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On holomorphic solution for space- and time-fractional telegraph equations in complex domain. (English) Zbl 1250.35176

Summary: We consider some classes of space- and time-fractional telegraph equations in the complex domain in the sense of the Riemann-Liouville fractional operators for time and the Srivastava-Owa fractional operators for space. The existence and uniqueness of holomorphic solutions is established. We illustrate our theoretical results by examples.

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
45K05 Integro-partial differential equations
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[1] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA, 2000. · Zbl 0998.26002 · doi:10.1142/9789812817747
[3] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003.
[4] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[5] J. Sabatier, O. P. Agrawal, and J. A. Machado, Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. · Zbl 1116.00014 · doi:10.1007/978-1-4020-6042-7
[6] D. Baleanu, B. Guvenc Ziya, and J. A. Tenreiro, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, NY, USA, 2010. · Zbl 1196.65021 · doi:10.1007/978-90-481-3293-5
[7] R. W. Ibrahim, “On holomorphic solutions for nonlinear singular fractional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1084-1090, 2011. · Zbl 1233.35200 · doi:10.1016/j.camwa.2011.04.037
[8] 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 · doi:10.1016/j.jmaa.2011.03.001
[9] R. W. Ibrahim, “On generalized Srivastava-Owa fractional operators in the unit disk,” Advances in Difference Equations, vol. 2011, article 55, 2011. · Zbl 1273.35295
[10] R. W. Ibrahim, “Ulam stability for fractional differential equation in complex domain,” Abstract and Applied Analysis, vol. 2012, Article ID 649517, 8 pages, 2012. · Zbl 1239.34106 · doi:10.1155/2012/649517
[11] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, Mass, USA, 1997. · Zbl 0892.35001
[12] A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus, London, UK, 1993.
[13] S. Momani, “Analytic and approximate solutions of the space- and time-fractional telegraph equations,” Applied Mathematics and Computation, vol. 170, no. 2, pp. 1126-1134, 2005. · Zbl 1103.65335 · doi:10.1016/j.amc.2005.01.009
[14] J. Biazar, H. Ebrahimi, and Z. Ayati, “An approximation to the solution of telegraph equation by variational iteration method,” Numerical Methods for Partial Differential Equations, vol. 25, no. 4, pp. 797-801, 2009. · Zbl 1169.65335 · doi:10.1002/num.20373
[15] R. Yulita 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 · doi:10.1016/j.nonrwa.2008.02.026
[16] A. Sevimlican, “An approximation to solution of space and time fractional telegraph equations by He’s variational iteration method,” Mathematical Problems in Engineering, vol. 2010, Article ID 290631, 10 pages, 2010. · Zbl 1191.65137 · doi:10.1155/2010/290631
[17] 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 · doi:10.1016/j.camwa.2006.12.037
[18] G. M. d. Araújo, R. B. Gúzman, and S. B. d. Menezes, “Periodic solutions for nonlinear telegraph equation via elliptic regularization,” Computational & Applied Mathematics, vol. 28, no. 2, pp. 135-155, 2009. · Zbl 1180.35043 · doi:10.1590/S1807-03022009000200001
[19] M. A. E. Herzallah and D. Baleanu, “On abstract fractional order telegraph equation,” Journal of Computational and Nonlinear Dynamics, vol. 5, no. 2, pp. 1-5, 2010. · Zbl 1209.34003
[20] H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1989. · Zbl 0683.00012
[21] E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley-Interscience, New York, NY, USA, 1976. · Zbl 0343.34007
[22] B. Baeumer and M. M. Meerschaert, “Stochastic solutions for fractional Cauchy problems,” Fractional Calculus & Applied Analysis, vol. 4, no. 4, pp. 481-500, 2001. · Zbl 1057.35102
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