A new type of fractional linear multi-step method with improved stability

Abstract

We present a new type of implicit fractional linear multi-step method (FLMM) of order two for fractional initial value problems. The method is obtained from the second order super convergence of the Grünwald-Letnikov form of the fractional derivative at a non-integer shift point in the domain. The proposed method coincides with the classical BDF method of order two for ordinary initial value problems when the fractional order of the derivative is one. The weight coefficients of the proposed method are obtained from the Grünwald weights and hence computationally efficient compared with the fractional backward difference formula of order two (FBDF2). The stability region of the FLMM is larger than that of the fractional Adams-Moulton method of order two and the fractional trapezoidal method, and is very much closer in size to the FBDF2. Numerical result and illustrations are presented to justify the claims.

Keywords

• Lubich Generating functions
• super convergence
• A-stability
• fractional Adams-Moulton methods
• Trapezoidal rule.