Method of characteristics

The method of characteristics is a technique for solving particular partial differential equations. It applies most directly to first-order equations, although characteristic curves can also be found for hyperbolic and parabolic equations. The idea is to reduce a partial differential equation to a family of ordinary differential equations along special curves, called characteristic curves, so that the solution can be integrated along each curve from initial data given on a suitable hypersurface. Once the ordinary differential equation along a characteristic is solved, the result is transformed back into a solution of the original partial differential equation. The article develops the method first for a quasilinear equation in two independent variables, where the characteristics follow from the requirement that a chosen vector field be tangent to the graph of the solution, and then for more general first-order equations.