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On fractional partial differential equations related to quantum mechanics. (English) Zbl 1211.26009

The authors investigate solutions of generalized fractional partial differential equations involving the Caputo time-fractional derivative and the Liouville space-fractional derivatives. These may be regarded as extended one dimensional space-time fractional diffusion, wave and Schrödinger equations. The initial conditions associated with the Laplace transform of Caputo derivative are more practical and convenient. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms in a closed and computational form in terms of the Green functions involving Mittag-Leffler functions. Several special cases as solutions of one-dimensional non-homogeneous fractional equations occurring in fluid and quantum mechanics (diffusion, wave and Schrödinger equations) are presented in the concluding section. The results given earlier by L. Debnath [Fract. Calc. Appl. Anal. 6, No. 2, 119–155 (2003; Zbl 1076.35095)], R. K. Saxena et al. [Appl. Math. Comput. 216, No. 5, 1412–1417 (2010; Zbl 1190.65157)] and G. Pagnini and F. Mainardi [J. Comput. Appl. Math. 233, No. 6, 1590–1595 (2010; Zbl 1179.82008)] follow as special cases of the findings here.

MSC:

26A33 Fractional derivatives and integrals
44A10 Laplace transform
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
35J10 Schrödinger operator, Schrödinger equation
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