Sugimoto, N. Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. (English) Zbl 0721.76011 J. Fluid Mech. 225, 631-653 (1991). Summary: This paper deals with initial-value problems for the Burgers equation with the inclusion of a hereditary integral known as the fractional derivative of order 1/2. Emphasis is placed on the difference between the local and global dissipation due to the second-order and the half-order derivatives, respectively. Exploiting the smallness of the coefficient of the second-order derivative, an asymptotic analysis is first developed. When a discontinuity appears, the matched-asymptotic expansion method is employed to derive a uniformly valid solution. If the coefficient of the half-order derivative is also small, as is usually the case, the evolution comprises three stages, namely a lossless near field, an intermediate Burgers region, and a hereditary far field. In view of these results, the equation is then solved numerically, under various initial conditions, by finite-difference and spectral methods. It is revealed that the effect of the fractional derivative accumulates slowly to give rise to a significant dissipation and distortion of the waveform globally, which is to be contrasted with the effect of the second-order derivative, significant only locally, in a thin ‘shock layer’. Cited in 89 Documents MSC: 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76Q05 Hydro- and aero-acoustics 35Q35 PDEs in connection with fluid mechanics Keywords:initial-value problems; Burgers equation; hereditary integral; half-order derivatives; shock layer PDFBibTeX XMLCite \textit{N. Sugimoto}, J. Fluid Mech. 225, 631--653 (1991; Zbl 0721.76011) Full Text: DOI References: [1] Crighton, Phil. Trans. R. Soc. Lond. 292 pp 101– (1979) [2] DOI: 10.1017/S0022112064000040 · Zbl 0129.19504 [3] DOI: 10.1016/0045-7930(86)90036-8 · Zbl 0612.76031 [4] Miksis, Adv. Appl. Mech. 323 pp 173– (1990) [5] DOI: 10.1007/BF00946746 · Zbl 0471.76074 [6] DOI: 10.1143/JPSJ.39.237 · Zbl 1334.76018 [7] Lee-Bapty, Phil. Trans. E. Soc. Lond. 323 pp 173– (1987) [8] Gittler, Z. Angew. Math. Mech. 69 pp 578– (1989) [9] DOI: 10.1016/0021-9991(73)90128-9 · Zbl 0267.65074 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.