  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.
