Special Affine Fourier Transform on Tempered Distribution and Its Application

The main aim of this work is to develop a theoretical framework for generalized pseudo-differential operators involving the special affine Fourier transform (SAFT), associated with a symbol delta(mu,eta)$$ \delta \left(\mu, \eta \right) $$. Some important properties of the SAFT are established, and it is proved that the product of two generalized pseudo-differential operators is shown to be a generalized pseudo-differential operator. Further, we explore the practical applications of the SAFT in solving generalized partial differential equations, such as the generalized telegraph and wave equations, providing closed-form solutions. Furthermore, graphical visualizations for these solutions are illustrated via MATLAB R2023b.