scholarly journals Accurate Estimations of Any Eigenpairs of N-th Order Linear Boundary Value Problems

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2663
Author(s):  
Pedro Almenar ◽  
Lucas Jódar
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Linear Boundary ◽  
Point Boundary ◽  
Eigenvalues And Eigenfunctions ◽  
Order Linear ◽  
Integral Problem ◽  
Selection Of

This paper provides a method to bound and calculate any eigenvalues and eigenfunctions of n-th order boundary value problems with sign-regular kernels subject to two-point boundary conditions. The method is based on the selection of a particular type of cone for each eigenpair to be determined, the recursive application of the operator associated to the equivalent integral problem to functions belonging to such a cone, and the calculation of the Collatz–Wielandt numbers of the resulting functions.

scholarly journals The principal eigenvalue of some nth order linear boundary value problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pedro Almenar ◽  
Lucas Jódar
Keyword(s):  
Integral Operator ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Principal Eigenvalue ◽  
Green Functions ◽  
Linear Boundary ◽  
Order Linear

AbstractThe purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasi-derivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz–Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided.

scholarly journals Fast Approximation of Over-Determined Second-Order Linear Boundary Value Problems by Cubic and Quintic Spline Collocation

Robotics ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 48
Author(s):  
Philipp Seiwald ◽  
Daniel J. Rixen
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Test System ◽  
Second Order ◽  
Linear Boundary ◽  
Spline Collocation ◽  
Initial State ◽  
Quintic Spline ◽  
Order Linear

We present an efficient and generic algorithm for approximating second-order linear boundary value problems through spline collocation. In contrast to the majority of other approaches, our algorithm is designed for over-determined problems. These typically occur in control theory, where a system, e.g., a robot, should be transferred from a certain initial state to a desired target state while respecting characteristic system dynamics. Our method uses polynomials of maximum degree three/five as base functions and generates a cubic/quintic spline, which is C 2 / C 4 continuous and satisfies the underlying ordinary differential equation at user-defined collocation sites. Moreover, the approximation is forced to fulfill an over-determined set of two-point boundary conditions, which are specified by the given control problem. The algorithm is suitable for time-critical applications, where accuracy only plays a secondary role. For consistent boundary conditions, we experimentally validate convergence towards the analytic solution, while for inconsistent boundary conditions our algorithm is still able to find a “reasonable” approximation. However, to avoid divergence, collocation sites have to be appropriately chosen. The proposed scheme is evaluated experimentally through comparison with the analytical solution of a simple test system. Furthermore, a fully documented C++ implementation with unit tests as example applications is provided.

scholarly journals Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ji Lin ◽  
Yuhui Zhang ◽  
Chein-Shan Liu
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Initial Conditions ◽  
Boundary Value ◽  
Accurate Solution ◽  
Point Boundary ◽  
Third Order ◽  
Boundary Shape ◽  
Free Function ◽  
New Algorithms

AbstractFor nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

Perturbation of two-point boundary value problems with eigenvalue parameter in the boundary conditions

1983 ◽  
Vol 94 (3-4) ◽  
pp. 213-219 ◽  
Author(s):  
Lawrence Turyn
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Banach Spaces ◽  
Boundary Value ◽  
Simple Eigenvalue ◽  
Point Boundary ◽  
Eigenvalue Parameter

SynopsisWe discuss smooth changes of eigenvalues under perturbation of the boundary value problems given in the title. The simple eigenvalue criterion is developed in the setting of Banach spaces, so very general perturbations of both the differential equation and the boundary conditions are allowed. Further, we need no assumptions about self-adjointness of the original or perturbed problems. The discussion is concluded with the application of the simple eigenvalue criterion to two examples.

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