A Short Account of Relaxation Methods

The relaxation method is a process of steadily improved approximation for the solution of simultaneous equations, and any problem that can be formulated in terms of simultaneous equations can, theoretically, be solved by this method. After the first section of this account, in which the physical basis leading to the title and vocabulary of relaxation is discussed, the method is presented as a simple mathematical technique. The solution of ordinary simultaneous equations is illustrated by an example, and devices are suggested for increased convergence. Application to the problem of vibrating structures is next described, in which Rayleigh's principle and relaxation are used to obtain the frequencies and modes of vibration. An example is included. The use of finite differences is then demonstrated, whereby differential equations are replaced by finite difference equivalents, and the solution of a differential equation is effected by solving a set of simultaneous equations, the unknowns being the values of the required function at pivotal points of a range or ranges of integration. Attention is focused on partial differential equations of the second order in two variables, and practical details are illustrated by an example of the solution of Laplace's equation, functional values being specified on a closed boundary. The application to other partial differential equations, including the biharmonic equation and the equations of vibration of membranes and of flat plates, is briefly considered, and the account ends with a short summary of problems already successfully solved by relaxation methods.