scholarly journals Reflection Method for Mathematical Modeling of Potential Fields in Multi-Layer Media

2019 ◽  
Vol 224 ◽  
pp. 01004
Author(s):  
Oleg Yaremko ◽  
Natalia Yaremko
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Field Potential ◽  
Computer Code ◽  
Potential Fields ◽  
Reflection Method ◽  
Dirichlet Problems ◽  
Thermal Fields ◽  
Dirichlet Model ◽  
Fourth Kind

In this paper, potential fields in areas with plane and circular symmetry have been studied. In this case, the field potential is defined as the sum of solutions of Dirichlet model boundary value problems. The reflection method is used for the modeling of stationary thermal fields in multilayer media. By applying the reflection method, we found analytical solutions of boundary value problems with boundary conditions of the fourth kind for the Laplace equations and developed new computational algorithms. The developed algorithms can be easily implemented and transformed into a computer code. First of all, these algorithms implement consistent solutions of the Dirichlet problems for the model domains that allows using libraries of subroutines. Secondly, they have high algorithmic efficiency. It has been shown that reflection method is identical to the method of transformation operators and proved that transformation operator can be decomposed into a series of successive reflections from the external and internal boundaries. Finally, a physical interpretation of the reflection method has been discussed in detail.

scholarly journals Stability of Non-Linear Dirichlet Problems with ϕ-Laplacian

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 647
Author(s):  
Michał Bełdziński ◽  
Marek Galewski ◽  
Igor Kossowski
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Dirichlet Boundary Conditions ◽  
Laplacian Operator ◽  
Dirichlet Problems ◽  
Non Linear ◽  
The Stability

We study the stability and the solvability of a family of problems −(ϕ(x′))′=g(t,x,x′,u)+f* with Dirichlet boundary conditions, where ϕ, u, f* are allowed to vary as well. Applications for boundary value problems involving the p-Laplacian operator are highlighted.

On variational impulsive boundary value problems

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Marek Galewski
Keyword(s):  
Boundary Value Problems ◽  
Variational Approach ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Classical Solutions ◽  
Dirichlet Problems ◽  
Functional Parameters ◽  
Impulsive Boundary Value Problems ◽  
Right Hand

AbstractUsing the variational approach, we investigate the existence of solutions and their dependence on functional parameters for classical solutions to the second order impulsive boundary value Dirichlet problems with L1 right hand side.

scholarly journals Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains

1995 ◽  
Vol 125 (1) ◽  
pp. 99-114 ◽  
Author(s):  
Gianni Dal Maso ◽  
Annalisa Malusa
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Radon Measure ◽  
Boundary Value ◽  
Approximation Procedure ◽  
Dirichlet Problems ◽  
Perforated Domains ◽  
Positive Radon Measure

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

scholarly journals New Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value Problems Using Chebyshev Polynomials of Third and Fourth Kinds

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
E. H. Doha ◽  
Y. H. Youssri
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
Multipoint Boundary ◽  
Nonlinear Algebraic Equations ◽  
Multipoint Boundary Value Problems ◽  
Fourth Kind

This paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinear multipoint boundary value problems. The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth-kind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. The obtained numerical results are comparing favorably with the analytical known solutions.

scholarly journals Boundary value problems of singular elliptic partial differential equations

1972 ◽  
Vol 13 (2) ◽  
pp. 111-118 ◽  
Author(s):  
Chi Yeung Lo
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Axially Symmetric ◽  
Dirichlet Problems ◽  
The Neumann Problem ◽  
Image Position ◽  
Generalized Axially Symmetric Potentials

In a recent paper [6], this author has extended the method of the kernel function [1] to the boundary value problems of the generalized axially symmetric potentialsThis method can also be applied to a more general class of singular differential equations, namelyor, equivalently,We shall derive in the sequel explicit formulas for the Dirichlet problems of (1.1) in the first quadrant of the x-y plane in terms of sufficiently smooth boundary data, and obtain an error-bound for their approximate solutions. We shall also indicate how the Neumann problem can be solved.

Sign in / Sign up

Export Citation Format

Share Document