Natural Transform for Solving Fractional Models

Ahmed Safwat Abdel-Rady¹, Saad Zagloul Rida¹, Anas Ahmed Mohamed Arafa², Hamdy Ragab Abedl-Rahim^1*

¹Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt.
²Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said, Egypt.

DOI: 10.4236/jamp.2015.312188                

Abstract

In this paper, we present a novel technique to obtain approximate analytical solution of fractional physical models. The new technique is a combination of a domain decomposition method and natural transform method called a domain decomposition natural transform method (ADNTM). The fractional derivatives are considered in Caputo sense. To illustrate the power and reliability of the method some applications are provided.

Keywords

Fractional Calculus, Natural Transform, A Domain Decomposition Natural Transform Method (ADNTM), Fokker-Planck Equation, Schrödinger Equation, Kelin-Gorden Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Young, G.O. (1995) Definition of Physical Consistent Damping Laws with Fractional Derivatives. Zeitschrift fur Angewandte Mathematik und Mechanik, 75, 623-635. http://dx.doi.org/10.1002/zamm.19950750820
[2]	He, J.H. (1999) Some Applications of Nonlinear Fractional Differential Equations and Their Approximations. Bulletin of Science and Technology, 15, 86-90.
[3]	He, J.H. (1998) Approximate Analytic Solution for Seepage Flow with Fractional Derivatives in Porous Media. Computer Methods in Applied Mechanics and Engineering, 167, 57-68. http://dx.doi.org/10.1016/S0045-7825(98)00108-X
[4]	Hilfer, R. (2000) Applications of Fractional Calculus in Physics. World Scientific Publishing Company, Singapore, 87-130. http://dx.doi.org/10.1142/9789812817747_0002
[5]	Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.
[6]	Mainardi, F., Luchko, Y. and Pagnini, G. (2001) The Fundamental Solution of the Space-Time Fractional Diffusion Equation. Fractional Calculus and Applied Analysis, 4, 153-192.
[7]	Debnath, L. (2003) Fractional Integrals and Fractional Differential Equations in Fluid Mechanics. Fractional Calculus and Applied Analysis, 6, 119-155.
[8]	Caputo, M. (1969) Elasticita e Dissipazione. Zani-Chelli, Bologna.
[9]	Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.
[10]	Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York.
[11]	Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
[12]	Wazwaz, A.M. (2009) Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Beijing and Springer-Verlag, Berlin Heidelberg.
[13]	Silambarasan, R. and Belgacem, F.B.M. (2012) Theory of Natural Transform. Mathematics in Engineering, Science and Aerospace (MESA), 3, 99-124.
[14]	Baskonus, H.M., Bulut, H. and Pandir, Y. (2014) The Natural Transform Decomposition Method for Linear. Mathematics in Engineering, Science and Aerospace (MESA), 5, 111-126.
[15]	Loonker, D. and Banerji, P.K. (2013) Solution of Fractional Ordinary Differential Equations by Natural Transform. International Journal of Mathematical Engineering and Science, 2, 2277-6982.
[16]	Risken, H. (1996) The Fokker-Planck Equation: Methods and Applications. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-642-61544-3_4
[17]	Chandler, D. (1987) Introduction to Modern Statistical Mechanics. Oxford University Press, New York.
[18]	Haken, H. (2004) Synergetics: Introduction and Advanced Topics. Springer, Berlin. http://dx.doi.org/10.1007/978-3-662-10184-1
[19]	Reif, F. (1965) Fundamentals of Statistical and Thermal Physics. McGraw-Hill Book Company, New York.
[20]	Terletskii, Y.P. (1971) Statistical Physics. North-Holland Publishing Company, Amsterdam.
[21]	Franck, T.D. (2004) Stochastic Feedback, Nonlinear Families of Markov Process and Nonlinear Fokker-Planck Equation. Physical A, 331, 391-408. http://dx.doi.org/10.1016/j.physa.2003.09.056
[22]	Tatari, M., Dehghan, M. and Razzaghi, M. (2007) Application of Adomain Decomposition Method for the Fokker-Planck Equation. Mathematical and Computer Modelling, 45, 639-650. http://dx.doi.org/10.1016/j.mcm.2006.07.010
[23]	Sadhigi, A., Ganji, D.D. and Sabzehmeidavi, Y. (2007) A Study on Fokker-Planck Equation by Variational Iteration Method. International Journal of Nonlinear Sciences, 4, 92-102.
[24]	Biazar, J., Hosseini, K. and Gholamin, P. (2008) Homotopy Perturbation Method Fokker-Planck Equation. International Mathematical Forum, 19, 945-954.
[25]	Kanth, A.S.V.R. and Aruna, K. (2009) Two-Dimensional Differential Transform Method for Solving Linear and Non-Linear Schrödinger Equation. Chaos, Solution and Fractals, 41, 2277-2281.
[26]	Ayati, Z., Biazar, J. and Ebrahimi, S. (2014) A New Homotopy Perturbation Method for Solving Linear and Nonlinear Schrödinger Equations. Journal of Interpolation and Approximation in Scientific Computing, 2014, 1-8.
[27]	Wazwaz, A.M. (2002) Partial Differential Equations: Methods and Applications. Balkema, Leiden.
[28]	Whitham, G.B. (1976) Linear and Nonlinear Waves. John Wiley, New York.
[29]	Mohyud-Din, S.T. and Yildirim, A. (2010) Variational Iteration Method for Solving Klein-Gordon Equations. Journal of Applied Mathematics, Statistics and Informatics, 6, 99-106.
[30]	Singh, J., Kumar, D. and Rathore, S. (2012) Application of Homotopy Perturbation Transform Method for Solving Linear and Nonlinear Klein-Gordon Equations. Information and Computation, 7, 131-139.