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Type: Text × Subject: integro-differential equations ×

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    High order discretization methods for spatial-dependent epidemic models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Takács, Bálint; Hadjimichael, Yiannis
    In this paper, an SIR model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of integro-differential equations. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different choices of spatial and temporal discretizations are employed, and step-size restrictions for population conservation, positivity, and monotonicity preservation of the discrete model are investigated. We provide sufficient conditions under which high order numerical schemes preserve the discrete properties of the model. Computational experiments verify the convergence and accuracy of the numerical methods.
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    The Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ros-Oton, Xavier; Valdinoci, Enrico
    We study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of one-dimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the spectral measure is singular, we construct an explicit counterexample.