scholarly journals A Regularization Method for the Elliptic Equation with Inhomogeneous Source

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Tuan H. Nguyen ◽  
Binh Thanh Tran
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Regularization Method ◽  
Dirichlet Boundary ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Ill Posed ◽  
Sharp Error

We consider the following Cauchy problem for the elliptic equation with inhomogeneous source in a rectangular domain with Dirichlet boundary conditions at x=0 and x=π. The problem is ill-posed. The main aim of this paper is to introduce a regularization method and use it to solve the problem. Some sharp error estimates between the exact solution and its regularization approximation are given and a numerical example shows that the method works effectively.

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

scholarly journals HP estimation for the Cauchy problem for nonlinear elliptic equation

2018 ◽  
Vol 1 (T5) ◽  
pp. 193-202
Author(s):  
Thang Duc Le
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Bounded Domain ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Regularized Solution

In this paper, we investigate the Cauchy problem for a ND nonlinear elliptic equation in a bounded domain. As we know, the problem is severely ill-posed. We apply the Fourier truncation method to regularize the problem. Error estimates between the regularized solution and the exact solution are established in Hp space under some priori assumptions on the exact solution.

A mollification method with Dirichlet kernel to solve Cauchy problem for two-dimensional Helmholtz equation

2019 ◽  
Vol 17 (05) ◽  
pp. 1950029 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Numerical Experiment ◽  
Helmholtz Equation ◽  
Regularization Method ◽  
Dirichlet Kernel ◽  
Two Dimensional ◽  
Strip Domain ◽  
Ill Posed ◽  
Mollification Method

In this paper, the ill-posed Cauchy problem for the Helmholtz equation is investigated in a strip domain. To obtain stable numerical solution, a mollification regularization method with Dirichlet kernel is proposed. Error estimate between the exact solution and its approximation is given. A numerical experiment of interest shows that our procedure is effective and stable with respect to perturbations of noise in the data.

scholarly journals Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 705 ◽  
Author(s):  
Fan Yang ◽  
Ping Fan ◽  
Xiao-Xiao Li
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Regularization Method ◽  
Three Dimensional ◽  
Wave Number ◽  
Modified Helmholtz Equation ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Regularized Solution

In this paper, the Cauchy problem of the modified Helmholtz equation (CPMHE) with perturbed wave number is considered. In the sense of Hadamard, this problem is severely ill-posed. The Fourier truncation regularization method is used to solve this Cauchy problem. Meanwhile, the corresponding error estimate between the exact solution and the regularized solution is obtained. A numerical example is presented to illustrate the validity and effectiveness of our methods.

Multiplicity and Existence Results for a Nonlinear Elliptic Equation With Sobolev Exponent

2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Zakaria Bouchech ◽  
Hichem Chtioui
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Nonlinear Elliptic Equation ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic ◽  
Sobolev Exponent

AbstractIn this paper we consider the following nonlinear elliptic equation with Dirichlet boundary conditions: -Δu = K(x)u

Dual fat boundary method: the fat boundary method in elasticity with an extension of the application scope

2020 ◽  
pp. 108128652096449
Author(s):  
Kui Liu ◽  
Ang Zhao ◽  
Zhendong Hu
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Dirichlet Boundary ◽  
Mathematical Proof ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Stress Concentrations ◽  
Conditional Convergence ◽  
Boundary Method ◽  
Wide Range

The fat boundary method (FBM) is a fictitious domain method, proposed to solve Poisson problems in a domain with small perforations. It can achieve higher accuracy around holes, which makes it very suitable to solve elasticity problems because stress concentrations often appear around holes. However, there are some strict restrictions of the FBM limiting the wide range of applications. For example, the original FBM deals with perforated rectangular domain with only Dirichlet boundary conditions. Furthermore, because the global domain is extended to the holes, analytical solutions in holes corresponding to the Dirichlet boundary conditions around holes are required. This limits both the boundary conditions around holes and the shape of holes, because for arbitrary holes it is difficult to get the analytical solutions. This article makes an attempt to break these limitations and apply the FBM to elasticity. Firstly, we review the FBM and introduce Neumann boundary conditions to the rectangular domain. A mathematical proof of the conditional convergence of the algorithm is presented. Furthermore, the FBM is compared with the Lagrange multiplier method to clarify that the FBM is one kind of weak imposition method. Then we apply the FBM to linear elasticity and the dual fat boundary method is proposed to solve problems without analytical solutions in holes. Some numerical examples are presented to verify the method proposed here.

scholarly journals A uniqueness result for some singular semilinear elliptic equations

2016 ◽  
Vol 18 (06) ◽  
pp. 1550084 ◽  
Author(s):  
Annamaria Canino ◽  
Berardino Sciunzi
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Open Subset ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Uniqueness Result ◽  
Dirichlet Boundary Conditions ◽  
Semilinear Elliptic ◽  
Bounded Open Subset

Given [Formula: see text] a bounded open subset of [Formula: see text], we consider non-negative solutions to the singular semilinear elliptic equation [Formula: see text] in [Formula: see text], under zero Dirichlet boundary conditions. For [Formula: see text] and [Formula: see text], we prove that the solution is unique.

scholarly journals Two regularized solutions of an ill-posed problem for the elliptic equation with inhomogeneous source

Filomat ◽  
2014 ◽  
Vol 28 (10) ◽  
pp. 2091-2110 ◽  
Author(s):  
Huy Tuan ◽  
Tran Binh
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Numerical Examples ◽  
Source Data ◽  
Regularized Methods ◽  
Ill Posed ◽  
Convergence Estimates ◽  
The Given

In this paper, we address a Cauchy problem for elliptic equations with inhomogeneous source data. The problem is shown to be ill-posed as the solution exhibits an unstable dependence on the given data functions. Here, we shall deal with this problem by using two different regularized methods. Moreover, convergence estimates are established under some priori assumptions on the exact solution. Some numerical examples are given to illuminate the effect of our methods.

scholarly journals A Lyapunov-Type Inequality for a Laplacian System on a Rectangular Domain with Zero Dirichlet Boundary Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 850 ◽  
Author(s):  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Type Inequality ◽  
Coupled System ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Necessary Condition ◽  
Laplacian Operator ◽  
Lyapunov Type Inequality

We consider a coupled system of partial differential equations involving Laplacian operator, on a rectangular domain with zero Dirichlet boundary conditions. A Lyapunov-type inequality related to this problem is derived. This inequality provides a necessary condition for the existence of nontrivial positive solutions.

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