Semilinear Boundary Value Problems Research Articles

A superconvergent finite node method for semilinear elliptic problems

This paper proposes and analyzes a superconvergent finite node method (SFPM) for meshless solution of semilinear boundary value problems with variable coefficients. Smoothed derivatives of the moving least squares approximation are adopted in the SFPM to achieve the high accuracy and favorable superconvergence. Theoretical accuracy analysis of the SFPM and the traditional FPM is detailedly presented through local truncation error analysis. Some numerical results are provided to demonstrate the superconvergence and theoretical results of the method.

On the spectrum of a mixed-type operator with applications to rotating wave solutions

We study rotating wave solutions of the nonlinear wave equation ∂t2v-Δv+mv=|v|p-2vinR×Bv=0onR×∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\partial _{t}^2 v - \\Delta v + m v = |v|^{p-2} v &{} \ ext {in }{\\mathbb {R} \ imes {\ extbf{B}}} \\\\ v = 0 &{} \\hbox {on }\\mathbb {R} \ imes \\partial {\ extbf{B}} \\end{array} \\right. \\end{aligned}$$\\end{document}where 2<p<infty , m in mathbb {R} and {textbf{B}} subset mathbb {R}^2 denotes the unit disk. If the angular velocity alpha of the rotation is larger than 1, this leads to a semilinear boundary value problem on textbf{B} involving a mixed-type operator, whose spectrum is related to the zeros of Bessel functions and could generally be badly behaved. Based on new estimates for these zeros, we find values of alpha such that the spectrum only consists of eigenvalues with finite multiplicity and has no accumulation point. Combined with suitable spectral estimates, this allows us to formulate an appropriate indefinite variational setting and find ground state solutions of the reduced equation for p in (2,4). Using a minimax characterization of the ground state energy, we ultimately show that these ground states are nonradial and thus yield nontrivial rotating waves, provided m is sufficiently large.

Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder

Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.

The domain derivative for semilinear elliptic inverse obstacle problems

&lt;p style='text-indent:20px;'&gt;We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.&lt;/p&gt;

Optimization of boundary value problems for higher order differential inclusions and duality

The paper is mainly devoted to the theory of duality of boundary value problems (BVPs) for differential inclusions of higher orders. For this, on the basis of the apparatus of locally conjugate mappings in the form of Euler–Lagrange-type inclusions and transversality conditions, sufficient optimality conditions are obtained. Wherein remarkable is the fact that inclusions of Euler–Lagrange type for prime and dual problems are “duality relations”. To demonstrate this approach, the optimization of some third-order semilinear BVPs and polyhedral fourth-order BVPs is considered. These problems show that sufficient conditions and dual problems can be easily established for problems of any order.

Semi-linear impulsive higher order boundary value problems

Semi-linear impulsive higher order boundary value problems

On the Existence of Three Solutions for Some Classes of Two-Point Semi-linear and Quasi-linear Differential Equations

A general theorem concerning the three critical points for some classes of coercive functionals depending on a real parameter is established, which may derive existence’s results of three solutions with various sufficient conditions for some classes of two-point semi-linear boundary value problems. Moreover, by applying known three existence theorems, we derive multiple existence results for a class of quasi-linear differential equation.

Existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem

This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles.&#x0D; &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; doi:10.1017/S1446181119000129

EXISTENCE AND UNIQUENESS OF WEAK AND CLASSICAL SOLUTIONS FOR A FOURTH-ORDER SEMILINEAR BOUNDARY VALUE PROBLEM

This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles.

Maximum norm error estimates for Neumann boundary value problems on graded meshes

AbstractThis paper deals with a priori pointwise error estimates for the finite element solution of boundary value problems with Neumann boundary conditions in polygonal domains. Due to the corners of the domain, the convergence rate of the numerical solutions can be lower than in the case of smooth domains. As a remedy, the use of local mesh refinement near the corners is considered. In order to prove quasi-optimal a priori error estimates, regularity results in weighted Sobolev spaces are exploited. This is the first work on the Neumann boundary value problem where both the regularity of the data is exactly specified and the sharp convergence order $h^{2} \lvert \ln h \rvert $ in the case of piecewise linear finite element approximations is obtained. As an extension we show the same rate for the approximate solution of a semilinear boundary value problem. The proof relies in this case on the supercloseness between the Ritz projection to the continuous solution and the finite element solution.

Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain

Abstract A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞(ε). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε → 0. Namely, we derive the limit problem (ε = 0) in the corresponding graph, define other terms of the asymptotic approximation and prove energetic and uniform pointwise estimates. These estimates allow us to observe the impact of the aneurysm on some properties of the solution.

Semilinear Hyperbolic Boundary Value Problem Associated to the Nonlinear Generalized Viscoelastic Equations

Consider a semilinear hyperbolic boundary value problem associated to the nonlinear generalized viscoelastic equations with Direchlet-Neumann boundary conditions. Then, the global existence of a weak solution is established. The uniqueness of the solution has been obtained by eliminating some hypotheses that have been imposed by other authors for different particular problems.

Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations

In this article, we investigate the application of pseudo‐transient‐continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, use the PTC‐methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction‐type PTC‐method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC ‐Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2005–2022, 2017

Nonlocal semi-linear fractional-order boundary value problems with strip conditions

This paper is concerned with the question of existence of solutions for one-dimensional higher-order semi-linear fractional differential equations supplemented with nonlocal strip type boundary conditions. The nonlocal strip condition addresses a situation where the linear combination of the values of unknown function at two nonlocal points, located to the left and right hand sides of the strip, respectively, is proportional to its strip value. The case of Stieltjes type strip condition is also discussed. Our results, relying on some standard fixed point theorems are supported with illustrative examples.

Positivity for fourth-order semilinear problems related to the Kirchhoff–Love functional

We study the ground states of the following generalization of the Kirchhoff-Love functional, $$J_\sigma(u)=\int_\Omega\dfrac{(\Delta u)^2}{2} - (1-\sigma)\int_\Omega det(\nabla^2u)-\int_\Omega F(x,u),$$ where $\Omega$ is a bounded convex domain in $\mathbb{R}^2$ with $C^{1,1}$ boundary and the nonlinearities involved are of sublinear type or superlinear with power growth. These critical points correspond to least-energy weak solutions to a fourth-order semilinear boundary value problem with Steklov boundary conditions depending on $\sigma$. Positivity of ground states is proved with different techniques according to the range of the parameter $\sigma\in\mathbb{R}$ and we also provide a convergence analysis for the ground states with respect to $\sigma$. Further results concerning positive radial solutions are established when the domain is a ball.

Singularly perturbed third-order semilinear boundary value problems with discontinuous coefficients

Singularly perturbed third-order semilinear boundary value problems with discontinuous coefficients

A NUMERICAL STUDY ON THE DIFFERENCE SOLUTION OF SINGULARLY PERTURBED SEMILINEAR PROBLEM WITH INTEGRAL BOUNDARY CONDITION

The present study is concerned with the numerical solution, using finite difference method on a piecewise uniform mesh (Shishkin type mesh) for a singularly perturbed semilinear boundary value problem with integral boundary condition. First we discuss the nature of the continuous solution of singularly perturbed differential problem before presenting method for its numerical solution. The numerical method is constructed on piecewise uniform Shishkin type mesh. We show that the method is first-order convergent in the discrete maximum norm, independently of singular perturbation parameter except for a logarithmic factor. We give effective iterative algorithm for solving the nonlinear difference problem. Numerical results which support the given estimates are presented.

Bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws

This paper is devoted to the study of semilinear degenerate elliptic boundary value problems arising in combustion theory that obey a general Arrhenius equation and a general Newton law of heat exchange. Our degenerate boundary conditions include as particular cases the isothermal condition (Dirichlet condition) and the adiabatic condition (Neumann condition). We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production. More precisely, we give sufficient conditions for our semilinear boundary value problems to have three positive solutions, which suggests that the bifurcation curves are S-shaped.

A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

We are considering a semilinear singular perturbation reaction-diffusion boundary value problem which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is \(\epsilon\)-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.

Newton’s method and a mesh-independence principle for certain semilinear boundary-value problems

We exhibit an algorithm which computes an ϵ-approximation of the positive solutions of a family of boundary-value problems with Neumann boundary conditions. Such solutions arise as the stationary solutions of a family of semilinear parabolic equations with Neumann boundary conditions. The algorithm is based on a finite-dimensional Newton iteration associated with a suitable discretized version of the problem under consideration. To determine the behavior of such a discrete iteration we establish an explicit mesh-independence principle. We apply a homotopy-continuation algorithm to compute a starting point of the discrete Newton iteration, and the discrete Newton iteration until an ϵ-approximation of the stationary solution is obtained. The algorithm performs roughly O((1/ϵ)1/2) flops and function evaluations.