scholarly journals An iterative method based on boundary integrals for elliptic Cauchy problems in semi-infinite domains

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
R. Chapko ◽  
B.T. Johansson
Keyword(s):  
Temperature Field ◽  
Iterative Method ◽  
Laplace Operator ◽  
Boundary Integral ◽  
Iteration Step ◽  
Integral Approach ◽  
Infinite Domains ◽  
Mixed Problems ◽  
Infinite Region ◽  
The Laplace Operator

In this study, we investigate the problem of reconstruction of a stationary temperature field from given temperature and heat flux on a part of the boundary of a semi-infinite region containing an inclusion. This situation can be modelled as a Cauchy problem for the Laplace operator and it is an ill-posed problem in the sense of Hadamard. We propose and investigate a Landweber-Fridman type iterative method, which preserve the (stationary) heat operator, for the stable reconstruction of the temperature field on the boundary of the inclusion. In each iteration step, mixed boundary value problems for the Laplace operator are solved in the semi-infinite region. Well-posedness of these problems is investigated and convergence of the procedures is discussed. For the numerical implementation of these mixed problems an efficient boundary integral method is proposed which is based on the indirect variant of the boundary integral approach. Using this approach the mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing that stable and accurate reconstructions of the temperature field on the boundary of the inclusion can be obtained also in the case of noisy data. These results are compared with those obtained with the alternating iterative method.

Uniqueness of solutions of improperly posed problems for singular ultrahyperbolic equations

1974 ◽  
Vol 18 (1) ◽  
pp. 97-103 ◽  
Author(s):  
Eutiquio C. Young
Keyword(s):  
Laplace Operator ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Neumann Problems ◽  
Mixed Problems ◽  
Ultrahyperbolic Equation ◽  
Summation Convention ◽  
Image Position ◽  
Necessary And Sufficient ◽  
The Laplace Operator

In [1], Owen gave sufficient conditions for the uniqueness of certain mixed problems having elliptic and hyperbolic nature for the ultrahyperbolic equation. Recently, Diaz and Young [2] has obtained necessary and sufficient conditions for the uniqueness of solutions of the Dirichlet and Neumann problems involving the more general ultrahyperbolic equation The purpose of this paper is to present corresponding uniqueness conditions for the Dirichlet and Neumann problems for the singular ultrahyperbolic equation for all values of the parameter, α −∞ > α > ∞. The symbol Δ denotes the Laplace operator in the variables x1, …, xm, Dj indicates partial differentiation with respect to the variable yj (1 ≧ J ≧ n), and the summation convention is adopted for repeated indices including (iu2), where denotes differentiation with respect to the variable xi.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

scholarly journals A faster iterative method for solving temperature field of mass concrete with cooling pipes

2017 ◽  
Author(s):  
Keshi You ◽  
Feng Wang ◽  
Liujiang Wang ◽  
Zhongwei Zhao ◽  
Yun Liu
Keyword(s):  
Temperature Field ◽  
Iterative Method ◽  
Mass Concrete ◽  
Cooling Pipes

scholarly journals Classification of Some Special Types Ruled Surfaces in Simply Isotropic 3-Space

2017 ◽  
Vol 55 (1) ◽  
pp. 87-98 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Dae Won Yoon ◽  
Nural Yuksel
Keyword(s):  
Real Number ◽  
Laplace Operator ◽  
Fundamental Form ◽  
Three Dimensional ◽  
Ruled Surfaces ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
The Laplace Operator ◽  
First Fundamental Form

AbstractIn this paper, we classify two types ruled surfaces in the three dimensional simply isotropic space I13under the condition ∆xi= λixiwhere ∆ is the Laplace operator with respect to the first fundamental form and λ is a real number. We also give explicit forms of these surfaces.

scholarly journals Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Ihor Borachok ◽  
Roman Chapko ◽  
B. Tomas Johansson
Keyword(s):  
Cauchy Problem ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Single Layer ◽  
Normal Derivative ◽  
Boundary Integral ◽  
Iteration Step ◽  
3 Dimensional ◽  
Boundary Surfaces

AbstractWe consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert’s method [

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