Nonlinear models and problems in applied sciences from differential quadrature to generalized collocation methods

This paper deals with the developments of mathematical methods for the discretization of continuous models and the solution of nonlinear problems of interest in applied sciences. In particular, the contents refer to developments of the differential quadrature method proposed by Bellmann, Kashef and Casti, which leads to the so called generalized collocation methods. The contents is in three parts. The first one is a general description of the method for the solution of initial-boundary value problems. The second part is on recent developments of the method both towards the solution of different classes of problems, e.g., solution of integro-differential equations, domain decomposition and stochastic problems. The third part is on improvements of solution algorithms, on computation of error estimates, and research perspectives. The whole content is constantly referred to the solution of nonlinear problems in applied sciences.