Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the third paper, the analytical analysis of multiscale phenomena inherent in the 2r-th order linear differential equations with constant coefficients and subjected to general boundary conditions is addressed. The limitation of the algebraical polynomial interpolation based composite Fourier series method is first discussed. This leads to a new formulation of the composite Fourier series method, where homogeneous solutions of the differential equations are adopted as interpolation functions. Consequently, the theoretical framework of the Fourier series multiscale method is provided, in which decomposition structures of solutions of the differential equations are specified and implementation schemes for application are detailed. The Fourier series multiscale method has not only made full use of existing achievements of the Fourier series method, but also given prominence to the fundamental position of structural decomposition of solutions of the differential equations, which results in perfect integration of consistency and flexibility of the Fourier series solution.
翻译:将在本系列论文中制定一套简单、高效的多尺度方法,即多尺度计算分析方法;第三份文件分析分析具有不变系数和受一般边界条件制约的第二阶线性差分方程式所固有的多尺度现象;首先讨论基于代数的多元多层次间插混合法的局限性;这导致以内插功能方式采用差异方程同质解决办法的复合方程组合法新拟订法;因此,提供了Fourier系列多尺度方法的理论框架,其中具体说明了差异方程解决办法的分解结构,并详细说明了应用办法;四阶系列多尺度方法不仅充分利用了Fourier系列方法的现有成就,而且还突出了差异方程解决办法结构分解的基本位置,从而完美地整合了四阶方程解决办法的一致性和灵活性。