Treatment of fractional multi-order/multi-term differential equations: utilizing fractional shifted Lucas polynomials

Abstract

In this work, novel type of fractional polynomials are proposed as the generalization to shifted Lucas polynomials, which are called as fractional shifted Lucas polynomials. Useful operational matrices are developed here utilizing the newly established analytical formula for the construction of the numerical scheme. Further, a theorem for the calculation of Caputo fractional derivative is proved and an useful remark is provided for integer order derivative. At the core of the work, the task is to use collocation points for tackling the multi-term differential equations/multi-order differential equations (MODEs/MTDEs). To develop the strong background for the proposed scheme, error analysis is performed. The algorithm of the scheme is examined through some test examples of MODEs/MTDEs, and comparisons are made with other existing methods.