Modeling dynamic systems by using Ck Spline functions: A case study

In this work, the principal algebraic, arithmetic and geometric properties of the C
^k
Spline functions are presented. In this way C
^k
spline functions can be defined as the interpolating functions of the set of the all Taylor-Maclaurin expansion up to degree k defined at each point of discretization of the considered studying function. The case study in this article concerns a loading characteristics of three phase Axial Flux Permanent Magnet Synchronous Machine (AFPMSM) with broad air-gap. It shows that this characteristic leads to algebraic differential equations (ADE), which we know recently, the properties and integrated numerical means. These equations are governed by a multidistributed value problem (MDVP) which makes this problem a difficult numerical one, non accessible by traditional solvers. This article shows thanks to the properties of C
^k
spline functions that we can integrated such problems by judicious choices of the initial vectors and simulated annealing methods. The first results of simulations will be given. Then traditional differential and integral calculus lead in the C
^k
spline functional spaces to new functional and invariant calculus.