Quadratic/linear rational spline collocation for linear boundary value problems

2018 ◽  
Vol 125 ◽  
pp. 143-158 ◽  
Author(s):  
Erge Ideon ◽  
Peeter Oja
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Linear Boundary ◽  
Spline Collocation ◽  
Rational Spline

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.

A new method for calculating eigenvalues with applications to gravity-capillary waves with edge constraints

1983 ◽  
Vol 94 (3) ◽  
pp. 553-564 ◽  
Author(s):  
James Graham-Eagle
Keyword(s):  
Boundary Value Problems ◽  
Variational Formulation ◽  
Boundary Value ◽  
Capillary Waves ◽  
Original Study ◽  
New Method ◽  
Linear Boundary ◽  
Hydrodynamic Problem ◽  
Alternative Means ◽  
Standard Linear

The method to be described provides an alternative means of dealing with certain non-standard linear boundary-value problems. It is developed in several applications to the theory of gravity-capillary waves. The analysis is based on a variational formulation of the hydrodynamic problem, being motivated by and extending the original study by Benjamin and Scott [3].

scholarly journals Multi-point Taylor approximations in one-dimensional linear boundary value problems

2009 ◽  
Vol 207 (2) ◽  
pp. 519-527 ◽  
Author(s):  
José L. López ◽  
Ester Pérez Sinusía ◽  
Nico M. Temme
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Linear Boundary ◽  
One Dimensional
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