Dimension Reduction and Error Estimates

Small perturbations in the type of boundary conditions for an elliptic operator

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

Generic bifurcation of steady-state solutions

We study the large time behavior of the solution of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Dirichlet or Neumann boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term.

Establishing variational formulation is an effective way to study the existence and uniqueness of the solution of certain elliptic partial differential equation with boundary condition. For the solution of certain elliptic partial differential equation with boundary condition, we know that the numerical solution obtained by the finite element method approximates the solution of this equation. Moreover, to avoid gridding overly complex domains, we can use the Chimera method to decompose the domain into several overlapping sub-domains. In this paper, we study Poisson’s equation with the homogeneous Dirichlet boundary condition. By analyzing the existence and uniqueness of the solution of the corresponding variational formulation, we know the existence and uniqueness of the solution of Poisson’s equation with the homogeneous Dirichlet boundary condition. We use the Chimera method and the finite element method to deal with Poisson’s equation with the homogeneous Dirichlet boundary condition by constructing two iterative sequences and analyzing their properties.

An asymptotically stable two-stage difference scheme applied previously to a homogeneous parabolic equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial condition is extended to the case of an inhomogeneous parabolic equation with an inhomogeneous Dirichlet boundary condition. It is shown that, in the class of schemes with two stages (at every time step), this difference scheme is uniquely determined by ensuring that high-frequency spatial perturbations are fast damped with time and the scheme is second-order accurate and has a minimal error. Comparisons reveal that the two-stage scheme provides certain advantages over some widely used difference schemes. In the case of an inhomogeneous equation and a homogeneous boundary condition, it is shown that the extended scheme is second-order accurate in time (for individual harmonics). The possibility of achieving second-order accuracy in the case of an inhomogeneous Dirichlet condition is explored, specifically, by varying the boundary values at time grid nodes by O(τ 2), where τ is the time step. A somewhat worse error estimate is obtained for the one-dimensional heat equation with arbitrary sufficiently smooth boundary data, namely, $O\left( {\tau ^2 \ln \frac{T} {\tau }} \right) $ , where T is the length of the time interval.

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

In this paper we present a residual-based a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems. In particular, we use as a model problem the Laplace eigenvalue problem on bounded domains in ℝd, d = 2, 3, with homogeneous Dirichlet boundary conditions. Analogous error estimators can be easily obtained for more complicated elliptic eigenvalue problems. We prove the reliability and efficiency of the residual-based error estimator also for non-convex domains and use numerical experiments to show that, under an hp-adaptation strategy driven by the error estimator, exponential convergence can be achieved, even for non-smooth eigenfunctions.

The nonlinear Schrödinger equation in the finite line

Let N be a Riemannian manifold, M ⊂ N be a domain with smooth boundary, μ a positive measure on M such that M has unit μ-volume. Consider the evolution driven by the p-Laplace-type operator (p > 2) associated to the natural p-energy functional E (p) con- structed from μ, homogeneous Dirichlet boundary conditions on ∂M being assumed. Assume that a single suitable logarithmic inequality holds for E (p) .Then we show that the evolution brings any data belonging to the natural domain of the evolution instantaneously into L q for any q > 2, with quantitative bounds on the Lq norms. (q > 2 arbitrary) for the evolution equation associated to (possibly degenerate or sin- gular) p-Laplacian-like operators on finite volume domains of Riemannian manifolds, Dirichlet boundary conditions being assumed, provided the associated energy func- tional satisfy a single logarithmic Sobolev inequality. This parallels, in the present case, the results discovered by L. Gross in his cel- ebrated paper (11) for the linear case (see also (12) and, without any claim of com- pleteness, the fundamental papers of Federbush, Nelson, Simon and Hoegh-Krohn(9), (13), (15)), but shows a substantial and unexpected difference with that situation, in which it is well known that no more than a L 2 -L p(t) regularization holds, with p(t) smooth and increasing, p(0 )= 2, p(t) → +∞ as t → +∞. The Ornstein-Uhlenbeck semigroup shows the sharpness of that result in the linear case, this being particularly evident in the fact that the eigenfunctions of such operator are unbounded. To start with we shall introduce our setting and the corresponding notation. We consider a connected, smooth Riemannian manifold (N,g) of dimension n endowed with the associated Riemannian measure m .L etM ⊂ N be an open domain with smooth boundary and consider a measurable function V on M. It will assumed here- after, and will be crucial in what follows, that e V is a probabilitymeasure on M ,s o that the μ-volume of M is one, where we set dμ := e V dm .A ll L p spaces and norms will

A version of fractional diffusion on bounded domains, subject to 'homogeneous Dirichlet boundary conditions' is derived from a kinetic transport model with homogeneous inflow boundary conditions. For nonconvex domains, the result differs from standard formulations. It can be interpreted as the forward Kolmogorow equation of a stochastic process with jumps along straight lines, remaining inside the domain.

This paper deals with the controllability problem of a linearized Korteweg-de Vries equation on bounded interval. The system has a homogeneous Dirichlet boundary condition and a homogeneous Neumann boundary condition at the right end-points of the interval, a non homogeneous Dirichlet boundary condition at the left end-point which is the control. We prove the null controllability by using a backstepping approach, a method usually used to handle stabilization problems.

In this paper, we consider an elliptic problem with the homogeneous Dirichlet boundary condition and introduce discontinuous Galerkin approximations of the problem. Optimal error estimates of discontinuous Galerkin approximations are obtained.

Asymptotic estimates in Thompson’s metric for semilinear parabolic equations

This chapter is devoted to the mathematical analysis of the model obtained in Sect. 3.3 and consists of four subsections. In Sect. 4.1 we verify the obtained model and prove the well posedness of the PDE-ODE systems corresponding to it. Having ordinary differential equations describing the evolution in time of three subpopulations of mitochondria indicates that we assume the mitochondria in the test tube, as well as within cells, do not move in any direction and hence the spatial effects are only introduced by the calcium evolution obeying a partial differential equation (PDE), namely a reaction-diffusion equation. We study the longtime dynamics of solutions of the PDE-ODE coupling. We obtain the complete classifications of the limiting profile of solutions and study partial and complete classification scenarios depending on the given data. Note that these scenarios, namely partial and complete swelling scenarios, have been observed in experiments. Section 4.3 deals with the numerical simulation (in silico) of PDE-ODE systems. One of the remarkable results here is the clearly visible spreading calcium wave. If we compare the dynamics with those of simple diffusion without any positive feedback, the numerical results show that the resulting calcium evolution induced by mitochondria swelling is indeed completely different. Our numerical simulations show that a small change in the initial distribution of calcium is enough to shift the behavior from partial to complete swelling behavior. In Sect. 4.4 we continue our analysis of a coupled PDE-ODE model of calcium induced mitochondria swelling in vitro. More precisely, we study the longtime dynamics of solutions of PDE-ODE systems under homogeneous Dirichlet boundary conditions. Note that, biologically, this kind of boundary condition appears if we put some chemical material on the test-tube wall that binds calcium ions and hence removes it as a swelling inducer. We especially emphasize that the analytical machinery that was developed in Sect. 4.1 is not applicable under homogeneous Dirichlet boundary conditions and therefore must be extended. In this section we show that the calcium ion concentration will tend to zero and that, in general, complete swelling will not occur as time goes to infinity. This distinguishes the situation under Dirichlet boundary conditions from the situation under Neumann boundary conditions that were analyzed in Sect. 4.1. In Sect. 4.5, we carry out numerical simulations validating the analytical results of Sect. 4.4.