Method of characteristics

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. a ( x , y , z ) ∂ z ∂ x + b ( x , y , z ) ∂ z ∂ y = c ( x , y , z ) . {displaystyle a(x,y,z){frac {partial z}{partial x}}+b(x,y,z){frac {partial z}{partial y}}=c(x,y,z).}     (1) d x i d s = a i ( x 1 , … , x n , u ) {displaystyle {frac {dx_{i}}{ds}}=a_{i}(x_{1},dots ,x_{n},u)}     (2) d u d s = c ( x 1 , … , x n , u ) . {displaystyle {frac {du}{ds}}=c(x_{1},dots ,x_{n},u).}     (3) F ( x 1 , … , x n , u , p 1 , … , p n ) = 0 {displaystyle F(x_{1},dots ,x_{n},u,p_{1},dots ,p_{n})=0}     (4) 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. For a first-order PDE (partial differential equation), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear PDE of the form Suppose that a solution z is known, and consider the surface graph z = z(x,y) in R3. A normal vector to this surface is given by As a result, equation (1) is equivalent to the geometrical statement that the vector field is tangent to the surface z = z(x,y) at every point, for the dot product of this vector field with the above normal vector is zero. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation and are given by the Lagrange-Charpit equations A parametrization invariant form of the Lagrange-Charpit equations is: