Calculation in the unsymmetrical bending problem of thin plates by spline boundary layer method

Abstract

The weighted-residual and perturbation methods are usually used to solve the unsymmetrical bending problem of thin plates. Here a mixed method of the weighted-residual and singular perturbation is considered. It is known that there are many ways from which a combination of the two methods can be chosen to obtain numerical solutions which are uniformly valid throughout the region, but they are very different in theory. In this article the main purpose is to give a description of a mixed method for calculation in the unsymmetrical bending problem of plates, so only the singular perturbation problem for the differential equation involving a small parameter in higher derivatives will be discussed. Called the spline boundary layer method for short, boundary layer solutions are constituted with the spline function in order to overcome the difficulties of obtaining the boundary layer solution. An example is given. Not only is the method simple and convenient for computations with a computer, but also the expression for the outer solution is offered simultaneously, and if necessary the expression of the boundary layer solution can also be given without using any complicated matching principle. Therefore the applications of the singular perturbation method in engineering calculations can be expanded.