Inmathematics, themethod of characteristics is a technique for solving particularpartial differential equations. Typically, it applies tofirst-order equations, though in generalcharacteristic curves can also be found forhyperbolic andparabolic partial differential equation. The method is to reduce a partial differential equation (PDE) to a family ofordinary differential equations (ODEs) along which the solution can be integrated from some initial data given on a suitablehypersurface.

For a first-order PDE, the method of characteristics discovers so calledcharacteristic curves along which the PDE becomes an ODE.^[1][2] 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 initially direct our attention to the case of a function of two independent variablesx andy. Consider aquasilinear PDE of the form^[3]

${\displaystyle a(x,y,u)\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+b(x,y,u)\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=c(x,y,u).}$

1

For a differentiable function${\displaystyle (x,y)\mapsto u(x,y)}$, consider thegraph ofu, which is the set$${\displaystyle \mathrm{gph}(u)=\{(x,y,z)\in {\mathbb{R}}^3\mid z=u(x,y)\}}$$ Anormal vector to${\displaystyle \mathrm{gph}(u)}$ is given by^[4]

$${\displaystyle n(x,y)=\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}(x,y),\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x,y),-1\right).}$$

Consider the vector field

${\displaystyle (x,y,z)\mapsto \left[\begin{array}{c}a(x,y,z)\\ b(x,y,z)\\ c(x,y,z)\end{array}\right].}$

2

Thedot product of the vector field (2) with the normal vector to${\displaystyle \mathrm{gph}(u)}$ at each${\displaystyle (x,y,u(x,y))\in \mathrm{gph}(u)}$ is$${\displaystyle \left[\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}}(x,y)\\ {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }y}}(x,y)\\ -1\end{array}\right]\cdot \left[\begin{array}{c}a(x,y,u(x,y))\\ b(x,y,u(x,y))\\ c(x,y,u(x,y))\end{array}\right]=a(x,y,u(x,y))\frac{\mathrm{\partial }u}{\mathrm{\partial }x}(x,y)+b(x,y,u(x,y))\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x,y)-c(x,y,u(x,y)).}$$

Comparing the right-hand side of the above equation with (1), it is evident the following statements are equivalent:

• the right-hand side of the above equation is zero;
• ${\displaystyle u}$ is a solution to (1);
• the vector field (2) is orthogonal to the normal vectors of${\displaystyle \mathrm{gph}(u)}$ at every point${\displaystyle (x,y,z)\in \mathrm{gph}(u)}$;
• the vector field (2) is tangent to the surface${\displaystyle \mathrm{gph}(u)}$ at every point${\displaystyle (x,y,z)\in \mathrm{gph}(u)}$;

In other words, the graph of the solution to (1) is the union ofintegral curves of the vector field (2). Each integral curve is called acharacteristic curve of the PDE (1) equation and follow as the solutions of thecharacteristic equations:^[3]

A parametrization invariant form of theLagrange–Charpit equations is:^[5]

${\displaystyle \frac{dx}{a(x,y,z)}=\frac{dy}{b(x,y,z)}=\frac{dz}{c(x,y,z)}.}$

Consider now a PDE of the form

${\displaystyle \sum _{i=1}^n{a}_i(x_1,\dots ,x_n,u)\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}=c(x_1,\dots ,x_n,u).}$

For this PDE to belinear, the coefficientsa_i may be functions of the spatial variables only, and independent ofu. For it to be quasilinear,^[6]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

${\displaystyle (x_1,\dots ,x_n,u)=(X_1(s),\dots ,X_n(s),U(s))}$

${\displaystyle u(\mathbf{X}(s))=U(s)}$

for some univariate functions${\displaystyle s\mapsto (X_i(s){)}_i,U(s)}$of one real variable${\displaystyle s}$satisfying the following system of ordinary differential equations

${\displaystyle X_i^{\prime }=a_i(X_1,\dots ,X_n,U)\text{ for }i=1,\dots ,n}$

4

${\displaystyle U^{\prime }=c(X_1,\dots ,X_n,U).}$

5

Equations (4) and (5) give the characteristics of the PDE.

Consider the partial differential equation

${\displaystyle F(x_1,\dots ,x_n,u,p_1,\dots ,p_n)=0}$

6

where the variablesp_i are shorthand for the partial derivatives

${\displaystyle p_i=\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}.}$

Let${\displaystyle s\mapsto (x_1(s),\dots ,x_n(s),u(s),p_1(s),\dots ,p_n(s))}$ be a curve inR²ⁿ⁺¹. Suppose thatu is any solution, and that

${\displaystyle u(s)=u(x_1(s),\dots ,x_n(s)).}$

The derivatives with respect to${\displaystyle s}$ of${\displaystyle x_i,}$${\displaystyle u,}$ and${\displaystyle p_i}$ are written as${\displaystyle {\dot{x}}_i,}$,${\displaystyle \dot{u},}$ and${\displaystyle {\dot{p}}_i,}$ respectively.Along a solution, differentiating (6) with respect tos gives^[7]

${\displaystyle \sum _i(F_{x_i}+F_u{p}_i){\dot{x}}_i+\sum _i{F}_{p_i}{\dot{p}}_i=0}$

${\displaystyle \dot{u}-\sum _i{p}_i{\dot{x}}_i=0}$

${\displaystyle \sum _i({\dot{x}}_{i}d{p}_i-{\dot{p}}_{i}d{x}_i)=0.}$

The second equation follows from applying thechain rule to a solutionu, and the third follows by taking anexterior derivative of the relation${\displaystyle du-\sum _i{p}_{i}d{x}_i=0}$. Manipulating these equations gives

where λ is a constant. Writing these equations more symmetrically, one obtains the Lagrange–Charpit equations for the characteristic

${\displaystyle \frac{{\dot{x}}_i}{F_{p_i}}=-\frac{{\dot{p}}_i}{F_{x_i}+F_u{p}_i}=\frac{\dot{u}}{\sum p_i{F}_{p_i}}.}$

Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that theMonge cone of the differential equation should everywhere be tangent to the graph of the solution.

As an example, consider theadvection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs).

${\displaystyle a\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=0}$

where${\displaystyle a}$ is constant and${\displaystyle u}$ is a function of${\displaystyle x}$ and${\displaystyle t}$. We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form

${\displaystyle \frac{d}{ds}u(x(s),t(s))=F(u,x(s),t(s)),}$

where${\displaystyle (x(s),t(s))}$ is a characteristic line. First, we find

${\displaystyle \frac{d}{ds}u(x(s),t(s))=\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\frac{dx}{ds}+\frac{\mathrm{\partial }u}{\mathrm{\partial }t}\frac{dt}{ds}}$

by the chain rule. Now, if we set${\displaystyle \frac{dx}{ds}=a}$ and${\displaystyle \frac{dt}{ds}=1}$ we get

${\displaystyle a\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }t}}$

which is the left hand side of the PDE we started with. Thus

${\displaystyle \frac{d}{ds}u=a\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=0.}$

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

• ${\displaystyle \frac{dt}{ds}=1}$, letting${\displaystyle t(0)=0}$ we know${\displaystyle t=s}$,
• ${\displaystyle \frac{dx}{ds}=a}$, letting${\displaystyle x(0)=x_0}$ we know${\displaystyle x=as+x_0=at+x_0}$,
• ${\displaystyle \frac{du}{ds}=0}$, letting${\displaystyle u(0)=f(x_0)}$ we know${\displaystyle u(x(t),t)=f(x_0)=f(x-at)}$.

In this case, the characteristic lines are straight lines with slope${\displaystyle a}$, and the value of${\displaystyle u}$ remains constant along any characteristic line.

LetX be adifferentiable manifold andP a lineardifferential operator

${\displaystyle P:C^{\mathrm{\infty }}(X)\to C^{\mathrm{\infty }}(X)}$

of orderk. In a local coordinate systemxⁱ,

${\displaystyle P=\sum _{|\alpha |\le k}{P}^{\alpha }(x)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\alpha }}}$

in whichα denotes amulti-index. The principalsymbol ofP, denotedσ_P, is the function on thecotangent bundle T^∗X defined in these local coordinates by

${\displaystyle {\sigma }_P(x,\xi )=\sum _{|\alpha |=k}{P}^{\alpha }(x){\xi }_{\alpha }}$

where theξ_i are the fiber coordinates on the cotangent bundle induced by the coordinate differentialsdxⁱ. Although this is defined using a particular coordinate system, the transformation law relating theξ_i and thexⁱ ensures thatσ_P is a well-defined function on the cotangent bundle.

The functionσ_P ishomogeneous of degreek in theξ variable. The zeros ofσ_P, away from the zero section of T^∗X, are the characteristics ofP. A hypersurface ofX defined by the equationF(x) = c is called a characteristic hypersurface atx if

${\displaystyle {\sigma }_P(x,dF(x))=0.}$

Invariantly, a characteristic hypersurface is a hypersurface whoseconormal bundle is in the characteristic set ofP.

Characteristics are also a powerful tool for gaining qualitative insight into a PDE.

One can use the crossings of the characteristics to findshock waves forpotential flow in acompressible fluid. Intuitively, we can think of each characteristic line implying a solution to${\displaystyle u}$ 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.^[8]

Characteristics may fail to cover part of the domain of the PDE. This is called ararefaction, and indicates the solution typically exists only in aweak, 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 whichfinite difference scheme is best for the problem.

• Method of quantum characteristics

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• Prof. Scott Sarra tutorial on Method of Characteristics
• Prof. Alan Hood tutorial on Method of Characteristics

• Partial differential equations
• Hyperbolic partial differential equations

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