2022/09/01 by Wenlin Qiu, Qiu, Wenlin
Mathematics · Engineering · #Fractional Differential Equations Solutions #Differential Equations and Numerical Methods #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2209.00235
In this work, the z-transform is presented to analyze time-discrete solutions for Volterra integrodifferential equations (VIDEs) with nonsmooth multi-term kernels in the Hilbert space, and this class of continuous problem was first considered and analyzed by Hannsgen and Wheeler (SIAM J Math Anal 15 (1984) 579-594). This work discusses three cases of kernels βq(t) included in the integrals for the multi-term VIDEs, from which we use corresponding numerical techniques to approximate the solution of multi-term VIDEs in different cases. Firstly, for the case of β1(t), β2(t) ∈ L1(ℝ+), the Crank-Nicolson (CN) method and interpolation quadrature (IQ) rule are applied to time-discrete solutions of the multi-term VIDEs; secondly, for the case of β1(t)∈ L1(ℝ+) and β2(t)∈ L1,loc(ℝ+), second-order backward differentiation formula (BDF2) and second-order convolution quadrature (CQ) are employed to discretize the multi-term problem in the time direction; thirdly, for the case of β1(t), β2(t)∈ L1,loc(ℝ+), we utilize the CN method and trapezoidal CQ (TCQ) rule to approximate temporally the multi-term problem. Then for the discrete solution of three cases, the long-time global stability and convergence are proved based on the z-transform and certain appropriate assumptions. Furthermore, the long-time estimate of the third case is confirmed by the numerical tests.