On the Existence, Uniqueness and a Numerical Approach to the Solution of Fractional Cauchy–Euler Equation

In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order $0<\alpha <2$. The problem also constitutes a class of examples of the Cauchy problem of the Bagley–Torvik equation with variable coefficients. For proving the existence and uniqueness of the solution of the given problem, the contraction mapping principle is utilized. Furthermore, a numerical method and an algorithm are developed for obtaining the approximate solution. Also, convergence analyses are studied, and simulations on some test problems are given. It is shown that the proposed method and the algorithm are easy to implement on a computer and efficient in computational time and storage.