A new approach to shooting methods for terminal value problems of fractional differential equations

TL;DR

This paper introduces a novel proportional shooting method for terminal value problems of fractional differential equations with Caputo derivatives, significantly improving convergence speed and accuracy over traditional methods.

Contribution

The paper develops a new proportional shooting technique that enhances the efficiency and precision of solving fractional differential equations of order  0,1, especially for terminal value problems.

Findings

Converges quickly and accurately to solutions.

Achieves a speedup factor of 4 to 10 over bisection.

Numerical experiments validate the method's effectiveness.

Abstract

For terminal value problems of fractional differential equations of order $\alpha \in (0,1)$ that use Caputo derivatives, shooting methods are a well developed and investigated approach. Based on recently established analytic properties of such problems, we develop a new technique to select the required initial values that solves such shooting problems quickly and accurately. Numerical experiments indicate that this new proportional secting technique converges very quickly and accurately to the solution. Run time measurements indicate a speedup factor of between 4 and 10 when compared to the standard bisection method.