海洋工程与环境问题的分形级数解

Fractal Series Solution for Problems of Ocean Engineering and Environment

给出戴劳级数、三角级数等的推广-分形级数,并讨论海洋工程与环境问题的分形级数解。在戴劳级数、三角级数等级数中,各项的指数及角度的系数等均为非负整数,而在分形级数中,各项的指数及角度的系数等均为任意实数。文中讨论了简支梁。五阶斯托克斯波浪理论,强非线性KdV方程等问题的分形级数解。在确定分形级数时,可应用加权残值法(如最小二乘法等)。当求解受均布荷载简支梁的挠度曲线时,原有的幂级数解的形式为Cx(1-x),而本文分形级数解的形式为C′x^(1.05)(1-x^(1.05)),分形级数解更接近精确解。

This paper presents the fractal series, i.e., the expanded form of Taylor series, Fourier series and the like, and discusses the fractal series solution for the problems of ocean engineering and environment. In Taylor series, Fourier series and the like, for each term the index and the angular coefficient are all the non - negative integral numbers, while in the fractal series they can be taken as any real numbers. In this paper the fractal series solutions for simply supported beam, fifth order Stokes wave theory, strongly non - linear KdV equation are discussed. In determining the fractal series solution, the method of weighted residuals such as LSM can be used. For the sag curve of simply supported beam acted by uniform load, the existing Taylor series solution reads C x (1 - x), while the fractal series solution reads C'x1.05(1-x1.05), the fractal series solution is closer to the accurate solution.