Factorization à la Dirac applied to the time-fractional telegraph equation

• Coupled time-fractional diffusion Dirac system derived via factorization method • Novel Fourier transform pairs with bivariate Mittag-Leffler and Fox H-functions • Explicit solutions, enabling analysis in both frequency and time-space, are derived • The long and short-term (asymptotic) behaviour of solutions is rigorously analysed • Factorization allows alternative triplets of Pauli matrices, yielding related solutions This paper examines a coupled system of two-term time-fractional diffusion Dirac-type equations. The system is derived by factorizing the multi-dimensional time-fractional telegraph equation with Hilfer fractional derivatives, using the Dirac method and a triplet of Pauli matrices. Solutions are obtained using operational methods provided by the combination of the Fourier transform in the space variable and the Laplace transform in the time variable. Key results include the discovery of novel Fourier transform pairs. These pairs relate specific Fourier kernels of bivariate Mittag-Leffler functions to Fox H-functions of two variables. This allows to obtain explicit solutions of the system in both Fourier-time and space-time domains. The asymptotic behaviour of these solutions is rigorously analysed, and graphical representations are generated. Further, we show that the factorization allows for the use of alternative triplets of Pauli matrices yielding related solutions. The results obtained can be generalised to the case of ψ -Hilfer derivatives.