On fractional partial differential equations related to quantum mechanics

Abstract

In this paper, we investigate the solutions of generalized fractional partial differential equations involving the Caputo time-fractional derivative and the Liouville space-fractional derivatives. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms. Several special cases as solutions of one-dimensional non-homogeneous fractional equations occurring in quantum mechanics are presented in the concluding section. The results given earlier by Debnath (2003 Fract. Calc. Appl. Anal. 6 119?55), Saxena et al (2010 Appl. Math. Comput. 216 1412?7) and Pagnini and Mainardi (2010 J. Comput. Appl. Math. 233 1590?5) follow as special cases of our findings.