期刊名称:American Journal of Computational Mathematics
印刷版ISSN:2161-1203
电子版ISSN:2161-1211
出版年度:2014
卷号:04
期号:04
页码:317-328
DOI:10.4236/ajcm.2014.44028
语种:English
出版社:Scientific Research Publishing
摘要:In this paper, we proposed a Extension Definition to derive, simultaneously, the first, second and high order generalized derivatives for non-smooth functions, in which the involved functions are Riemann integrable but not necessarily locally Lipschitz or continuous. Indeed, we define a functional optimization problem corresponding to smooth functions where its optimal solutions are the first and second derivatives of these functions in a domain. Then by applying these functional optimization problems for non-smooth functions and using this method we obtain generalized first derivative (GFD) and generalized second derivative (GSD). Here, the optimization problem is approximated with a linear programming problem that by solving of which, we can obtain these derivatives, as simple as possible. We extend this approach for obtaining generalized high order derivatives (GHODs) of non-smooth functions, simultaneously. Finally, for efficiency of our approach some numerical examples have been presented.
关键词:Generalized Derivative; Smooth and Nonsmooth Functions; Nonsmooth Optimization Problem; Linear Programming