Singularity subtraction for nonlinear weakly singular integral equations of the second kind

The singularity subtraction technique for computing an approximate solution of a linear weakly singular Fredholm integral equation of the second kind is generalized to the case of a nonlinear integral equation of the same kind. Convergence of the sequence of approximate solutions to the exact one is proved under mild standard hypotheses on the nonlinear factor, and on the sequence of quadrature rules used to build an approximate equation. A numerical example is provided with a Hammerstein operator to illustrate some practical aspects of effective computations.The third author was partially supported by CMat (UID/MAT/00013/2013), and the second and fourth authors were partially supported by CMUP (UID/ MAT/ 00144/2013), which are funded by FCT (Portugal) with national funds (MCTES) and European structural funds (FEDER) under the partnership agreement PT2020

The decoupled potential integral equation for time harmonic electromagnetic scattering

This is the peer reviewed version of the following article: "Vico, F., Ferrando, M., Greengard, L. and Gimbutas, Z. (2016), The Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scattering. Commun. Pur. Appl. Math., 69: 771–812", which has been published in final form at http://dx.doi.org/10.1002/cpa.21585. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[EN] We present a new formulation for the problem of electromagnetic scattering from perfect electric conductors. While our representation for the electric and magnetic fields is based on the standard vector and scalar potentials A, phi in the Lorenz gauge, we establish boundary conditions on the potentials themselves rather than on the field quantities. This permits the development of a wellconditioned second-kind Fredholm integral equation that has no spurious resonances, avoids low-frequency breakdown, and is insensitive to the genus of the scatterer. The equations for the vector and scalar potentials are decoupled. That is, the unknown scalar potential defining the scattered field, phi(scat), is determined entirely by the incident scalar potential phi(inc). Likewise, the unknown vector potential defining the scattered field, A scat, is determined entirely by the incident vector potential A(inc). This decoupled formulation is valid not only in the static limit but for arbitrary omega >= 0. (C) 2016 Wiley Periodicals, Inc.This work was supported in part by the Applied Mathematical Sciences Program of the U.S. Department of Energy under Contract DE-FGO288ER25053 (L.G.), by the Office of the Assistant Secretary of Defense for Research and Engineering and AFOSR under NSSEFF Program Award FA9550-10-1-0180 (L.G. and Z.G.), and in part by the Spanish Ministry of Science and Innovation (Ministerio de Ciencia e Innovacion) under projects CSD2008-00068 and TEC2010-20841-C04-01. The authors thank A. Klockner and M. O'Neil for many useful discussions.Vico Bondía, F.; Ferrando Bataller, M.; Greengard, L.; Gimbutas, Z. (2016). The decoupled potential integral equation for time harmonic electromagnetic scattering. Communications on Pure and Applied Mathematics. 69(4):771-812. https://doi.org/10.1002/cpa.21585S77181269

Hopf bifurcation in a gene regulatory network model:Molecular movement causes oscillations

Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper we analyse a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA in a 1-dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behaviour and this is the case for our model, shown initially via computational simulations. In order to investigate this behaviour more deeply, we next solve our system using Green's functions and then undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. This has implications for transcription factors such as p53, NF-$\kappa$B and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer