Long-time behavior of numerical solutions to nonlinear fractional ODEs
Publication in refereed journal



摘要In this work, we study the long time behavior, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By means of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grunwald-Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including convolution quadrature schemes based on classical second order BDF and product integration schemes based on quadratic interpolation approximation, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.
著者Dongling WANG, Aiguo XIAO, Jun ZOU
期刊名稱ESAIM: Mathematical Modelling and Numerical Analysis
頁次335 - 358
關鍵詞Fractional ODEs, contractivity, dissipativity, fractional BDFs

上次更新時間 2020-14-09 於 23:41