Abstract
Fractional differential problems are widely used in applied sciences. For this reason, there is a great interest in the construction of efficient numerical methods to approximate their solution. The aim of this paper is to describe in detail a collocation method suitable to approximate the solution of dynamical systems with time derivative of fractional order. We will highlight all the steps necessary to implement the corresponding algorithm and we will use it to solve some test problems. Two Mathematica Notebooks that can be used to solve these test problems are provided.
Highlights
Fractional differential problems are well established models to describe a great variety of real-world phenomena [1,2,3,4]
The aim of this paper is to describe in detail a collocation method suitable to approximate the solution of dynamical systems with time derivative of fractional order
The aim of this paper is to describe in detail how to apply the fractional collocation method introduced in [8] for the numerical solution of nonlinear dynamical systems with time derivative of fractional order
Summary
Fractional differential problems are well established models to describe a great variety of real-world phenomena [1,2,3,4]. Since their analytical solution can be obtained in just a few cases, numerical methods are required for its approximation. A key point to construct efficient numerical methods is the ability to reproduce the nonlocal behavior of the fractional derivative This is a challenging goal especially in cases of nonlinear problems. In the literature several methods, such as finite difference methods, spectral methods, finite element methods, collocation methods, were proposed to solve fractional differential problems [5,6,7].
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