Search results for "Boundary problem"

Numerical solution of a class of nonlinear boundary value problems for analytic functions

1982

We analyse a numerical method for solving a nonlinear parameter-dependent boundary value problem for an analytic function on an annulus. The analytic function to be determined is expanded into its Laurent series. For the expansion coefficients we obtain an operator equation exhibiting bifurcation from a simple eigenvalue. We introduce a Galerkin approximation and analyse its convergence. A prominent problem falling into the class treated here is the computation of gravity waves of permanent type in a fluid. We present numerical examples for this case.

Generalized differential transform method for nonlinear boundary value problem of fractional order

2015

Abstract In this paper the generalized differential transform method is applied to obtain an approximate solution of linear and nonlinear differential equation of fractional order with boundary conditions. Several numerical examples are considered and comparisons with the existing solution techniques are reported. Results show that the method is effective, easier to implement and very accurate when applied for the solution of fractional boundary values problems.

An asymptotic holomorphic boundary problem on arbitrary open sets in Riemann surfaces

2020

Abstract We show that if U is an arbitrary open subset of a Riemann surface and φ an arbitrary continuous function on the boundary ∂ U , then there exists a holomorphic function φ ˜ on U such that, for every p ∈ ∂ U , φ ˜ ( x ) → φ ( p ) , as x → p outside a set of density 0 at p relative to U . These “solutions to a boundary problem” are not unique. In fact they can be required to have interpolating properties and also to assume all complex values near every boundary point. Our result is new even for the unit disc.

On a singular boundary value problem for a second order ordinary differential equation

2000

A free boundary problem stemmed from combustion theory. Part II: Stability, instability and bifurcation results

2002

AbstractWe deal with a free boundary problem, depending on a real parameter λ, in a infinite strip in R2, providing stability, instability and bifurcation.

Bifurcation of traveling waves in a Keller–Segel type free boundary model of cell motility

2018

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of …

Trapping Horizons as inner boundary conditions for black hole spacetimes

2007

We present a set of inner boundary conditions for the numerical construction of dynamical black hole space-times, when employing a 3+1 constrained evolution scheme and an excision technique. These inner boundary conditions are heuristically motivated by the dynamical trapping horizon framework and are enforced in an elliptic subsystem of the full Einstein equation. In the stationary limit they reduce to existing isolated horizon boundary conditions. A characteristic analysis completes the discussion of inner boundary conditions for the radiative modes.

Drops moving in flow with chernical reaction

1994

We propose a free boundary model described by coupled Navier-Stokes and chemical reaction equations with discontinuous coefRcients to simulate the chemical re- ¿ctions in viscous drops moving in a viscous incompressible ûuid. Approximation of the solution by a special ñnite element method (FEM) with a method of mapping is discussed. Several numerical resulùs åre presented.

Minimizers for the Thin One‐Phase Free Boundary Problem

2021

We consider the “thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0001 plus the area of the positivity set of that function in urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0002. We establish full regularity of the free boundary for dimensions urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0003, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for…

\( L^{1} \) existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions

2007

Abstract In this paper we study the questions of existence and uniqueness of weak and entropy solutions for equations of type − div a ( x , D u ) + γ ( u ) ∋ ϕ , posed in an open bounded subset Ω of R N , with nonlinear boundary conditions of the form a ( x , D u ) ⋅ η + β ( u ) ∋ ψ . The nonlinear elliptic operator div a ( x , D u ) is modeled on the p-Laplacian operator Δ p ( u ) = div ( | D u | p − 2 D u ) , with p > 1 , γ and β are maximal monotone graphs in R 2 such that 0 ∈ γ ( 0 ) and 0 ∈ β ( 0 ) , and the data ϕ ∈ L 1 ( Ω ) and ψ ∈ L 1 ( ∂ Ω ) .