cours-maths-enseee

Chapter IIIClassi cation ofpartial di eren tial equations intoelliptic, parabolic and hyperbolictypesThe previous chapters have displayed examples of partial di eren tial equations in various eldsof mathematical physics. Attention has been paid to the interpretation of these equations in1the speci c contexts they were presented.In fact, we have delineated three types of eld equations, namely hyperbolic, parabolic andelliptic. The basic idea that the mathematical nature of these equations was fundamental totheir physical signi cance has been creeping throughout.Still, the formats in which these three types were presented correspond to their canonicalforms, that is, a form that one recognizes at rst glance. Such is not the general case. Forexample, it is not obvious (to this author at least!) that the following second order equation,2 2 2@ u @ u @ u @u2 4 6 + = 0;2 2@x @x@t @t @xis of hyperbolic type. In other words, it shares essential physical properties with the waveequation,2 2@ u @ u= 0:2 2@x @tIndeed, this is the aim of the present chapter to show that all equations of mathematicalphysics can be recast in these three fundamental types. By the same token, we introduce a newnotion, that of a characteristic curve. A method to solve IBVPs based on characteristics willbe exposed in the next chapter.The terminology used to coin the three types of PDEs borrows from geometry, as thecriterion will be seen to rely on the nature of the roots of ...