Boundary Value Methods for Caputo Fractional Differential Equations

Abstract

This paper deals with the numerical computation and analysis for Caputo fractional differential equations (CFDEs). By combining the $p$-order boundary value methods (BVMs) and the $m$-th Lagrange interpolation, a type of extended BVMs for the CFDEs with $\gamma$-order ($0 <\gamma < 1$) Caputo derivatives are derived. The local stability, unique solvability and convergence of the methods are studied. It is proved under the suitable conditions that the convergence order of the numerical solutions can arrive at $\min\left\{p, m-\gamma+1\right\}$. In the end, by performing several numerical examples, the computational efficiency, accuracy and comparability of the methods are further illustrated.