scholarly journals On Oscillation Properties of Self-Adjoint Boundary Value Problems of Fourth Order

2021 ◽  
Author(s):  
A. A. Vladimirov ◽  
A. A. Shkalikov
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Nontrivial Solutions ◽  
New Approach ◽  
Oscillatory Problem ◽  
Derivatives Of ◽  
Inertia Index ◽  
Oscillation Properties

Abstract The connection between the number of internal zeros of nontrivial solutions to fourth-order self-adjoint boundary value problems and the inertia index of these problems is studied. We specify the types of problems for which such a connection can be established. In addition, we specify the types of problems for which a connection between the inertia index and the number of internal zeros of the derivatives of nontrivial solutions can be established. Examples demonstrating the effectiveness of the proposed new approach to an oscillatory problem are considered.

A Superconvergent Local Discontinuous Galerkin Method for Nonlinear Fourth-Order Boundary-Value Problems

2019 ◽  
Vol 17 (07) ◽  
pp. 1950035 ◽  
Author(s):  
Mahboub Baccouch
Keyword(s):  
Exact Solutions ◽  
Boundary Value Problems ◽  
Discontinuous Galerkin ◽  
Fourth Order ◽  
Boundary Value ◽  
Local Discontinuous Galerkin Method ◽  
Rates Of Convergence ◽  
Piecewise Polynomials ◽  
Local Discontinuous Galerkin ◽  
Derivatives Of

In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form [Formula: see text]. We prove optimal [Formula: see text] error estimates for the solution and for the three auxiliary variables that approximate the first, second, and third-order derivatives. The order of convergence is proved to be [Formula: see text], when piecewise polynomials of degree at most [Formula: see text] are used. Our numerical experiments demonstrate optimal rates of convergence. We further prove that the derivatives of the LDG solutions are superconvergent with order [Formula: see text] toward the derivatives of Gauss–Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order [Formula: see text] toward Gauss–Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is [Formula: see text]. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree [Formula: see text] and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

scholarly journals Existence and Nonexistence of Solutions for Fourth-Order Nonlinear Difference Boundary Value Problems via Variational Methods

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Xia Liu ◽  
Tao Zhou ◽  
Haiping Shi
Keyword(s):  
Variational Methods ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Nonlinear Difference Equation ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions

This paper is concerned with boundary value problems for a fourth-order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence ofndistinct pairs of nontrivial solutions, and nonexistence of solutions. Some examples are provided to show the effectiveness of the main results.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

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