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Optimal derivative

Study of stability using the optimal derivative

Approximation of nonlinear systems

Cooperation : Université Béchar, I.R.D- Bondy (France), Université Bordeaux II

It was realized long ago that most of the phenomena of mathematical physics are nonlinear case, the most famous being the Boltzmann equation in static mechanics, the Navier-Stokes equations of fluid mechanics (equations are Moreover, an approximation of the Boltzmann equation), the equations of Von Karman flat plates with large displacements, etc. .... However, before the possibility of using systematic-and almost «trivial» computers to compute approximate solutions of the system state, accurate results could generally be obtained in the linear case.

Some physical problems can be modeled directly (ie without approximation) by linear equations: the case of the neutron transport equation in particular. Other phenomena may be deducted from the "true" nonlinear systems is neglecting certain terms (which is valid in certain situations: "small" movements, movements, movements "slow").

Our team offers a method that has given his evidence through a large number of results (doctoral theses, Magisters, Publications .....).

Our goal is to use the optimal derivation developed by Arino, Benouaz (1995-1996-2001) and supplemented by Bohner, Benouaz (2007-2009-2011) for approximation and study the stability of certain classes of function non-linear, in order to test this method and demonstrate its capabilities.

Objectives:

- Validation of the method through:
- International publications.
- PhD thesis: they will be facing several nonlinear models Energetic.

- Develop a user-friendly software to popularize the method.

It presents a comparison with the continuation software Auto 2000.

Bifurcation diagram showing the variation of the L2 norm depending on the parameter a (gain) solid solution stable, dotted unstable solution, PB pitchfork bifurcation, HB: Hopf bifurcation, PD: Bifurcation to split time. Information from the qualitative analysis conducted using the optimal derivation applied to bifurcation points are indicated by arrows circled red on the diagram

Different types of points and symbols encountered in the bifurcation diagram for the execution of software auto2000 are defined according to the following parameters:

TY: Abbreviation of the name of the type of solution

TY: Number of solution

BR: Number of branches

PT: Number of points

LAB: Label Solution

TY LAB meaning

BP 1 bifurcation point

HB 3 Hopf bifurcation

7 PD period doubling bifurcation to

EP 9 beginning and end of normal execution

PRM (1) is the parameter for continuation, here U (1), U (2), and U (3) are x, y, z.

MORE INFORMATION: SEE PhD thesis and Habilitation of Dr CHIKHAOUI A..