A New Definition for Generalized First Derivative of Nonsmooth Functions

Abstract

In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.

A New Approach for the Generalized First Derivative and Extension It to the Generalized Second Derivative of Nonsmooth Functions

“…In what follows, the problem (1) is approximated as the following finite dimensional problem ( [18]):…”

“…By using this problem for nonsmooth functions, we approximate the second order derivative of these functions on an interval which is GSD. Firstly, we consider again Lemma II.1 in [18] and state the following theorem and prove it where …”

“…In this stage, we find the GSD of smooth and nonsmooth functions in several examples using problem (18 (18) is solved for functions in these examples using Simplex method in MATLAB software. Attend that in our approach points in …”

“…The function () fx is illustrated in Fig.1. The exact second derivative of () fx is shown in Fig.2 and Fig.3 shows the GSD of () fx which has been obtained by using the problem (18).The absolute error of GSD is presented in Fig.4.…”

“…It is therefore natural that we achieve a useful and practical approach for defining GD. This paper is basically an extended version of the results presented in [18]. Our intention here is to propose a useful and practical definition for GFD and in what follows, we will extend it to a new GSD (see [1] for application of practical new GFD in order to solve nonsmooth ordinary differential equations).…”