## Table of Content Open Access

ARTICLE

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

Shixiong Wang1,∗, Jianhua He1, Chen Wang2, Xitong Li1

1 School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710129, China
2 Department of Computer Science, University College London, London WC1E 6BT, UK

* Corresponding Author: E-mail: Computer Systems Science and Engineering 2018, 33(5), 379-387. https://doi.org/10.32604/csse.2018.33.379

## Abstract

Many Engineering Problems could be mathematically described by FinalValue Problem, which is the inverse problem of InitialValue 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.

## Keywords

S. Wang, J. He, C. Wang and X. Li, "The definition and numerical method of final value problem and arbitrary value problem," Computer Systems Science and Engineering, vol. 33, no.5, pp. 379–387, 2018. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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