<p>Wavelets and the wavelet transform have gained significant popularity in recent years and are widely used in the numerical solution of differential and integral equations. Fractional calculus is another discipline currently in vogue due to its applications in modeling certain systems and processes. In this paper, the continuous wavelet transform with the Poisson wavelet is applied to the Liouville and Caputo fractional calculus operators, resulting in new theorems. These results are then used to obtain the analytical solution of the inhomogeneous Bagley–Torvik equation, defined in the Caputo sense. Additionally, theorems are developed to represent the Liouville fractional differential operator in orthogonal wavelet bases by utilizing the theory of multiresolution analysis and properties of the Fourier transform. Examples of some wavelets suitable for this purpose are provided, along with an approximation of the fractional derivatives of a function using this method.</p>