In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with noninteger exponents. We apply the method to equations resembling generalizations of the heat equations and discuss the possibility of extending the procedure to the relativistic Schr\"odinger and Dirac equations.

A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view and the relevant impact on the study of the properties of special functions is carefully discussed.

We use the split-operator technique (SOT) to solve evolution equations of the Liouville type. The method we propose is based on an iterative application of the evolution operator, associated with the equation under study, on the initial distribution. The SOT approximation of the evolution operator leads to analytical expressions that can be easily programmed. We discuss the validity of the method, solving the Liouville equation governing the longitudinal phase-space dynamics of an e beam undergoing free-electron-laser interaction and the phase-space evolution of a quartic anharmonic oscillator.

The Lamb-Bateman integral equation was introduced to study the solitary wave diffraction and its solution was written in terms of an integral transform. We prove that it is essentially the Abel integral equation and its solution can be obtained using the formalism of fractional calculus.