A Stage-Structured Population Model with Time-Dependent Delay in an Almost Periodic Environment

Qiang, Lizhong; Wang, Bin-Guo; Zhao, Xiao-Qiang

doi:10.1007/s10884-020-09827-6

Cited by 12 publications

(2 citation statements)

References 28 publications

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“…Note: The functions 𝜓 ki are defined in (22); K, M, N are arbitrary bounded linear operators on Banach space X; and B is arbitrary bilinear mapping from X × X to X.…”

Section: No Condition Ordermentioning

confidence: 99%

“…Various classes of differential equations, such as reaction-diffusion equations with delays in the nonlinear reaction terms, and age-dependent population equations, can be reformulated into this abstract form Wu [21]. The ubiquity of DDEs has also been demonstrated through their numerous applications, spanning from biology and medicine to climate and social sciences [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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Convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations

Zhan,

Fang

2023

Math Methods in App Sciences

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In this work, we extend previous research on exponential integrators for stiff semilinear delay differential equations to the nonlinear case. In addition to the stiffness, there are two new issues that should be handled properly: nonlinear term and delay term. For nonlinear problems, a badly chosen linearization can cause a severe step size restriction. In this work, we linearize the equation along the numerical solution in each step. For the delay term, the interpolation based on the numerical values at the mesh points rather than inner stage values is adopted to significantly reduce the number of stiff order conditions. We focus on the construction and convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations. The main result of this paper is that under the framework of strongly continuous semigroup, the explicit exponential Rosenbrock method is proved to be stiffly convergent of order even if the order conditions of order hold in a weak form. Moreover, by pointing that there does not exist fifth‐order method with less than or equal to four stages, we present the construction of a fifth‐order method with five stages. Finally, numerical tests are carried out to validate the theoretical results and to demonstrate the superiority of high‐order methods.

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“…Note: The functions 𝜓 ki are defined in (22); K, M, N are arbitrary bounded linear operators on Banach space X; and B is arbitrary bilinear mapping from X × X to X.…”

Section: No Condition Ordermentioning

confidence: 99%