A kind of linearity-invariance under homotopy and some simple applications o fit in mechanics

Abstract

A kind of linearity-invariance under homotopy and some simple applications o fit in mechanics

A kind of linearity-invariance under homotopy and some simple applications o fit in mechanics: Neither numerical techniques nor analytical methods of nonlinear problems are satisfactory. The iterative techniques are sensitive not only to the initial solutions but also to the number of unknowns. On the other side, perturbation expansion method depends upon small or great parameters. These limit their applications. This paper is a prelimilary attempt to improve both analytical and numerical techniques of general nonlinear problems, i.e., to overcome the limitations of perturbation methods and iterative techniques described above. Based on a kind of linear property of continuous mapping ( mathematically speaking, a kind of linearity-invariance under homotopy), a kind of analytical method for non linear problems, namely, Process Analysis Method, is described, which does not depend upon small or great parameters. And based on the same property of continuous mapping (homotopy), a kind of numerical technique, called Finite Process Method, is developed, which can avoid the use of iterative techniques to solve non linear problems.The essence of solving nonlinear problems and the differences and relations of linear and nonlinear problems are also simply discussed. Some simple nonlinear problems in mechanics, for instance, the falling of a ball in fluid, the motion of a simple pendulum, 2D nonlinear water waves and so on, are used to introduce and examine the both methods.