The general boundary element method and its further generalizations

Abstract

In this paper, the basic ideas of the general boundary element method (BEM) proposed by Liao [in Boundary Elements XVII, Computational Mechanics Publications, Southampton, MA, 1995, pp. 67–74; Int. J. Numer. Methods Fluids, 23, 739–751 (1996), 24, 863–873 (1997); Comput. Mech., 20, 397–406 (1997)] and Liao and Chwang [Int. J. Numer. Methods Fluids, 23, 467–483 (1996)] are further generalized by introducing a non‐zero parameter $\hbar$. Some related mathematical theorems are proposed. This general BEM contains the traditional BEM in logic, but is valid for non‐linear problems, including those whose governing equations and boundary conditions have no linear terms. Furthermore, the general BEM can solve non‐linear differential equations by means of no iterations. This disturbs the absolutely governing place of iterative methodology of the BEM for non‐linear problems. The general BEM can greatly enlarge application areas of the BEM as a kind of numerical technique. Two non‐linear problems are used to illustrate the validity and potential of the further generalized BEM. Copyright © 1999 John Wiley & Sons, Ltd.

Notes on the homotopy analysis method: Some definitions and theorems

“…For example, the so-called ''general boundary element method" [70][71][72][73], which is based on the HAM, gives accurate convergent results of the viscous driven flows (governed by the exact Navier-Stokes equations) in a square cavity with the high Reynolds number Re ¼ 7500, as shown by Zhao and Liao [74]. Currently, Wu and Cheung [49] applied the HAM to give an explicit numerical approach for Riemann problems related to nonlinear shallow water equations.…”

A General Approach to Obtain Series Solutions of Nonlinear Differential Equations

“…Currently, Liao [28] replaced a unsteady boundary-layer flow problem by means of an infinite number of steady-state linear boundary-value problems. Besides, this kind of freedom can be used to develop some new numerical techniques for nonlinear problems, such as the so-called "generalized boundary element method" [50][51][52][53][54]. All of these indicate that we indeed have much larger freedom and flexibility to solve nonlinear problems than we thought traditionally.…”

Some notes on the general boundary element method for highly nonlinear problems

“…It is easy to prove that many well-known iterative schemes are only special cases of (19) when N ¼ 1. It is found [36] that, at each iteration and for a given initial guess, the higher the order N of iteration formula, the better the approximate result. Thus, the higher-order iterative formula (with larger N ) has better property of convergence.…”

“…Liao and his co-authors generalized the traditional nonlinear BEM schemes and proposed the so-called general boundary element method [30][31][32][33][34][35][36][37][38][39][40], which can overcome two of the abovementioned limitations of the traditional nonlinear boundary element method. In the following sections the basic ideas of the general boundary element method are briefly described and some examples are given to show its validity.…”