Гиперболикалық дербес дифференциал теңдеулерді шешу әдісі

Technique for solving hyperbolic partial differential equations

In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface.

Characteristics are also a powerful tool for gaining qualitative insight into a PDE. One can use the crossings of the characteristics to find shock waves for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to along itself. Thus, when two characteristics cross, the function becomes multi valued resulting in a non physical solution. Physically, this contradiction is removed by the formation of a shock wave, a tangential discontinuity or a weak discontinuity and can result in non potential flow, violating the initial assumptions. Characteristics may fail to cover part of the domain of the PDE. This is called a rarefaction, and indicates the solution typically exists only in a weak, i. e. integral equation, sense. The direction of the characteristic lines indicates the flow of values through the solution, as the example above demonstrates. This kind of knowledge is useful when solving PDEs numerically as it can indicate which finite difference scheme is best for the problem.