The Method of Φ-Laplace Adomian Decomposition for Φ-Caputo Fractional Bloch Equations

Abstract

This article studies the Bloch equations (BEs), which form a system of macroscopic equations used for the simulation of nuclear magnetization as a function of time, when the relaxation times T<inf>1</inf> and T<inf>2</inf> are given. These equations have been applied to describe nuclear magnetic resonance (NMR), electron spin resonance (ESR), and magnetic resonance imaging (MRI). In this work, we present analytic solutions to the fractional Bloch equations (FBEs). The fractional derivatives in the Bloch equations under consideration are in the sense of Φ-Caputo; we use the Φ-Laplace Adomian decomposition procedure (Φ-LADP) to solve the FBEs. This procedure combines both the Adomian decomposition and Φ-Laplace transform methods. To explain the analytical solutions of the system of Φ-Caputo fractional Bloch equations (Φ-CFBEs) of the order η with known initial conditions, we apply the two-dimensional and three-dimensional phase portraits. We compare these solutions by considering diverse functions in place of Φ(t) and values of 0 < η ≤ 1. Finally, to show the usefulness of our proposed method, we discuss the advantages of the new method compared to the existing methods for solving Caputo FBEs. © 2025 Elsevier B.V., All rights reserved.