Approximation of multiple integrals by simple integrals

Abstract

The Alienor method, based on α ‐dense curves, has been developed by Yves Cherruault and collaborators, to solve optimization problems. It can be coupled with the decomposition method of Adomian to solve optima control problems also. But α ‐dense curves can be used in many other problems. Gives an application of α ‐dense curves for calculating multiple integrals by means of simple integrals.

Approximation method error of multiple integrals by simple integrals

“…If we introduce the function g defined on [0, 1] by:Equation 14it is well known that g is integrable on [0, 1] (Bourbaki, 1967), and by using Lebesgue theorem (Bourbaki, 1973), we can conclude thatEquation 15that is to say:Equation 16Recall thatEquation 17If we set:Equation 18then:Equation 19by where:Equation 20andEquation 21For m = n , as seen in Benabidallah et al (2003), we obtain:Equation 22Using:Equation 23we obtain:Equation 24…”

“…First we give an approximation of a multiple integral in [0,1] d by means of a simple integral, generalising the results obtained in Benabidallah et al (2003) for calculating double and triple integrals by densifying the integration domain.…”

Approximation of multiple integrals by length of α‐dense curves

“…In contrast with the results proved by Benabidallah et al , to be published and Benabidallah et al (2001), this new method leads to an integral of f with weight:Equation 1The integral of f on [0,1] d can be then obtained by introducing an auxiliary function g =f· p , where p is the weight in the previous formulae.…”

“…Using α ‐dense curves (Cherruault, 1998, 1999), multiple integrals can be approximated by simple integrals (Benabidallah et al , to be published, 2001). In this paper, we propose a new approach (Cherrault et al , 2002), consisting in densifying the set Hf ={( x , y ), x∈[0,1] d , 0≤ y ≤f( x )} for a real value function f defined on [0,1] d , d∈ N x .…”

Approximation of multiple integrals by simple integrals involving periodic functions

“…The theory of a-dense curves developed by Cherruault (1998Cherruault ( , 1999, and has permitted us to approximate multiple integrals by simple integrals of oscillatory functions. In Benabidallah et al (2001), the one-variable functions obtained by using the reducing transformations are periodic but discontinuous, while in Benabidallah et al (2002) the integrands are just continuous and not periodic.…”