The asymptotic solutions of two-term linear fractional differential equations via Laplace transform

Abstract

In this paper, the asymptotic solutions about the origin and infinity are formulated via Laplace transform for a two-term linear Caputo fractional differential equation. The asymptotic expansion about the origin describes the complete singular information of the solution, which is also a good approximation of the solution near the origin. The expansion at infinity exhibits the structure of the solution, as well as the stable or unstable property of the solution, which becomes more accurate as the variable tends to larger. Based on the asymptotic solution about the origin, a singularity-separation Legendre collocation method is designed to validate the methods in this paper. Numerical examples show the easy calculation and high accuracy of the truncated expansions and their Padé approximations when the variable is suitably small or sufficiently large. As an application, the method is used to solve the initial value problem of the Bagley–Torvik equation, and the oscillatory property of the solution is displayed.