On the theoretical basis of memory-free approaches for fractional differential equations. Issue 4 (20th May 2019)

• Record Type: Journal Article
• Authors: Liu, Q.X.; Liu, J.K.; Chen, Y.M.
• Is Part Of: Engineering computations. Volume 36:Issue 4(2019)
• Page Start: 1201
• Page End: 1218
• Publication Date: 2019-05-20
• Subjects: Fractional differential equation -- Theoretical basis -- Yuan–Agrawal memory-free method; Computer-aided engineering -- Periodicals; Computer graphics -- Periodicals; 620.00285
• DOI: 10.1108/EC-08-2018-0389
• Languages: English
• ISSNs: 0264-4401

Abstract

Abstract : Purpose: A nonclassical method, usually called memory-free approach, has shown promising potential to release arithmetic complexity and meets high memory-storage requirements in solving fractional differential equations. Though many successful applications indicate the validity and effectiveness of memory-free methods, it has been much less understood in the rigorous theoretical basis. This study aims to focus on the theoretical basis of the memory-free Yuan–Agrawal (YA) method [Journal of Vibration and Acoustics 124 (2002), pp. 321-324]. Design/methodology/approach: Mathematically, the YA method is based on the validity of two fundamental procedures. The first is to reverse the integration order of an improper quadrature deduced from the Caputo-type fractional derivative. And, the second concerns the passage to the limit under the integral sign of the improper quadrature. Findings: Though it suffices to verify the integration order reversibility, the uniform convergence of the improper integral is proved to be false. Alternatively, this paper proves that the integration order can still be reversed, as the target solution can be expanded as Taylor series on [0, ∞). Once the integration order is reversed, the paper presents a sufficient condition for the passage to the limit under the integral sign such that the target solution is continuous on [0, ∞). Both positive and counter examples are presented to illustrate and validate the theoretical analysis results. Originality/value: This study presents some useful results for the real performance for the YA and some similar memory-free approaches. In addition, it opens a theoretical question on sufficient and necessary conditions, if any, for the validity of memory-free approaches.