The Method of Least Squares on the Boundary for the Biharmonic Equation (for the Problem of Wall-beams). The Trefftz Method of the Solution of the Dirichlet Problem for the Laplace Equation

Abstract

Methods discussed in this book till now consisted — roughly speaking — in the fact that the approximate solution u_n(x) of the considered problem satisfied (at least in some generalized sense) the boundary conditions but did not satisfy the given differential equation, in general. In this chapter, we shall show some methods which are based on a in a certain sense converse process: We look for the approximate solution u_n(x) of the given problem in the form

$${u_n}\left( x \right) = \sum\limits_{i = 1}^n {{a_{ni}}} {z_i}\left( x \right),$$

(43.1)

where the function u_n(x) satisfies the given differential equation for arbitrary values of the constants a _ni . The constants are determined from the condition that the boundary conditions be best satisfied, in a certain sense. It is obvious that such methods are used when the given equation is linear and homogeneous. (If nonhomogeneous, a suitable particular solution has to be found first.) Thus, if we choose the functions z _i (x) so that each of them be a solution of this equation, then every linear combination (43.1) also is a solution of the given equation.