TY - JOUR

T1 - A Matrix Framework for the Solution of ODEs

T2 - Initial-, Boundary-, and Inner-Value Problems

AU - Harker, Matthew

AU - O'Leary, Paul

PY - 2013/4/11

Y1 - 2013/4/11

N2 - A matrix framework is presented for the solution of ODEs, including initial-, boundary and inner-value problems. The framework enables the solution of the ODEs for arbitrary nodes. There are four key issues involved in the formulation of the framework: the use of a Lanczos process with complete reorthogonalization for the synthesis of discrete orthonormal polynomials (DOP) orthogonal over arbitrary nodes within the unit circle on the complex plane; a consistent definition of a local differentiating matrix which implements a uniform degree of approximation over the complete support --- this is particularly important for initial and boundary value problems; a method of computing a set of constraints as a constraining matrix and a method to generate orthonormal admissible functions from the constraints and a DOP matrix; the formulation of the solution to the ODEs as a least squares problem. The computation of the solution is a direct matrix method. The worst case maximum number of computations required to obtain the solution is known a-priori. This makes the method, by definition, suitable for real-time applications. The functionality of the framework is demonstrated using a selection of initial value problems, Sturm-Liouville problems and a classical Engineering boundary value problem. The framework is, however, generally formulated and is applicable to countless differential equation problems.

AB - A matrix framework is presented for the solution of ODEs, including initial-, boundary and inner-value problems. The framework enables the solution of the ODEs for arbitrary nodes. There are four key issues involved in the formulation of the framework: the use of a Lanczos process with complete reorthogonalization for the synthesis of discrete orthonormal polynomials (DOP) orthogonal over arbitrary nodes within the unit circle on the complex plane; a consistent definition of a local differentiating matrix which implements a uniform degree of approximation over the complete support --- this is particularly important for initial and boundary value problems; a method of computing a set of constraints as a constraining matrix and a method to generate orthonormal admissible functions from the constraints and a DOP matrix; the formulation of the solution to the ODEs as a least squares problem. The computation of the solution is a direct matrix method. The worst case maximum number of computations required to obtain the solution is known a-priori. This makes the method, by definition, suitable for real-time applications. The functionality of the framework is demonstrated using a selection of initial value problems, Sturm-Liouville problems and a classical Engineering boundary value problem. The framework is, however, generally formulated and is applicable to countless differential equation problems.

KW - math.NA

KW - 15B02, 30E25, 65L60, 65L10, 65L15, 65L80

M3 - Article

JO - Journal of Mathematical Imaging and Vision

JF - Journal of Mathematical Imaging and Vision

SN - 0924-9907

ER -