Random Collocation Method (RCM)

Recently, young people do not show much interest on theory, and a lack of communication is occurring between the older and younger generations. This tells us that a much simpler numerical procedure directly related to the governing equations is strongly required and the effort answering the requirement should be continued. If there is a numerical procedure that is friendlier with theory, the distance between theory and calculation would be decreased much, and the interaction among people in both fields would become more active. When the geometry of the domain is simple, the traditional analytical method using function expansion is very convenient in many numerical problems. In many problems, it has given very useful solutions for various problems. However, its effectiveness is usually limited to simple geometries of the domain. In the past, a fusion of the analytical approach and computational one has not been pursued sufficiently. If it becomes possible, it may give a different paradigm for obtaining the numerical solution. In the present paper, a very simple and innovative idea named Random Collocation Method (RCM) is discussed on how to overcome the weak point of the traditional method by combining it with computational method. It is the purpose of the present paper to develop the simplest numerical method and to make the distance between the theory and numerical method as small as possible.