NUMERICAL STUDY FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS VIA THE LAPLACE–ADOMIAN DECOMPOSITION METHOD
Keywords:
Fractional differential equations, Caputo derivative, Laplace transform, Adomian decomposition, nonlinear systemsAbstract
Laplace-Adomian Decomposition Method (LADM) is an effective semi-analytical system of solving systems of fractional differential equations (SFDEs) and is the subject of investigation in this paper. Fractional models offer a potent system of describing memory and hereditary characteristics of the dynamical systems, but precise answers are certainly unattainable unless the system is of simple linear nature. The given method is based on the combination of Laplace transform, transforming the fractional operator into an algebraic expression, and Adomian decomposition, which is the systematic expansion of nonlinear terms into a convergent series. The hybrid approach does not discretize, perturb or linearize, and yields fast convergent approximations.
Theoretical bases are laid by the application of the method to a generic system of Caputo-type SFDEs. Convergence conditions are obtained via contraction mapping principles and practical error estimates are obtained by examining the residual of truncated series solutions. This is then followed by an explicit algorithm, which describes each of the steps, starting with the initialisation to the iterative construction of terms used as solutions.
The efficacy of LADM is demonstrated by both numerical studies of the linear and nonlinear systems. The method essentially reproduces exact solutions in case of linear forced systems by Laplace inversion. In nonlinear systems, including the case of the fractional predator-prey, the series converges rapidly using only a small number of terms, but also distinctly displays the effects of the fractional order α on the behavior of a system. Findings confirm rapid decay of residual norms as truncation order increases, and that N = 6 is usually adequate to find accurate solutions.
In general, it can be concluded that LADM is a computational method that is simple, accurate and efficient, hence, a promising tool to be deployed in a variety of applications of fractional modeling.
