cours-maths-enseee

Chapter IVSolving linear and nonlinearpartial di eren tial equationsby the method of characteristicsChapter III has brought to light the notion of characteristic curves and their signi cance in theprocess of classi cation of partial di eren tial equations.Emphasis will be laid here on the role of characteristics to guide the propagation of infor-mation within hyperbolic equations. As a tool to solve PDEs, the method of characteristicsrequires, and provides, an understanding of the structure and key aspects of the equationsaddressed. It is particularly useful to inspect the e ects of initial conditions, and/or boundaryconditions.While the method of characteristics may be used as an alternative to methods based ontransform techniques to solve linear PDEs, it can also address PDEs which we call quasi-linear(but that one usually coins as nonlinear). In that context, it provides a unique tool to handlespecial nonlinear features, that arise along shock curves or expansion zones.As a model problem, the method of characteristics is rst applied to solve the wave equationdue to disturbances over in nite domains so as to avoid re ections. The situation is morecomplex in semi-in nite or nite bodies where waves get re ected at the boundaries. The issueis examined in Exercise IV.2.Basic features of scalar conservation laws are next addressed with emphasis on under- andover-determined characteristic network, associated with expansion zone and shock curves.Finally ...