Generalized pseudospectral method: Theory and applications

In this study, we provide a new method, namely the Generalized Pseudospectral Method (GPM), for solving the linear and nonlinear ordinary/partial differential equations. Initially, we introduce a new class of functions, namely the Generalized Lagrange Functions (GLFs), so that they satisfy in the property of the Kronecker delta at the collocation points. Then, for these GLFs, the differentiation matrix of D-(1) is calculated. Also, it is shown that by using D-(1) and a generalized theorem, all differentiation matrices of D-(m) for all m is an element of N can be calculated, that is, we have generalized the classical Lagrange theorem for this new class of functions. Finally, in order to demonstrate the efficiency and convergence of the GPM, some well-known linear and nonlinear differential equations, which are applicable in engineering and applied sciences, are investigated based on many classes of the collocation points. (C) 2019 Elsevier B.V. All rights reserved.