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

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

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.

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 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.]

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

Peskin’s Immersed Boundary (IB) model and method are among the most important modeling tools and numerical methods. The IB method has been known to be first order accurate in the velocity for incompressible Stokes and Navier–Stokes equations. The convergence of the IB method for Stokes equations with periodic boundary conditions is discussed in Liu & Mori (2014, $L^p$ convergence of the immersed boundary method for stationary Stokes problems. SIAM J. Numer. Anal., 52, 496–514) and Mori (2008, Convergence proof of the velocity field for a Stokes flow immersed boundary method. Comm. Pure Appl. Math., 61, 1213–1263). However, almost no rigorous theoretical proof can be found in the literature for Stokes equations with a prescribed velocity boundary condition. In this paper, it has been shown that the pressure of the Stokes equation has a convergence order $O(\sqrt{h})$ in the $L^{2}$ norm, while the velocity has an $O(h |\!\log h| )$ convergence order in the infinity norm in two-space dimensions. The proofs are based on splitting the singular source terms, discrete Green functions on finite lattices with homogeneous Dirichlet and Neumann boundary conditions, properties of discrete delta functions and the convergence proof of the IB method for elliptic interface problems (Li (2015) On convergence of the immersed boundary method for elliptic interface problems. Math. Comp., 84, 1169–1188). The conclusion in this paper can apply to problems with different boundary conditions as long as the problems are well-posed. The proof process also provides an efficient way to decouple the Stokes equations into three Helmholtz/Poisson equations without affecting the order of convergence. Nontrivial numerical examples are also provided to confirm the theoretical analysis.

A diffusion-convection equation is a partial differential equation featuring two important physical processes. In this paper, we establish the theory of solving a 1D diffusion-convection equation, subject to homogeneous Dirichlet, Robin, or Neumann boundary conditions and a general initial condition. Firstly, we transform the diffusion-convection equation into a pure diffusion equation. Secondly, using a separation of variables technique, we obtain a general solution formula for each boundary type case, subject to transformed boundary and initial conditions. While eigenvalues in the cases of Dirichlet and Neumann boundary conditions can be constructed easily, the Robin boundary condition necessitates solving a transcendental algebraic equation to determine all the eigenvalues. Thirdly, we use Python to construct and illustrate the solutions for all the cases based on the newly developed solution formulas. Finally, we share all the associated Python code for public access.

This paper compares computer implementations of numerical algorithms for solving the equations of mathematical models of the dynamics of gas suspensions with viscous heat-conducting, inviscid heatconducting and ideal carrier media. Mathematical models are developed within the framework of the continuum technique for modeling the dynamics of multiphase media. In the study, the process of interaction of a shock wave moving from a homogeneous gas into a gas suspension, which is often encountered in the mining industry, was modeled. The relevance of the study of this flow of inhomogeneous media is associated with the shielding of industrial explosions by aerosol curtains. When modeling for a viscous medium, homogeneous Dirichlet boundary conditions were set, for an inviscid medium, homogeneous Neumann boundary conditions. The equations of the mathematical model were integrated by the McCormack finite difference method. To overcome numerical oscillations, a nonlinear scheme for correcting grid functions was used. The program that implements the continuum method for the dynamics of multiphase media consisted of a block for specifying boundary conditions, a block that implements a numerical solution, and a block for accounting for interfacial interaction. As a result of comparing numerical calculations of mathematical models of the dynamics of a gas suspension with an ideal, inviscid heat-conducting and viscous heat-conducting carrier media, it was found that during the movement of a gas suspension, the viscosity of the carrier medium of the gas suspension has the greatest influence on the intensity of interfacial momentum exchange.

We study asymptotically and numerically the fundamental gaps (i.e. the difference between the first excited state and the ground state) in energy and chemical potential of the Gross–Pitaevskii equation (GPE) – nonlinear Schrödinger equation with cubic nonlinearity – with repulsive interaction under different trapping potentials including box potential and harmonic potential. Based on our asymptotic and numerical results, we formulate a gap conjecture on the fundamental gaps in energy and chemical potential of the GPE on bounded domains with the homogeneous Dirichlet boundary condition, and in the whole space with a convex trapping potential growing at least quadratically in the far field. We then extend these results to the GPE on bounded domains with either the homogeneous Neumann boundary condition or periodic boundary condition.