scholarly journals On the Dirichlet Problem with Corner Singularity

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1870
Author(s):  
Viktor A. Rukavishnikov ◽  
Elena I. Rukavishnikova
Keyword(s):  
Approximate Solution ◽  
Dirichlet Problem ◽  
Numerical Methods ◽  
Elliptic Equation ◽  
Existence And Uniqueness ◽  
Generalized Solution ◽  
Corner Singularity

We consider the Dirichlet problem for an elliptic equation with a singularity. The singularity of the solution to the problem is caused by the presence of a re-entrant corner at the boundary of the domain. We define an Rν-generalized solution for this problem. This allows for the construction of numerical methods for finding an approximate solution without loss of accuracy. In this paper, the existence and uniqueness of the Rν-generalized solution in set W∘2,α1(Ω,δ) is proven. The Rν-generalized solution is the same for different parameters ν.

scholarly journals On the Existence and Uniqueness ofRv-Generalized Solution for Dirichlet Problem with Singularity on All Boundary

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
V. Rukavishnikov ◽  
E. Rukavishnikova
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Input Data ◽  
Generalized Solution ◽  
Two Dimensional ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation ◽  
Dimensional Domain

The existence and uniqueness of theRv-generalized solution for the first boundary value problem and a second order elliptic equation with coordinated and uncoordinated degeneracy of input data and with strong singularity solution on all boundary of a two-dimensional domain are established.

scholarly journals The Dirichlet Problem for the Perturbed Elliptic Equation

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2108 ◽  
Author(s):  
Ulyana Yarka ◽  
Solomiia Fedushko ◽  
Peter Veselý
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Riesz Basis ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Fourier Method ◽  
Eigenvalues And Eigenfunctions ◽  
Uniqueness Of The Solution ◽  
Basis System

In this paper, the authors consider the construction of one class of perturbed problems to the Dirichlet problem for the elliptic equation. The operators of both problems are isospectral, which makes it possible to construct solutions to the perturbed problem using the Fourier method. This article focuses on the Dirichlet problem for the elliptic equation perturbed by the selected variable. We established the spectral properties of the perturbed operator. In this work, we found the eigenvalues and eigenfunctions of the perturbed task operator. Further, we proved the completeness, minimal spanning system, and Riesz basis system of eigenfunctions of the perturbed operator. Finally, we proved the theorem on the existence and uniqueness of the solution to the boundary value problem for a perturbed elliptic equation.

scholarly journals Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 804
Author(s):  
Ioannis K. Argyros ◽  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Approximate Solution ◽  
Convergence Analysis ◽  
Banach Spaces ◽  
Recurrence Relation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Existence And Uniqueness ◽  
Numerical Illustration ◽  
Type Method ◽  
Local Convergence Analysis

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

scholarly journals Best Proximity Point for Generalized Proximal Weak Contractions in Complete Metric Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Erdal Karapınar ◽  
V. Pragadeeswarar ◽  
M. Marudai
Keyword(s):  
Approximate Solution ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Best Proximity Point ◽  
Complete Metric Space ◽  
Optimal Approximate Solution ◽  
Weak Contraction ◽  
Complete Metric Spaces ◽  
New Class

We introduce a new class of nonself-mappings, generalized proximal weak contraction mappings, and prove the existence and uniqueness of best proximity point for such mappings in the context of complete metric spaces. Moreover, we state an algorithm to determine such an optimal approximate solution designed as a best proximity point. We establish also an example to illustrate our main results. Our result provides an extension of the related results in the literature.

Method of corrections by higher order differences for elliptic equations with variable coefficients

2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Givi Berikelashvili ◽  
Bidzina Midodashvili
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Convergence Rate ◽  
Difference Scheme ◽  
Elliptic Equations ◽  
Variable Coefficients ◽  
Order Accuracy ◽  
Corrected Scheme

AbstractWe consider the Dirichlet problem for an elliptic equation with variable coefficients, the solution of which is obtained by means of a finite-difference scheme of second order accuracy. We establish a two-stage finite-difference method for the posed problem and obtain an estimate of the convergence rate consistent with the smoothness of the solution. It is proved that the solution of the corrected scheme converges at rate

scholarly journals Norm Comparison Estimates for the Composite Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xuexin Li ◽  
Yong Wang ◽  
Yuming Xing
Keyword(s):  
Elliptic Equation ◽  
Quasilinear Elliptic Equation ◽  
Differential Forms ◽  
Generalized Solution ◽  
Composite Operator ◽  
Potential Operator ◽  
Norm Estimates ◽  
Quasilinear Elliptic ◽  
Littlewood Maximal Operator ◽  
Norm Comparison

This paper obtains the Lipschitz and BMO norm estimates for the composite operator𝕄s∘Papplied to differential forms. Here,𝕄sis the Hardy-Littlewood maximal operator, andPis the potential operator. As applications, we obtain the norm estimates for the Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.

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