Application of SubIval in solving initial value problems with fractional derivatives

Abstract

The application of a numerical method for the approximation of the fractional derivative (in Riemann–Liouville and Caputo definitions) in initial value problems is discussed. The method (previously known as the subinterval-based method) is now referred to by its acronym, SubIval, for simpler future references.It is dependent on subinterval partitions (performed according to a proposed algorithm), interpolations using selected time axis nodes (i.e. nodes where solutions have been computed) and analytical monomial integrodifferentiation formulae.Two exemplary circuit problems have been introduced as a test for the method. These problems have analytical solutions available in literature. The evaluations of these solutions have been compared with results obtained through an adaptive step size predictor-corrector scheme, where the core computations relied on the proposed numerical method.SubIval has been implemented into an ActiveX Dynamic-Link Library (DLL). The paper contains instructions on how the predictor-corrector algorithm can be implemented into a Computer Algebra System, where the SubIval library is applied for the fractional derivative approximation. Examples of this are given in the form of scripts in MATLAB and Mathematica. These scripts generally allow to solve systems of fractional order state equations, to which the introduced circuit examples can be brought to.