Burgers Equation And It’s Applications

Résumé: T the last are generally denoted in abbreviation PDEs. o understand the physical phenomena, we use in general, partial differential equations, In everyday life, the majority of mechanical engineers as well as in physics encounter problems related to the phenomenon of fluid dynamics and mass transport, and in particular, the biological phenomena are describe by PDEs models. In general it is very difficult to solve these problems, but in some cases we can find analytical solution in particular forms. In this work we shall study the Burgers equation with her different forms: the first type, the second type and the third type which called the forced Burgers equation. These types of Burgers equation, are studied by many authors [20] ,[5],[3],[9], some class of methods are used by these authors: -The reduction method, which transform the PDEs to EDOs, and finding a particular solutions, such as self similar solutions and travelling wave solutions. - The Cole-Hopf transformation, is used particularly for all types of Burgers equations, which transform them to a linear equations and in particular the heat equation. Our objectif in this thesis, is to develop all these methods for resolving the three types of the Burgers equations. This work is organized in three chapters: The first chapter, we present some ideas about partial differential equations, heat equation, wave equation, definition of the self-similar solution, some notions about differential equations and reduction methods for PDEs. We also gave some biological models for the PDE. In the second chapter, we give the definition of the Burgers equation and it types, Cole-Hopf transformation and we find by reduction methods ,the self similar solution, the travelling wave solution for the Burgers equation with the following form: 5ut + uux = 0. (1) We also find a self similar solution and travelling wave solution and a solution based on the Cole-Hopf transformation, for the Burgers equation for the second type with the form: ut + uux = uxx. (2) In the third chapter, we introduce the forced Burgers equation in the form: ut − uux − uxx = Fx,t,u, (3) we study some particular cases, when F = C, is constant, and F = Fx which call stationary forcing, and finally, the case which F = Fx,t, which call transient forcing, and find an exact solution for it, using the Cole-Hopf transformation, and searching a travelling wave solutio

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