The Definition and Numerical Method of Final Value Problem and Arbitrary Value Problem

TL;DR

This paper introduces a comprehensive framework for final, arbitrary, and inner-interval value problems in ODEs, establishing their theoretical properties and developing stable numerical methods validated through experiments.

Contribution

It proposes the new concept of arbitrary value problems, extends to multivariate and high-order cases, and provides convergence and stability analysis for numerical solutions.

Findings

Existence and uniqueness of solutions proved for first-order problems.

Numerical methods with proven convergence and stability developed.

Validation through numerical experiments confirms effectiveness.

Abstract

Many Engineering Problems could be mathematically described by Final Value Problem, which is the inverse problem of Initial Value Problem. Accordingly, the paper studies the final value problem in the field of ODE problems and analyses the differences and relations between initial and final value problems. The more general new concept of the endpoints-value problem which could describe both initial and final problems is proposed. Further, we extend the concept into inner-interval value problem and arbitrary value problem and point out that both endpoints-value problem and inner-interval value problem are special forms of arbitrary value problem. Particularly, the existence and uniqueness of the solutions of final value problem and inner-interval value problem of first order ordinary differential equation are proved for discrete problems. The numerical calculation formulas of the problems are derived, and for each algorithm, we propose the convergence and stability conditions of them. Furthermore, multivariate and high-order final value problems are further studied, and the condition of fixed delay is also discussed in this paper. At last, the effectiveness of the considered methods is validated by numerical experiment.