scholarly journals Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

2020 ◽  
Author(s):  
Pauline Achieng ◽  
Fredrik Berntsson ◽  
Jennifer Chepkorir ◽  
Vladimir Kozlov
Keyword(s):  
Cauchy Problem ◽  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Iterative Algorithm ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
General Elliptic ◽  
The Cauchy Problem

Abstract The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers $$k^2$$ k 2 , in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of $$k^2$$ k 2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

Blowup properties for several diffusion systems with localised sources

2006 ◽  
Vol 48 (1) ◽  
pp. 37-56 ◽  
Author(s):  
Zhaoyin Xiang ◽  
Qiong Chen ◽  
Chunlai Mu
Keyword(s):  
Cauchy Problem ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Parabolic Systems ◽  
Neumann Boundary Conditions ◽  
Nonlocal Problems ◽  
Blowup Rate ◽  
Diffusion Systems ◽  
Blowup Criterion ◽  
The Cauchy Problem

AbstractThis paper investigates the Cauchy problem for two classes of parabolic systems with localised sources. We first give the blowup criterion, and then deal with the possibilities of simultaneous blowup or non-simultaneous blowup under some suitable assumptions. Moreover, when simultaneous blowup occurs, we also establish precise blowup rate estimates. Finally, using similar ideas and methods, we shall consider several nonlocal problems with homogeneous Neumann boundary conditions.

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone

1992 ◽  
Vol 122 (1-2) ◽  
pp. 137-160
Author(s):  
Chie-Ping Chu ◽  
Hwai-Chiuan Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Semilinear Elliptic Equations ◽  
Spherically Symmetric ◽  
Semilinear Elliptic ◽  
Symmetry Properties

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

Solutions for the microsensor thermistor equations in the small bias case

1993 ◽  
Vol 123 (6) ◽  
pp. 987-999 ◽  
Author(s):  
W. Allegretto ◽  
H. Xie
Keyword(s):  
Banach Space ◽  
Boundary Conditions ◽  
Implicit Function Theorem ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Implicit Function ◽  
Applied Bias

SynopsisThe behaviour of a microsensor thermistor is described by a system of nonlinear coupled elliptic equations subject to mixed Dirichlet-Neumann boundary conditions, to be solved on different domains. We employ the Implicit Function Theorem in Banach space to show that the system has a solution for small applied bias. It does not appear that earlier approaches for similar thermistor problems can be employed in this physically important situation. The fact that the problem is cast in a subset of R3 is significant in our presentation.

scholarly journals Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions

2019 ◽  
Vol 39 (2) ◽  
pp. 159-174 ◽  
Author(s):  
Gabriele Bonanno ◽  
Giuseppina D'Aguì ◽  
Angela Sciammetta
Keyword(s):  
Boundary Conditions ◽  
Elliptic Equations ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Mixed Boundary Conditions ◽  
Neumann Boundary Conditions ◽  
Laplacian Operator ◽  
Differential Problem ◽  
Nonlinear Elliptic

In this paper, a nonlinear differential problem involving the \(p\)-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.

scholarly journals A Collocation Method for Numerical Solution of Hyperbolic Telegraph Equation with Neumann Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. C. Mittal ◽  
Rachna Bhatia
Keyword(s):  
Boundary Conditions ◽  
Numerical Experiments ◽  
Neumann Boundary ◽  
Telegraph Equation ◽  
Neumann Boundary Conditions ◽  
Basis Functions ◽  
One Dimensional ◽  
First Order ◽  
Spline Basis ◽  
Good Agreement

We present a technique based on collocation of cubic B-spline basis functions to solve second order one-dimensional hyperbolic telegraph equation with Neumann boundary conditions. The use of cubic B-spline basis functions for spatial variable and its derivatives reduces the problem into system of first order ordinary differential equations. The resulting system subsequently has been solved by SSP-RK54 scheme. The accuracy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and in good agreement with the exact solution.

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