In mathematics, the method of characteristics is a technique for solving particular partial differential equations. Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equations. The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODEs) along which the solution can be integrated from some initial data given on a suitable hypersurface.
Characteristics of first-order partial differential equation For a first-order PDE, the method of characteristics discovers so called characteristic curves along which the PDE becomes an ODE. Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.
Two-dimensional quasilinear PDE For the sake of simplicity, we initially direct our attention to the case of a function of two independent variables x and y. Consider a quasilinear PDE of the form
For a differentiable function ( x , y ) ↦ u ( x , y ) {\displaystyle (x,y)\mapsto u(x,y)} , consider the graph of u, which is the set
gph ( u ) = { ( x , y , z ) ∈ R 3 ∣ z = u ( x , y ) } {\displaystyle \operatorname {gph} (u)=\{(x,y,z)\in \mathbb {R} ^{3}\mid z=u(x,y)\}} A normal vector to gph ( u ) {\displaystyle \operatorname {gph} (u)} is given by
n ( x , y ) = ( ∂ u ∂ x ( x , y ) , ∂ u ∂ y ( x , y ) , − 1 ) . {\displaystyle n(x,y)=\left({\frac {\partial u}{\partial x}}(x,y),{\frac {\partial u}{\partial y}}(x,y),-1\right).}
Consider the vector field
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