2021/12/05 by M. Bogoya, S. M. Grudsky, Bogoya, M. +5 · 1 citation
Mathematics · Physics and Astronomy · #15A18 #15B05 #26A33 #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2112.02685
openalex publication_date 2021/12/05 · openalex created_date 2023/02/18 · openalex updated_date 2026/07/28
The analysis of the spectral features of a Toeplitz matrix-sequence\n \Tn(f) \n\∈ mathbb N, generated by a symbol f\∈\nL1([-\π,\π]), real-valued almost everywhere (a.e.), has been provided in\ngreat detail in the last century, as well as the study of the conditioning,\nwhen f is nonnegative a.e. Here we consider a novel type of problem arising\nin the numerical approximation of distributed-order fractional differential\nequations (FDEs), where the matrices under consideration take the form n
mathcalTn=c0Tn(f0)+c1 hh Tn(f1)+c2 h2h\nTn(f2)+
cdots+cn-1 h(n-1)hTn(fn-1), c0,c1,\…,\ncn-1 \∈ [c_*,c^*], c^*\≥ c_*>0, independent of n, h=\(1)/(n),\nfj\∼ gj, gj=|\θ|2-jh, j=0,\…,n-1. Since the resulting\ngenerating function depends on n, the standard theory cannot be applied and\nthe analysis has to be performed using new ideas. Few selected numerical\nexperiments are presented, also in connection with matrices that come from\ndistributed-order FDE problems, and the adherence with the theoretical analysis\nis discussed together with open questions and future investigations.\n