A Matrix Framework for the Solution of ODEs: Initial-, Boundary-, and Inner-Value Problems

TL;DR

This paper introduces a versatile matrix framework for solving various types of ODEs using orthonormal polynomials, enabling real-time solutions for complex problems with arbitrary nodes.

Contribution

It presents a novel matrix-based approach utilizing discrete orthonormal polynomials and least squares formulation for solving initial, boundary, and inner-value ODE problems.

Findings

Framework handles arbitrary nodes within the unit circle

Solution computation is a direct matrix method

Suitable for real-time differential equation solving

Abstract

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.