In this paper, we propose a penalty-based recurrent neural network for solving a class of constrained optimization problems with generalized convex objective functions. The model has a simple structure described by using a differential inclusion. It is also applicable for any nonsmooth optimization problem with affine equality and convex inequality constraints, provided that the objective funct...

Finite-volume procedure is presented for solving the natural convection of the laminar  nanofluid flow in a Γ shaped microchannel in this article. Modified Navier-Stokes equations for nanofluids are the basic equations for this problem. Slip flow region, including the effects of velocity slip and temperature jump at the wall, are the main characteristics of flow in the slip flow region. Steady ...

Let (Gn) ∞ n=1 be a sequence of finite graphs, and let Yt be the length of a loop-erased random walk on Gn after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which Gn is the d-dimensional torus of size-length n for d ≥ 4, the process (Yt) ∞ t=0, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. O...

Let (Gn) ∞ n=1 be a sequence of finite graphs, and let Yt be the length of a loop-erased random walk on Gn after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which Gn is the d-dimensional torus of size-length n for d ≥ 4, the process (Yt) ∞ t=0, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. O...

This work considers the rigorous derivation of continuummodels of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.

We consider a model of population dynamics whose mortality function is unbounded. We approximate the solution of the model using a discontinuous Galerkin finite element for the age variable and a backward Euler for the time variable. We present several numerical examples. It is experimentally shown that the scheme converges at the rate of h3/2 in the case of piecewise linear polynomial space.

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general H initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in H towards a dissipative weak solution of Cam...

A standard finite element method and a finite element truncation method are applied to solve the boundary value problems of nonlinear elasticity with certain nonconvex stored energy functions such as those of St. Venant-Kirchhoff materials. Finite element solutions are proved to exist and to be in the form of minimizers in appropriate sets of admissible finite element functions for both methods...

In this paper, we propose a finite volume discretization for multidimensional nonlinear drift-diffusion system. Such a system arises in semiconductors modeling and is composed of two parabolic equations and an elliptic one. We prove that the numerical solution converges to a steady state when time goes to infinity. Several numerical tests show the efficiency of the method.

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general H1 initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in H1 towards a dissipative weak solution of C...