Abstract:

Given a second-order differential equation with Dirichlet conditions at the ends of an interval, it is possible to obtain an approximate solution by using the shooting method or the finite difference method. Another procedure is based on reformulating the initial problem, transforming it into the problem of minimizing a certain integral related to the differential equation in a set of sufficiently derivable functions that verify the initial conditions. The variational technique proposed by the Rayleigh-Ritz method consists of finding the minimum value in a smaller set formed by all the linear combinations of certain linearly independent basic functions that satisfy the boundary conditions. It is known that if linear functions are used piece by piece the solution obtained is continuous, but not derivable. A better alternative is to use basic functions constructed with cubic splines, which provides a solution with a second continuous derivative. Originally the method was proposed to obtain an approximate solution of the elasticity problem in its energy formulation.

In this paper we present a teaching experience carried out during the last course in the subject of Mathematical Methods of the Degree in Industrial Technologies of the Polytechnic University of Valencia. It is based on the development of a virtual laboratory (VL) designed as a graphical user interface of Matlab, dedicated to the study and understanding of the Rayleigh-Ritz method. Interactively, the approximate solution can be visualized and, in the case in which it is known, the exact solution together with the errors made. It is also possible to show the graphs of the basic functions. Both basic linear functions and those obtained using splines can be used.

The methodology that has been followed consists of:
(1) introduction of the necessary theory in the classroom classes,
(2) realization of a practice in the computer rooms, where a complete insight of the problem is known via working with the LV and this method can be compared to another and
(3) application to the particular case of the deformation model of a beam.

We have confirmed that the experience has been very well received by our students and excellent academic results have been achieved.

Keywords:

differential equation, contour problem, cubic spline, Rayleigh-Ritz method, variational technique, virtual laboratory, graphical interface.