Complex transforms for systems of fractional differential equations. (English) Zbl 1257.35194

Ibrahim, Rabha W.

Abstr. Appl. Anal. 2012, Article ID 814759, 15 p. (2012).

Summary: We provide a complex transform that maps the complex fractional differential equation into a system of fractional differential equations. The homogeneous and nonhomogeneous cases for equivalence equations are discussed and also nonequivalence equations are studied. Moreover, the existence and uniqueness of solutions are established and applications are illustrated.

MSC:

35R11	Fractional partial differential equations
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

References:

[1]	M. Cassol, S. Wortmann, and U. Rizza, “Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace transform techniques,” Environmental Modelling & Software, vol. 24, no. 1, pp. 144-151, 2009.
[2]	C. Delong, “A novel fingerprint encryption algorithm based on chaotic system and fractional Fourier transform,” International Conference on Machine Vision and Human-machine Interface, pp. 186-171, 2010.
[3]	A. Bekir, “New exact travelling wave solutions of some complex nonlinear equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1069-1077, 2009. · Zbl 1221.35238
[4]	P. R. Gordoa, A. Pickering, and Z. N. Zhu, “Bäcklund transformations for a matrix second Painlevé equation,” Physics Letters A, vol. 374, no. 34, pp. 3422-3424, 2010. · Zbl 1238.34155
[5]	S. Sivasubramanian, M. Darus, and R. W. Ibrahim, “On the starlikeness of certain class of analytic functions,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 112-118, 2011. · Zbl 1225.30016
[6]	H. M. Srivastava, M. Darus, and R. W. Ibrahim, “Classes of analytic functions with fractional powers defined by means of a certain linear operator,” Integral Transforms and Special Functions, vol. 22, no. 1, pp. 17-28, 2011. · Zbl 1207.30031
[7]	Z. Li and J. He, “Application of the fractional complex transform to fractional differential equations,” Nonlinear Science Letters A, vol. 2, no. 3, pp. 121-126, 2011.
[8]	R. Y. Molliq and B. Batiha, “Approximate analytic solutions of fractional Zakharov-Kuznetsov equations by fractional complex transform,” International Journal of Engineering and Technology, vol. 1, no. 1, pp. 1-13, 2012.
[9]	W. G. Glöckle and T. F. Nonnenmacher, “Fox function representation of non-Debye relaxation processes,” Journal of Statistical Physics, vol. 71, no. 3-4, pp. 741-757, 1993. · Zbl 0945.82559
[10]	R. W. Ibrahim, “Existence and uniqueness of holomorphic solutions for fractional Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 380, no. 1, pp. 232-240, 2011. · Zbl 1214.30027
[11]	R. W. Ibrahim, “On holomorphic solution for space- and time-fractional telegraph equations in complex domain,” Journal of Function Spaces and Applications, vol. 2012, Article ID 703681, 10 pages, 2012. · Zbl 1250.35176
[12]	R. W. Ibrahim, “Approximate solutions for fractional differential equation in the unit disk,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 64, pp. 1-11, 2011. · Zbl 1340.34018
[13]	S. Momani and Z. Odibat, “Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 910-919, 2007. · Zbl 1141.65398
[14]	R. Y. Molliq, M. S. M. Noorani, and I. Hashim, “Variational iteration method for fractional heat- and wave-like equations,” Nonlinear Analysis. Real World Applications, vol. 10, no. 3, pp. 1854-1869, 2009. · Zbl 1172.35302
[15]	K. Sayevand, A. Golbabai, and A. Yildirim, “Analysis of differetial equations of fractional order,” Applied Mathematical Modelling, vol. 36, pp. 4356-4364, 2012. · Zbl 1252.34007
[16]	H. Jafari and M. A. Firoozjaee, “Homotopy analysis method for solving KdV equations,” Surveys in Mathematics and its Applications, vol. 5, pp. 89-98, 2010. · Zbl 1198.65255
[17]	R. Metzler, E. Barkai, and J. Klafter, “Deriving fractional Fokker-Planck equations from a generalised master equation,” Europhysics Letters, vol. 46, no. 4, pp. 431-436, 1999.
[18]	E. K. Lenzi, L. R. Evangelista, and G. Barbero, “Fractional diffusion equation and impedance spectroscopy of electrolytic cells,” The Journal of Physical Chemistry B, vol. 113, no. 33, pp. 11371-11374, 2009.
[19]	J. R. Macdonald, L. R. Evangelista, E. K. Lenzi, and G. Barbero, “Comparison of impedance spectroscopy expressions and responses of alternate anomalous poisson-nernst-planck diffusion equations for finite-length situations,” The Journal of Physical Chemistry C, vol. 115, no. 15, pp. 7648-7655, 2011.
[20]	P. A. Santoro, J. L. de Paula, E. K. Lenzi, and L. R. Evangelista, “Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell,” Journal of Chemical Physics, vol. 135, Article ID 114704, 5 pages, 2011.
[21]	J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140-1153, 2011. · Zbl 1221.26002
[22]	R. Hilfer, R. Metzler, A. Blumen, and J. Klafter, “Preface,” Chemical Physics, vol. 284, no. 1-2, pp. 1-2, 2002.
[23]	J. Klafter, S. C. Lim, and R. Metzler, Eds., Fractional Dynamics: Recent Advances, World Scientific, Publishing Company, Singapore, 2011. · Zbl 1238.93005
[24]	R. Metzler and J. Klafter, “From a generalized Chapman-Kolmogorov equation to the fractional Klein-Kramers equation,” The Journal of Physical Chemistry B, vol. 104, pp. 3851-3857, 2000.
[25]	I. Podlubny, Fractional Differential Equations, Academic Press, London, UK, 1999. · Zbl 0924.34008
[26]	A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[27]	H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, John Wiley & Sons, New York, NY, USA, 1989. · Zbl 0683.00012
[28]	K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010. · Zbl 1215.34001
[29]	A. N. Kochubeĭ, “The Cauchy problem for evolution equations of fractional order,” Differential Equations, vol. 25, pp. 967-974, 1989. · Zbl 0696.34047
[30]	A. N. Kochubeĭ, “Diffusion of fractional order,” Differential Equations, vol. 26, no. 4, pp. 485-492, 1990. · Zbl 0729.35064
[31]	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. · Zbl 0984.82032
[32]	G. M. Zaslavsky, “Fractional kinetic equation for Hamiltonian chaos,” Physica D, vol. 76, no. 1-3, pp. 110-122, 1994. · Zbl 1194.37163
[33]	F. Mainardi, G. Pagnini, and R. Gorenflo, “Some aspects of fractional diffusion equations of single and distributed order,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 295-305, 2007. · Zbl 1122.26004
[34]	F. Mainardi, A. Mura, G. Pagnini, and R. Gorenflo, Sub-diffusion equations of fractional order and their fundamental solutions, Invited lecture by F. Mainardi at the 373. WEHeraeus- Seminar on Anomalous Transport: Experimental Results and Theoretical Challenges, Physikzentrum Bad-Honnef (Germany), July 2006.
[35]	T. Sandev, R. Metzler, and Z. Tomovski, “Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative,” Journal of Physics A, vol. 44, no. 25, pp. 1-21, 2011. · Zbl 1301.82042
[36]	R. W. Ibrahim, “Solutions of fractional diffusion problems,” Electronic Journal of Differential Equations, vol. 147, pp. 1-11, 2010. · Zbl 1205.35333
[37]	M. Merdan, “Solutions of time-fractional reaction-diffusion equation with modified Riemann-Liouville derivative,” International Journal of Physical Sciences, vol. 7, pp. 2317-2326, 2012.
[38]	J. Hu and W.-P. Li, Theory of Ordinary Differential Equations, The Hong Kong University of Science and Technology, 2005.
[39]	T. Wei and Z. Zhang, “Reconstruction of a time-dependent so urce term in a time-fractional diffusion equation,” Engineering Analysis with Boundary Elements, vol. 37, no. 1, pp. 23-31, 2013. · Zbl 1351.35267