In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to firstorder 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.
Contents

Characteristics of firstorder partial differential equation 1

Linear and quasilinear cases 1.1

Fully nonlinear case 1.2

Example 2

Characteristics of linear differential operators 3

Qualitative analysis of characteristics 4

See also 5

Notes 6

References 7

External links 8
Characteristics of firstorder partial differential equation
For a firstorder 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 motivation, 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

a(x,y,z) \frac{\partial z}{\partial x}+b(x,y,z) \frac{\partial z}{\partial y}=c(x,y,z).


(1)

Suppose that a solution z is known, and consider the surface graph z = z(x,y) in R^{3}. A normal vector to this surface is given by

\left(\frac{\partial z}{\partial x}(x,y),\frac{\partial z}{\partial y}(x,y),1\right).\,
As a result,^{[1]} equation (1) is equivalent to the geometrical statement that the vector field

(a(x,y,z),b(x,y,z),c(x,y,z))\,
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.
The equations of the characteristic curve may be expressed invariantly by the LagrangeCharpit equations^{[2]}

\frac{dx}{a(x,y,z)} = \frac{dy}{b(x,y,z)} = \frac{dz}{c(x,y,z)},
or, if a particular parametrization t of the curves is fixed, then these equations may be written as a system of ordinary differential equations for x(t), y(t), z(t):

\begin{array}{rcl} \frac{dx}{dt}&=&a(x,y,z)\\ \frac{dy}{dt}&=&b(x,y,z)\\ \frac{dz}{dt}&=&c(x,y,z). \end{array}
These are the characteristic equations for the original system.
Linear and quasilinear cases
Consider now a PDE of the form

\sum_{i=1}^n a_i(x_1,\dots,x_n,u) \frac{\partial u}{\partial x_i}=c(x_1,\dots,x_n,u).
For this PDE to be linear, the coefficients a_{i} may be functions of the spatial variables only, and independent of u. For it to be quasilinear, a_{i} may also depend on the value of the function, but not on any derivatives. The distinction between these two cases is inessential for the discussion here.
For a linear or quasilinear PDE, the characteristic curves are given parametrically by

(x_1,\dots,x_n,u) = (x_1(s),\dots,x_n(s),u(s))
such that the following system of ODEs is satisfied

\frac{dx_i}{ds} = a_i(x_1,\dots,x_n,u)


(2)


\frac{du}{ds} = c(x_1,\dots,x_n,u).


(3)

Equations (2) and (3) give the characteristics of the PDE.
Fully nonlinear case
Consider the partial differential equation

F(x_1,\dots,x_n,u,p_1,\dots,p_n)=0


(4)

where the variables p_{i} are shorthand for the partial derivatives

p_i = \frac{\partial u}{\partial x_i}.
Let (x_{i}(s),u(s),p_{i}(s)) be a curve in R^{2n+1}. Suppose that u is any solution, and that

u(s) = u(x_1(s),\dots,x_n(s)).
Along a solution, differentiating (4) with respect to s gives

\sum_i(F_{x_i} + F_u p_i)\dot{x}_i + \sum_i F_{p_i}\dot{p}_i = 0

\dot{u}  \sum_i p_i \dot{x}_i = 0

\sum_i (\dot{x}_i dp_i  \dot{p}_i dx_i)= 0.
The second equation follows from applying the chain rule to a solution u, and the third follows by taking an exterior derivative of the relation du  \sum_i p_i \, dx_i = 0. Manipulating these equations gives

\dot{x}_i=\lambda F_{p_i},\quad\dot{p}_i=\lambda(F_{x_i}+F_up_i),\quad \dot{u}=\lambda\sum_i p_iF_{p_i}
where λ is a constant. Writing these equations more symmetrically, one obtains the LagrangeCharpit equations for the characteristic

\frac{\dot{x}_i}{F_{p_i}}=\frac{\dot{p}_i}{F_{x_i}+F_up_i}=\frac{\dot{u}}{\sum p_iF_{p_i}}.
Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the Monge cone of the differential equation should everywhere be tangent to the graph of the solution.
Example
As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs).

a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0\,
where a\, is constant and u\, is a function of x\, and t\,. We want to transform this linear firstorder PDE into an ODE along the appropriate curve; i.e. something of the form

\frac{d}{ds}u(x(s), t(s)) = F(u, x(s), t(s)) ,
where (x(s),t(s))\, is a characteristic line. First, we find

\frac{d}{ds}u(x(s), t(s)) = \frac{\partial u}{\partial x} \frac{dx}{ds} + \frac{\partial u}{\partial t} \frac{dt}{ds}
by the chain rule. Now, if we set \frac{dx}{ds} = a and \frac{dt}{ds} = 1 we get

a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} \,
which is the left hand side of the PDE we started with. Thus

\frac{d}{ds}u = a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0.
So, along the characteristic line (x(s), t(s))\,, the original PDE becomes the ODE u_s = F(u, x(s), t(s)) = 0\,. That is to say that along the characteristics, the solution is constant. Thus, u(x_s, t_s) = u(x_0, 0)\, where (x_s, t_s)\, and (x_0, 0)\, lie on the same characteristic. So to determine the general solution, it is enough to find the characteristics by solving the characteristic system of ODEs:

\frac{dt}{ds} = 1, letting t(0)=0\, we know t=s\,,

\frac{dx}{ds} = a, letting x(0)=x_0\, we know x=as+x_0=at+x_0\,,

\frac{du}{ds} = 0, letting u(0)=f(x_0)\, we know u(x(t), t)=f(x_0)=f(xat)\,.
In this case, the characteristic lines are straight lines with slope a\,, and the value of u\, remains constant along any characteristic line.
Characteristics of linear differential operators
Let X be a differentiable manifold and P a linear differential operator

P : C^\infty(X) \to C^\infty(X)
of order k. In a local coordinate system x^{i},

P = \sum_{\alpha\le k} P^{\alpha}(x)\frac{\partial}{\partial x^\alpha}
in which α denotes a multiindex. The principal symbol of P, denoted σ_{P}, is the function on the cotangent bundle T^{∗}X defined in these local coordinates by

\sigma_P(x,\xi) = \sum_{\alpha=k} P^\alpha(x)\xi_\alpha
where the ξ_{i} are the fiber coordinates on the cotangent bundle induced by the coordinate differentials dx^{i}. Although this is defined using a particular coordinate system, the transformation law relating the ξ_{i} and the x^{i} ensures that σ_{P} is a welldefined function on the cotangent bundle.
The function σ_{P} is homogeneous of degree k in the ξ variable. The zeros of σ_{P}, away from the zero section of T^{∗}X, are the characteristics of P. A hypersurface of X defined by the equation F(x) = c is called a characteristic hypersurface at x if

\sigma_P(x,dF(x)) = 0.
Invariantly, a characteristic hypersurface is a hypersurface whose conormal bundle is in the characteristic set of P.
Qualitative analysis of characteristics
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 u\, along itself. Thus, when two characteristics cross, the function becomes multivalued resulting in a nonphysical solution. Physically, this contradiction is removed by the formation of a shock wave, a tangential discontinuity or a weak discontinuity and can result in nonpotential 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 indicate 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.
See also
Notes

^ John 1991

^ Delgado 1997
References


Delgado, Manuel (1997), "The LagrangeCharpit Method", SIAM Review 39 (2): 298–304,

Evans, Lawrence C. (1998), Partial Differential Equations, Providence: American Mathematical Society,


Polyanin, A. D.; Zaitsev, V. F.; Moussiaux, A. (2002), Handbook of First Order Partial Differential Equations, London: Taylor & Francis,

Polyanin, A. D. (2002), Handbook of Linear Partial Differential Equations for Engineers and Scientists, Boca Raton: Chapman & Hall/CRC Press,

Sarra, Scott (2003), "The Method of Characteristics with applications to Conservation Laws", Journal of Online Mathematics and its Applications .

Streeter, VL; Wylie, EB (1998), Fluid mechanics (International 9th Revised ed.), McGrawHill Higher Education
External links

Prof. Scott Sarra tutorial on Method of Characteristics

Prof. Alan Hood tutorial on Method of Characteristics
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