Analytical solutions for the fractional Klein-Gordon equation

Document Type : Research Paper


University of Tabriz


In this paper, we solve a inhomogeneous fractional Klein-Gordon equation by the method of separating variables. We apply the method for three boundary conditions, contain Dirichlet, Neumann, and Robin boundary conditions, and solve some examples to illustrate the effectiveness of the method.


[1] Y. Chalco-Cano, J. Nieto, A. Ouahab and H. Roman-Flores, Solution set for fractional differential equations with Riemann-Liouville derivative. Fractional Calculus and Applied Analysis, 16(3) (2013) 682-694.
[2] Y. F. Cheng and T. Q. Dai, Exact solutions of the Klein-Gordon equation with a ring-shaped modified kratzer potential. Chinese J. Phys., 45 (2007) 480-487.
[3] S. Dong, S. H. Dong, H. Bahlouli and V. B. Bezzerra, An algebraic method for the analytical solutions of the Klein-Gordon equation for any angular momentum for some diatomic potentials. Int. J. Mod. Phys E, 20 (2011) 55-68.
[4] K. A. Gepreel and M. S. Mohamed, Analytical approximate solution for nonlinear space-time fractional Klein-Gordon equation. Chinese physics B, 22(1) (2013) 010201.
[5] A. K. Golmankhaneh, A. K. Golmankhaneh, D. Baleanu, On nonlinear fractional Klein-Gordon equation. Signal Processing, 91 (2011) 446-451.
[6] I. Podlubny, Fractional differential equations. Academic Press, San Diego, Calif, USA, 1999.
[7] W. Liu and K. Chen, The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations. Pramana J. Phys, 81(2013) 377-384.
[8] Y. Luchko and R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnamica, 24 (1999) 207-233.
[9] S. Samko, A. Kilbas and O. Marichev, Fractional integrals and derivatives: Theory and applications. Gordon and Breach Sci. Publishers, Yverdon, Switzerland, 1993.
[10] V. Uchaikin, Method of fractional derivatives. Artishok-Press, Ulyanovsk, Russia, 2008.
[11] Q. F. Wang and D. Cheng, Numerical solution of damped nonlinear Klein-Gordon equations using variational method and finite element approach. Appl. Math. Comput., 162 (2005) 381-401.
[12] E. Yusufoglu, Variational iteration method for construction of some compact and noncompact structures of Klein-Gordon equations. International Journal of Nonlinear Sciences and Numerical Simulation, 8 (2007) 153-158.
[13] X. J. Yang, H. M. Srivastava, J. -H. He and D. Baleanu, Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Physics Letters A, 377(28) (2013) 1696-1700.
[14] L. Zhao and W. Deng, Jacobian-predictor-corrector approach for fractional differential equations. Advances in Computational Mathematics, 40(1) (2014) 137-165.
[15] M. Znojil, Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation. J.Phys. A: Math. Gen., 37 (2004) 9557-9571.