Search results for "Mathematical analysis"
showing 10 items of 2409 documents
Local and nonlocal weighted pLaplacian evolution equations with Neumann boundary conditions
2011
In this paper we study existence and uniqueness of solutions to the local diffusion equation with Neumann boundary conditions and a bounded nonhomogeneous diffusion coefficient g ≥ 0, {ut = div (g|∇u|p-2∇u) in ]0; T[×Ωg|∇u|p-2u·n = 0 on ]0; T[×∂Ω; for 1 ≤ p < ∞. We show that a nonlocal counterpart of this diffusion problem is ut(t; x)= ∫ω J(x-y)g(x+y/2)|u(t; y)-u(t; x)| p-2 (u(t; y)-u(t; x)) dy in ]0; T[× Ω,where the diffusion coefficient has been reinterpreted by means of the values of g at the point x+y/2 in the integral operator. The fact that g ≥ 0 is allowed to vanish in a set of positive measure involves subtle difficulties, specially in the case p = 1.
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions
2008
In this paper we study the nonlocal p-Laplacian type diffusion equation,ut (t, x) = under(∫, Ω) J (x - y) | u (t, y) - u (t, x) |p - 2 (u (t, y) - u (t, x)) d y . If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div (| ∇ u |p - 2 ∇ u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞ (0, T ; Lp (Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous t…
Sharp estimates for eigenfunctions of a Neumann problem
2009
In this paper we provide some bounds for the eigenfunctions of the Laplacian with homogeneous Neumann boundary conditions in a bounded domain Ω of R^n. To this aim we use the so-called symmetrization techniques and the obtained estimates are asymptotically sharp, at least in the bidimensional case, when the isoperimetric constant relative to Ω goes to 0.
Remark on a nonlocal isoperimetric problem
2017
Abstract We consider isoperimetric problem with a nonlocal repulsive term given by the Newtonian potential. We prove that regular critical sets of the functional are analytic. This optimal regularity holds also for critical sets of the Ohta–Kawasaki functional. We also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set.
High-quality discretizations for microwave simulations
2016
We apply high-quality discretizations to simulate electromagnetic microwaves. Instead of the vector field presentations, we focus on differential forms and discretize the model in the spatial domain using the discrete exterior calculus. At the discrete level, both the Hodge operators and the time discretization are optimized for time-harmonic simulations. Non-uniform spatial and temporal discretization are applied in problems in which the wavelength is highly-variable and geometry contains sub-wavelength structures. peerReviewed
Damage Identification of Beams Using Static Test Data
2003
A damage identification procedure for beams under static loads is presented. Damage is modelled through a damage distribution function which determines a variation of the beam stiffness with respect to a reference condition. Using the concept of the equivalent superimposed deformation, the equations governing the static problem are recast in a Fredholm’s integral equation of the second kind in terms of bending moments. The solution of this equation is obtained through an iterative procedure as well as in closed form. The latter is explicitly dependent from the damage parameters, thus, it can be conveniently used to set-up a damage identification procedure. Some numerical results are present…
The Heat Content for Nonlocal Diffusion with Non-singular Kernels
2017
Abstract We study the behavior of the heat content for a nonlocal evolution problem.We obtain an asymptotic expansion for the heat content of a set D, defined as ℍ D J ( t ) := ∫ D u ( x , t ) 𝑑 x ${\mathbb{H}_{D}^{J}(t):=\int_{D}u(x,t)\,dx}$ , with u being the solution to u t = J ∗ u - u ${u_{t}=J\ast u-u}$ withinitial condition u 0 = χ D ${u_{0}=\chi_{D}}$ . This expansion is given in terms of geometric values of D. As a consequence, we obtain that ℍ D J ( t ) = | D | - P J ( D ) t + o ( t ) ${\mathbb{H}^{J}_{D}(t)=\lvert D\rvert-P_{J}(D)t+o(t)}$ as t ↓ 0 ${t\downarrow 0}$ .We also recover the usual heat content for the heat equation when we rescale the kernel J in an appro…
Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations
2011
We analyze collocation methods for nonlinear homogeneous Volterra-Hammerstein integral equations with non-Lipschitz nonlinearity. We present different kinds of existence and uniqueness of nontrivial collocation solutions and we give conditions for such existence and uniqueness in some cases. Finally we illustrate these methods with an example of a collocation problem, and we give some examples of collocation problems that do not fit in the cases studied previously.
Discrete wavelet transform implementation in Fourier domain for multidimensional signal
2002
Wavelet transforms are often calculated by using the Mallat algorithm. In this algorithm, a signal is decomposed by a cascade of filtering and downsampling operations. Computing time can be important but the filtering operations can be speeded up by using fast Fourier transform (FFT)-based convolutions. Since it is necessary to work in the Fourier domain when large filters are used, we present some results of Fourier-based optimization of the sampling operations. Acceleration can be obtained by expressing the samplings in the Fourier domain. The general equations of the down- and upsampling of digital multidimensional signals are given. It is shown that for special cases such as the separab…
A numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions
2012
Abstract We study numerically the small dispersion limit for the Korteweg–de Vries (KdV) equation u t + 6 u u x + ϵ 2 u x x x = 0 for ϵ ≪ 1 and give a quantitative comparison of the numerical solution with various asymptotic formulae for small ϵ in the whole ( x , t ) -plane. The matching of the asymptotic solutions is studied numerically.