics, 162 (2000), 186–189.

• [40] C. Vidal, A. Pascault, Non-equilibrium Dynamics in Chemical Systems, Wiley, New York (1986)

• [41] A. T. Winfree, Spiral waves of chemical activity, Science, 175 (1972), 634–642.

• [42] A. Yildirim, Solution of BVPs for Fourth- Order Integro-Differential Equations by using Homotopy Perturbation Method, Computers & Mathematics with Applications, 56 (2008), 3175–3180.

• [43] A. Yildirim, An Algorithm for Solving the Fractional Nonlinear Schröndinger Equation by Means of the Homotopy Perturbation Method, International Journal of Nonlinear Science and Numerical Simulation, 10 (2009), 445–451.

• [44] A. Yildirim, He's homotopy perturbation method for solving the space- and time- fractional telegraph equations, International Journal of Computer Mathematics, 87(13) (2010), 2998–3006.

• [45] A. Yildirim, S. A. Sezer, Analytical solution of linear and non-linear space-time fractional reaction-diffusion equations, International Journal of Chemical Reactor Engineering, 8 (2010), 1–21.

• [46] Q. Yu, F. Liu, V. Anh, I. Turner, Solving linear and non-linear Space-time fractional reaction-diffusion equations by the Adomian decomposition method, Int. J. Numer. Meth. Engg., 74 (2008), 138–158.