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.

scholarly journals On fractional higher-order Dirichlet boundary value problems: Between the Laplacian and the bilaplacian

2021 ◽  
pp. 255-277
Author(s):  
Alberto Saldaña
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Natural Extension ◽  
Higher Order ◽  
Dirichlet Boundary Conditions ◽  
Maximum Principles ◽  
Limiting Behavior

The solutions of boundary value problems for the Laplacian and the bilaplacian exhibit very different qualitative behaviors. Particularly, the failure of general maximum principles for the bilaplacian implies that solutions of higher-order problems are less rigid and more complex. One way to better understand this transition is to study the intermediate Dirichlet problem in terms of fractional Laplacians. This survey aims to be an introduction to this type of problems; in particular, the different pointwise notions for these operators is introduced considering a suitable natural extension of the Dirichlet boundary conditions for the fractional setting. Solutions are obtained variationally and, in the case of the ball, via explicit kernels. The validity of maximum principles for these intermediate problems is also discussed as well as the limiting behavior of solutions when approaching the Laplacian or the bilaplacian case.

scholarly journals On solvability of elliptic boundary value problems via global invertibility

2020 ◽  
Vol 40 (1) ◽  
pp. 37-47
Author(s):  
Michał Bełdziński ◽  
Marek Galewski
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Dirichlet Boundary Conditions ◽  
Elliptic Boundary Value ◽  
Global Invertibility

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.

scholarly journals Modification of Lesnic’s Approach and New Analytic Solutions for Some Nonlinear Second-Order Boundary Value Problems with Dirichlet Boundary Conditions

2010 ◽  
Vol 65 (8-9) ◽  
pp. 692-696
Author(s):  
Abdelhalim Ebaid
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Analytic Solutions ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Boundary Value Problems ◽  
Troesch’S Problem

For solving nonlinear boundary value problems (BVPs), a main difficulty of using Adomian’s method is to find a canonical form which takes into account all the boundary conditions of the problem. This difficulty is overcome by using a modification for Lesnic’s approach developed in this paper. The effectiveness of the proposed procedure is verified by two nonlinear problems: the nonlinear oscillator equation and Troesch’s problem

scholarly journals ON SHOOTING AND FINITE DIFFERENCE METHODS FOR NON-LINEAR TWO POINT BOUNDARY VALUE PROBLEMS

2018 ◽  
Vol 6 (1) ◽  
pp. 23-35
Author(s):  
Ibrahim I. O. ◽  
Markus S.
Keyword(s):  
Finite Difference ◽  
Boundary Value Problems ◽  
Shooting Method ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Dirichlet Boundary Conditions ◽  
Point Boundary ◽  
Non Linear ◽  
Difference Methods

The paper investigates the efficacy of non-linear two point boundary value problems via shooting and finite difference methods. It was observed that the shooting method provides better result as when compared to the finite difference methods with dirichlet boundary conditions. It was observed that the accuracy of the shooting method is dependent upon the integrator adopted.

Wavelet Based Solution to Time-Dependent Higher Order Non-Linear Two-Point Initial Boundary Value Problems With Non-Periodic Boundary Conditions: KdV, Boussinesq Equations

2003 ◽  
Author(s):  
H. N. Narang ◽  
Rajiv K. Nekkanti
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Initial Boundary ◽  
Time Dependent ◽  
Initial Boundary Value Problems ◽  
Non Linear ◽  
Initial Boundary Value

The Wavelet solution for boundary-value problems is relatively new and has been mainly restricted to the solutions in data compression, image processing and recently to the solution of differential equations with periodic boundary conditions. This paper is concerned with the wavelet-based Galerkin’s solution to time dependent higher order non-linear two-point initial-boundary-value problems with non-periodic boundary conditions. The wavelet method can offer several advantages in solving the initial-boundary-value problems than the traditional methods such as Fourier series, Finite Differences and Finite Elements by reducing the computational time near singularities because of its multi-resolution character. In order to demonstrate the wavelet, we extend our prior research of solution to parabolic equations and problems with non-linear boundary conditions to non-linear problems involving KdV Equation and Boussinesq Equation. The results of the wavelet solutions are examined and they are found to compare favorably to the known solution. This paper on the whole indicates that the wavelet technique is a strong contender for solving partial differential equations with non-periodic conditions.

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