Extending spectral methods to solve time fractional-order Bloch equations using generalized Laguerre polynomials

Nuclear magnetic resonance (NMR) is a widely utilized physical phenomenon in the fields of chemistry, medicine, and engineering for the examination of complex materials. The fundamental basis of NMR lies in the Bloch equation, which establishes a connection between a macroscopic model of magnetization and the application of radiofrequency, gradient, and static magnetic fields. In cases where materials exhibit simple behavior, the classical first-order dynamics of precession and relaxation are described in the vector form of the Bloch equation. However, when dealing with more diverse experimental scenarios involving materials that are heterogeneous, porous, or composite, there is an opportunity to broaden the applicability of the Bloch equation through fractional-order derivative operators. In this study, we address the time fractional-order Bloch equations by extending the spectral methods approach to find a solution. Our proposed extension is based on the newly introduced generalized fractional-order integral operational matrix of Laguerre polynomials (LPs) in generalized Riemann–Liouville sense. Comparative analysis is also presented to highlight the efficiency of the method.