Abstract

In this paper, we introduce the theory of a generalized Fourier transform in order to solve differential equations with a generalized fractional derivative, and we state its main properties. In particular, we obtain the corresponding convolution, inverse and Plancherel formulas, and Hausdorff-Young inequality. We show that this generalized Fourier transform is useful in the study of fractional partial differential equations, by solving the fractional heat equation on the real line.