A novel numerical technique used in the solution of ordinary differential equations with a mixture of integer and fractional derivatives

TL;DR

This paper introduces new numerical algorithms combining fractional derivatives (Riemann-Liouville and Caputo) with classical derivatives to solve ordinary differential equations, addressing initial condition incorporation.

Contribution

It presents novel numerical methods that integrate fractional and classical derivatives, expanding solution techniques for differential equations with mixed derivatives.

Findings

Developed algorithms with four discrete Caputo derivative forms

Proposed three numerical techniques for solving ODEs

Demonstrated how to incorporate classical initial conditions with Riemann-Liouville derivatives

Abstract

Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some algorithms where four discrete forms of the Caputo derivative and three different numerical techniques of solving ordinary differential equations are proposed. We then illustrate how to introduce classical initial conditions into equations where the Riemann-Liouville derivative is included.