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This paper is concerned with the asymptotic stability of linear fractional-order neutral delay differential–algebraic systems described by the Caputo–Fabrizio (CF) fractional derivative. A novel characteristic equation is derived using the Laplace transform. Based on an algebraic approach, stability criteria are established. The effect of the index on such criteria is analyzed to ensure the asymptotic stability of the system. It is shown that asymptotic stability is ensured for the index-1 problems provided that a stability criterion holds for any delay parameter. Also, asymptotic stability is still valid for higher-index problems under the conditions that the system matrices have common eigenvectors and each pair of such matrices is simultaneously triangularizable so that a stability criterion holds for any delay parameter. An example is provided to demonstrate the effectiveness and applicability of the theoretical results.

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In this paper, the approximate solution of a reaction-diffusion system with fast reversible reaction is obtained by using Adomian decomposition method (ADM) and variational iteration method (VIM) which are two powerful methods that were recently developed. The VIM requires the evaluation of the Lagrange multiplier, whereas ADM requires the evaluation of the Adomian polynomials. The behavior of the approximate solutions and the effects of different values of are shown graphically.