Abstract:The integral representation of some biological phenomena consists in Volterra equations whose kernels involve a convolution term plus a non convolution one. Some significative applications arise in linearised models of cell migration and collective motion, as described in Di Costanzo et al. (Discrete Contin. Dyn. Syst. Ser. B 25 (2020) 443-472), Etchegaray et al. (Integral Methods in Science and Engineering (2015)), Grec et al. (J. Theor. Biol. 452 (2018) 35-46) where the asymptotic behaviour of the analytical… Show more
“…For linear evolutionary Volterra integro-differential equations in Hilbert space, the uniform behavior of numerical methods were derived in [45] and the references therein. The asymptotic stability for a quasi-covolution Volterra integral equation was investigated recently in [35]. The F-ODEs with α = 1 reduces to the classical ODEs and exponential decay rates of numerical solutions for LMMs was studied in [14].…”
The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3 (1990), 216-240), we are able to prove that both L1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on α-difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation and the decay rate of the continuous fractional resolvent operator. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations, fractional linear system and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.
“…For linear evolutionary Volterra integro-differential equations in Hilbert space, the uniform behavior of numerical methods were derived in [45] and the references therein. The asymptotic stability for a quasi-covolution Volterra integral equation was investigated recently in [35]. The F-ODEs with α = 1 reduces to the classical ODEs and exponential decay rates of numerical solutions for LMMs was studied in [14].…”
The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3 (1990), 216-240), we are able to prove that both L1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on α-difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation and the decay rate of the continuous fractional resolvent operator. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations, fractional linear system and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.
This paper describes the effect of perturbation of the kernel on the solutions of linear Volterra integral equations on time scales and proposes a new perspective for the stability analysis of numerical methods.
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