• Explosive solutions of a stochastic non-local reaction–diffusion equation arising in shear band formation

      Kavallaris, Nikos I.; University of Chester (Wiley, 2015-07-07)
      In this paper, we consider a non-local stochastic parabolic equation which actually serves as a mathematical model describing the adiabatic shear-banding formation phenomena in strained metals. We first present the derivation of the mathematical model. Then we investigate under which circumstances a finite-time explosion for this non-local SPDE, corresponding to shear-banding formation, occurs. For that purpose some results related to the maximum principle for this non-local SPDE are derived and afterwards the Kaplan's eigenfunction method is employed.
    • High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problem

      Yan, Yubin; Qiao, Leijie; Xu, Da; University of Chester, UK; Guangdong University of Technology, PR. China; Hunan Normal University, P. R. China (John Wiley & Sons Ltd, 2020-02-05)
      We use the generalized L1 approximation for the Caputo fractional deriva-tive, the second-order fractional quadrature rule approximation for the inte-gral term, and a classical Crank-Nicolson alternating direction implicit (ADI)scheme for the time discretization of a new two-dimensional (2D) fractionalintegro-differential equation, in combination with a space discretization by anarbitrary-order orthogonal spline collocation (OSC) method. The stability of aCrank-Nicolson ADI OSC scheme is rigourously established, and error estimateis also derived. Finally, some numerical tests are given
    • Quenching solutions of a stochastic parabolic problem arising in electrostatic MEMS control

      Kavallaris, Nikos I.; University of Chester (Wiley, 2016-09-15)
      In the current paper, we consider a stochastic parabolic equation which actually serves as a mathematical model describing the operation of an electrostatic actuated micro-electro-mechanical system (MEMS). We first present the derivation of the mathematical model. Then after establishing the local well-posedeness of the problem we investigate under which circumstances a {\it finite-time quenching} for this SPDE, corresponding to the mechanical phenomenon of {\it touching down}, occurs. For that purpose the Kaplan's eigenfunction method adapted in the context of SPDES is employed.