A new ordinary differential equation for the evaluation of the frequency-domain Green function

Abstract

Clément (2013) derived a second order ordinary differential equation (ODE) satisfied by the free-surface Green function in the frequency domain. Since then, similar ODEs for the gradient of the Green function have been developed. Unfortunately, all these ODEs degenerate at zero frequency. Therefore, it is not possible to initialize the numerical solution of these ODEs from this zero frequency. Alternative methods based on the shifting of the initial condition to frequencies strictly greater than zero have then been developed. The present paper describes an alternative approach to address this issue. It involves a new function which is the solution of a modified ODE which can be solved from the zero frequency. Finally, comparisons with evaluations of the Green function using the classical direct integration method are provided. They show that the new ODE can provide accurate estimates of the Green function.

Higher-order derivatives of the Green function in hyper-singular integral equations

“…transformation [14,15] 4. Solving ordinary differential equation [16,17,18] The accuracy and time-efficiency of different algorithms have been commented in a recent review by Xie et al [19]. Usually, the focus is placed on the Green function itself, whereas the derivatives are evaluated via straightforwardly differentiating the expressions of approximation.…”

“…The comparison between approximation via the direct differentiation (17) and the proposed formulation (11a) is considered. To achieve six-decimal accuracy, the series are truncated to n = 9 when d ≥ 15.…”

Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast

“…Bagd applied the Haar wavelength division operator matrix to the power system problem and promoted the application of wavelet in power systems [5]. Xie used the Haar wavelet method to solve the linear sonocity division, non-linear sub-division, high-order differential equations, one-dimensional diffusion equation, two-dimensional Poisson equation, and so on, as well as a change in the variable steps of the wavelet method [6]. Cooperation extends the Haar wavelet configuration method to linear integral equations, second types of Freholm non-linear integral equations and numerical methods for non-linear solution equations [7].…”

Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast

“…BAGD applied the HAAR wavelength division operator matrix to the power system problem, and promoted the application of wavelet in the power system [5]. XIE used the HAAR wavelet method to solve the linear sonocity division, nonlinear sub-division, high-order differential equations, onedimensional diffusion equation, two-dimensional Poisson equation, as well as a change in the variable steps Wavelet method [6]. A cooperation will extend the HAAR wavelet configuration method to linear integral equations, the second type of Freholm nonlinear integral equations, numerical methods for nonlinear solution equations [7].…”