2015/11/19 by Anna Oleynik, Arcady Ponosov, Oleynik, Anna +5
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Functional Analysis (math.FA) #Mathematical Biology Tumor Growth #Model Reduction and Neural Networks #math.AP #math.FA
paper · pdf · doi:10.48550/arxiv.1511.06364
arxiv created 2015/11/19 · openalex publication_date 2015/11/19 · arxiv updated 2015/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the existence of fixed points to a parameterized Hammertstain operator \cHβ, β∈ (0,∞], with sigmoid type of nonlinearity. The parameter β<∞ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case β=∞ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large β exist and can be approximated by the fixed points of \cH_∞. These results are of a high importance in biological applications where one often approximates the smooth sigmoid by discontinuous unit step function. Moreover, in order to achieve even better approximation than a solution of the limit problem, we employ the iterative method that has several advantages compared to other existing methods. For example, this method can be used to construct non-isolated homoclinic orbit of a Hamiltionian system of equations. We illustrate the results and advantages of the numerical method for stationary versions of the FitzHugh-Nagumo reaction-diffusion equation and a neural field model.