Galevkin Method for Autonomous Differential Equations

As for the periodic differential equations, M. Urabe [8] developed Galerkin method for numerical analysis of periodic solution. But, in the autonomous cases, the period of periodic solution is also unknown. Hence, how to deal with the unknown period is a problem. In the previous papers [4], [5], the author has proposed a Galerkin method for calculating the periodic solution and its period simultaneously to autonomous cases by making use of a boundary value problem. It is clear that, when x(t) is a solution of autonomous differential equation x(t+α) is also a solution for an arbitrary constant α. The fact tells us the Galerkin approximation to x(t) is not uniquely determined by the periodic boundary condition alone. Hence, in order to determine the Galerkin approximation uniquely, the author considered an additional linear functional and gave a rule how to choose the linear functional. In the present paper we shall give a mathematical foundation to the Galerkin method for autonomous differential equations, similar to the one for periodic cases given by M. Urabe [8], and summarize our results obtained in the previous papers [4], [5], [12]. It is worth stressing that, in autonomous cases, the quantity L (m) appeared in the inequalities (5.30) and (5.36) may vanish just as in periodic cases if we choose as l(u)=∫^<2π>_0 x(t)-cos pt dt (p@pre;m) the additional linear functional.

公開日:2010年1月24日で登録したコンテンツは、国立情報学研究所において電子化したものです。