In the present paper, we numerically simulate fractional-order model of the Bloch equation by using the Jacobi polynomials. It arises in chemistry, physics and nuclear magnetic resonance (NMR). It also arises in magnetic resonance imaging (MRI) and electron spin resonance (ESR). It is used for purity determination, provided that the molecular weight and structure of the compound is known. It can also be used for structural determination. By the study of NMR, chemists can determine the structure of many compounds. The obtained numerical results are compared and simulated with the known solutions. Accuracy of the proposed method is shown by providing tables for absolute errors and root mean square errors. Different orders of the time-fractional derivatives results are illustrated by using figures.

In this article, we present the Jacobi spectral colocation method to solve the fractional model of Liénard and Duffing equations with the Liouville–Caputo fractional derivative. These equations are the generalization of the spring–mass system equation and describe the oscillating circuit. The main reason for using this technique is high accuracy and low computational cost compared to some other methods. The main solution behaviors of these equations are due to fractional orders, which are explained graphically. The convergence analysis of the proposed method is also provided. A comparison is made between the exact and approximate solutions.