Generalized fractional Stockwell transform and its associated pseudo-differential operator

The novel fractional Stockwell transform is extended to the suitably defined Schwartz space and the generalized fractional Stockwell transform is introduced. The pseudo-differential operator associated with the fractional Stockwell transform is defined and it is proved to be continuous on the Schwartz space. Examples are illustrated for the generalized fractional Stockwell transform. The generalized convection-diffusion equation is solved using the fractional Stockwell transform.