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

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

The nonlinear Schrödinger equation in the finite line

This article deals with the optimization of the shape of the regions assigned to different types of boundary conditions in the definition of a ‘physical’ partial differential equation. At first, we analyze a model situation involving the solution $$u_\varOmega $$ to a Laplace equation in a domain $$\varOmega $$ ; the boundary $$\partial \varOmega $$ is divided into three parts $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ , supporting respectively homogeneous Dirichlet, homogeneous Neumann and inhomogeneous Neumann boundary conditions. The shape derivative $$J^\prime (\varOmega )(\theta )$$ of a general objective function $$J(\varOmega )$$ of the domain is calculated in the framework of Hadamard’s method when the considered deformations $$\theta $$ are allowed to modify the geometry of $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ (i.e. $$\theta $$ does not vanish on the boundary of these regions). The structure of this shape derivative turns out to depend very much on the regularity of $$u_\varOmega $$ near the boundaries of the regions $$\varGamma _D$$ , $$\varGamma $$ and $$\varGamma _N$$ . For this reason, in particular, $$J^\prime (\varOmega )$$ is difficult to calculate and to evaluate numerically when the transition $$\overline{\varGamma _D} \cap {\overline{\varGamma }}$$ between homogeneous Dirichlet and homogeneous Neumann boundary conditions is subject to optimization. To overcome this difficulty, an approximation method is proposed, in which the considered ‘exact’ Laplace equation with mixed boundary conditions is replaced with a ‘smoothed’ version, featuring Robin boundary conditions on the whole boundary $$\partial \varOmega $$ with coefficients depending on a small parameter $$\varepsilon $$ . We prove the consistency of this approach in our model context: the approximate objective function $$J_\varepsilon (\varOmega )$$ and its shape derivative converge to their exact counterparts as $$\varepsilon $$ vanishes. Although it is rigorously justified only in a model problem, this approximation methodology may be adapted to many more complex situations, for example in three space dimensions, or in the context of the linearized elasticity system. Various numerical examples are eventually presented in order to appraise the efficiency of the proposed approximation process.

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.

Generic bifurcation of steady-state solutions

We prove theorems on the existence and regularization of periodic solutions of the wave equation with variable coefficients on an interval with homogeneous Dirichlet and Neumann boundary conditions. The nonlinear term has a power-law growth or satisfies the nonresonance condition at infinity.

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.

A characteristic based volume penalization method for general evolution problems applied to compressible viscous flows

Non-degenerate solutions of boundary-value problems

Schwinger-type variational principles are presented for the scattered wave in the case of scalar wave scattering for an arbitrary incident field from an object of arbitrary shape with either homogeneous Dirichlet or homogeneous Neumann boundary conditions. Designating the distance from the scatterer to the observer by r, then the results are variationally invariant for all values of r ranging from the surface of the scatterer to the farfield. Explicit results are presented for the case when the scatterer is a sphere obeying homogeneous Dirichlet boundary conditions. Special attention is given to the selection of the trial fields that produce accurate results over a broad frequency range. [Work supported by the U.S. Navy.]

In this paper, we study a spatially diffusive SIR epidemic model with constant parameters in a bounded spatial domain and investigate the relationship between the basic reproduction number \(\mathcal{R}_0\) and the shape of the spatial domain. Under the homogeneous Neumann boundary conditions, \(\mathcal{R}_0\) is the same as that for the classical non-diffusive SIR epidemic model, and thus, it does not depend on the shape of the spatial domain. On the other hand, under the homogeneous Dirichlet boundary conditions, the next generation operator does not have a constant eigenvector, and \(\mathcal{R}_0\) depends on the shape of the spatial domain. By numerical simulation for the 2-dimensional rectangular domain \(\Omega = (0,p) \times (0,1/p)\), \(p > 0\) with constant area \(|\Omega | =1\), we show that such \(\mathcal{R}_0\) attains its maximum for \(p=1\) (that is, \(\Omega \) is square) and decreases as the shape of the domain \(\Omega \) becomes long and narrow. Moreover, we observe a similar relationship between \(\mathcal{R}_0\) and the shape of the spatial domain in a random 2-dimensional lattice model.

A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation

This work is concerned with the periodic solution of a doubly degenerate Allen-Cahn equation with nonlocal terms associated with Neumann boundary conditions. Firstly, we define a new associated auxiliary problem. Secondly, the topological degree theorem is applied to prove the existence of a limit point to the auxiliary problem, where this limit point represents a nontrivial nonnegative time-periodic solution of the main studied problem. It is observed that the topological degree theorem technique plays an important role in proving the desired results. Furthermore, this technique can be applied to other similar equations with homogeneous Dirichlet or Neumann boundary conditions.

This work is concerned with the periodic solution of a p-Laplacian Allen-Cahn equation with nonlocal terms associated with Neumann boundary conditions. Namely, the topological degree theorem is applied to prove the existence of a limit point to the auxiliary problem, which is also considered a nontrivial nonnegative time-periodic solution of the main studied problem. The results of this work can be extended to other similar equations with homogeneous Dirichlet or Neumann boundary conditions.

We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions, or combinations of either for different parts of the boundary. We use an inverse power plus Gauss-Seidel algorithm to solve the generalized eigenvalue problem. For Neumann boundary conditions the method is much more efficient than the equivalent finite difference algorithm. We checked the algorithm by comparing the cumulative level density of the spectrum obtained numerically with the theoretical prediction given by the Weyl formula. We found a systematic deviation due to the discretization, not to the algorithm itself.