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)=∫^_0 x(t)-cos pt dt (p@pre;m) the additional linear functional

On Numerically Integrable Solutions of Ordinary Differential Equations

The paper notes that the numerically integrable solutions of ordinary differential equations must not be numerically ill conditioned on the interval in question

On the Remainder Terms of the Optimal Formulas

The remainder term of the optimal multi-step formula, that is the stable formula of maximum order, is considered from the analytical point of view

Synthesis and circularly polarized luminescence properties of BINOL-derived bisbenzofuro[2,3-b:3′,2′-e]pyridines (BBZFPys)

A series of optically active bisbenzofuro[2,3-b:3′,2′-e]pyridine (BBZFPy) derivatives was synthesized starting with the readily available (S)- and (R)-1,1′-bi-2-naphthols through a palladium-catalyzed multiple intramolecular C-H/C-H coupling as the key ring-closure step. The effect of terminal tert-butyl substituents on the BBZFPy skeleton was systematically investigated to uncover a unique aggregation-induced enhancement of CPL characteristics in the solid state. The crystal structures of the coupling products were also evaluated by single crystal X-ray analysis and the well-ordered intermolecular stacking arrangements appeared to be responsible for the enhanced CPL.Beilstein J. Org. Chem. 2020, 16, 325–336. doi:10.3762/bjoc.16.3