Analysis of Fractional Linear Multi-Step Methods of Order Four from Super-Convergence

Abstract

We analyze two implicit fractional linear multi-step methods of order four for solving fractional initial value problems. The methods are derived from the Grunwald-Letnikov approximation of fractional derivative at a non-integer shift point with super-convergence. The weight coefficients of the methods are computed from fundamental G unwald weights, making them computationally efficient when compared with other known methods of order four. We also show that the stability regions are larger than that of the fractional Adams-Moulton and fractional backward difference formula methods. We present numerical results and illustrations to verify that the theoretical results obtained are indeed satisfied.