Editing Method of characteristics

{{Short description|Technique for solving hyperbolic partial differential equations}}
{{Differential equations}}

In [[mathematics]], the '''method of characteristics''' is a technique for solving particular [[partial differential equations]].  Typically, it applies to [[first-order partial differential equation|first-order equations]], though in general [[Partial_differential_equation#Method_of_characteristics|characteristic curves]] can also be found for [[Hyperbolic partial differential equation|hyperbolic]] and [[parabolic partial differential equation]].  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.{{sfn|Zachmanoglou|Thoe|1986|pp=112–152}}{{sfn|Pinchover|Rubinstein|2005|pp=25-28}} 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 [[partial differential equation#Linear and nonlinear equations|quasilinear PDE]] of the form{{sfn|John|1991|p=9}}

{{NumBlk|:|<math>a(x,y,z) \frac{\partial z}{\partial x}+b(x,y,z) \frac{\partial z}{\partial y}=c(x,y,z).</math>|{{EquationRef|1}}}}

Suppose that a solution ''z'' is known, and consider the surface graph ''z''&nbsp;=&nbsp;''z''(''x'',''y'') in '''R'''<sup>3</sup>.  A [[normal vector]] to this surface is given by{{sfn|Zauderer|2006|p=82}}

:<math>\left(\frac{\partial z}{\partial x}(x,y),\frac{\partial z}{\partial y}(x,y),-1\right).\,</math>

As a result, equation ({{EquationNote|1}}) is equivalent to the geometrical statement that the vector field

:<math>(a(x,y,z),b(x,y,z),c(x,y,z))\,</math>

is tangent to the surface ''z''&nbsp;=&nbsp;''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 curve]]s of this vector field.  These integral curves are called the characteristic curves of the original partial differential equation and follow as the solutions of the characteristic equations:{{sfn|John|1991|p=9}}

:<math>
\begin{align}
\frac{dx}{dt}&=a(x,y,z),\\[8pt]
\frac{dy}{dt}&=b(x,y,z),\\[8pt]
\frac{dz}{dt}&=c(x,y,z).
\end{align}
</math>

A parametrization invariant form of the ''Lagrange–Charpit equations'' is:{{sfn|Demidov|1982|pp=331–333}}

:<math>\frac{dx}{a(x,y,z)} = \frac{dy}{b(x,y,z)} = \frac{dz}{c(x,y,z)} .</math>

===Linear and quasilinear cases===
Consider now a PDE of the form

:<math>\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).</math>

For this PDE to be [[linear]], the coefficients ''a''<sub>''i''</sub> may be functions of the spatial variables only, and independent of ''u''.  For it to be quasilinear,<ref name="quasilinear">{{cite web| url = https://reference.wolfram.com/language/tutorial/DSolveLinearAndQuasiLinearFirstOrderPDEs.html |title = Partial Differential Equations (PDEs)—Wolfram Language Documentation}}</ref> ''a''<sub>''i''</sub> 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

:<math>(x_1,\dots,x_n,u) = (X_1(s),\dots,X_n(s),U(s))</math>
:<math>u(\mathbf{X}(s)) = U(s)</math>
for some univariate functions <math>s\mapsto (X_i(s))_i,U(s)</math>
of one real variable <math>s</math>
satisfying the following system of ordinary differential equations

{{NumBlk|:|<math>X_i' = a_i(X_1,\dots,X_n,U)
\text{ for }i=1,\dotsc,n</math>|{{EquationRef|2}}}}
{{NumBlk|:|<math>U' = c(X_1,\dots,X_n,U).</math>|{{EquationRef|3}}}}

Equations ({{EquationNote|2}}) and ({{EquationNote|3}}) give the characteristics of the PDE.

{{collapse top|title=Proof for quasilinear case}} 
In the quasilinear case, the use of the method of characteristics is justified by [[Grönwall's inequality]]. The above equation may be written as
<math display="block">\mathbf{a}(\mathbf{x},u) \cdot \nabla u(\mathbf{x}) = c(\mathbf{x},u) </math>

We must distinguish between the solutions to the ODE and the solutions to the PDE, which we do not know are equal ''a priori.'' Letting capital letters be the solutions to the ODE we find
<math display="block">\mathbf{X}'(s) = \mathbf{a}(\mathbf{X}(s),U(s)) </math>
<math display="block">U'(s) = c(\mathbf{X}(s), U(s)) </math>

Examining <math>\Delta(s) = |u(\mathbf{X}(s)) - U(s)|^2 </math>, we find, upon differentiating that
<math display="block">\Delta'(s) = 2\big(u(\mathbf{X}(s)) - U(s)\big) \Big(\mathbf{X}'(s)\cdot \nabla u(\mathbf{X}(s)) - U'(s)\Big) </math>
which is the same as
<math display="block">\Delta'(s) = 2\big(u(\mathbf{X}(s)) - U(s)\big) \Big(\mathbf{a}(\mathbf{X}(s),U(s))\cdot \nabla u(\mathbf{X}(s)) - c(\mathbf{X}(s),U(s))\Big) </math>

We cannot conclude the above is 0 as we would like, since the PDE only guarantees us that this relationship is satisfied for
<math>u(\mathbf{x})</math>, <math>\mathbf{a}(\mathbf{x},u) \cdot \nabla u(\mathbf{x}) = c(\mathbf{x},u)</math>,
and we do not yet know that <math>U(s) = u(\mathbf{X}(s))</math>.

However, we can see that
<math display="block">\Delta'(s) = 2\big(u(\mathbf{X}(s)) - U(s)\big) \Big(\mathbf{a}(\mathbf{X}(s),U(s))\cdot \nabla u(\mathbf{X}(s)) - c(\mathbf{X}(s),U(s))-\big(\mathbf{a}(\mathbf{X}(s),u(\mathbf{X}(s))) \cdot \nabla u(\mathbf{X}(s)) - c(\mathbf{X}(s),u(\mathbf{X}(s)))\big)\Big) </math>
since by the PDE, the last term is 0. This equals
<math display="block">\Delta'(s) = 2\big(u(\mathbf{X}(s)) - U(s)\big) \Big(\big(\mathbf{a}(\mathbf{X}(s),U(s))-\mathbf{a}(\mathbf{X}(s),u(\mathbf{X}(s)))\big)\cdot \nabla u(\mathbf{X}(s)) - \big(c(\mathbf{X}(s),U(s))-c(\mathbf{X}(s),u(\mathbf{X}(s)))\big)\Big) </math>

By the triangle inequality, we have
<math display="block">|\Delta'(s)| \leq 2\big|u(\mathbf{X}(s)) - U(s)\big| \Big(\big\|\mathbf{a}(\mathbf{X}(s),U(s))-\mathbf{a}(\mathbf{X}(s),u(\mathbf{X}(s)))\big\| \ \|\nabla u(\mathbf{X}(s))\| + \big|c(\mathbf{X}(s),U(s))-c(\mathbf{X}(s),u(\mathbf{X}(s)))\big|\Big) </math>

Assuming <math>\mathbf{a},c </math> are at least <math>C^1 </math>, we can bound this for small times. Choose a neighborhood <math>\Omega </math> around <math>\mathbf{X}(0), U(0) </math> small enough such that <math>\mathbf{a},c </math> are [[locally Lipschitz]]. By continuity, <math>(\mathbf{X}(s),U(s)) </math> will remain in <math>\Omega </math> for small enough <math>s
 </math>. Since <math>U(0) = u(\mathbf{X}(0)) </math>, we also have that <math>(\mathbf{X}(s), u(\mathbf{X}(s))) </math> will be in <math>\Omega </math> for small enough <math>s </math> by continuity. So, <math>(\mathbf{X}(s),U(s)) \in \Omega </math> and <math>(\mathbf{X}(s), u(\mathbf{X}(s))) \in \Omega </math> for <math>s \in [0,s_0] </math>. Additionally, <math>\|\nabla u(\mathbf{X}(s))\| \leq M </math> for some <math>M \in \R </math> for <math>s \in [0,s_0] </math> by compactness. From this, we find the above is bounded as
<math display="block">|\Delta'(s)| \leq C|u(\mathbf{X}(s)) - U(s)|^2 = C |\Delta(s)| </math>
for some <math>C \in \mathbb{R} </math>. It is a straightforward application of Grönwall's Inequality to show that since <math>\Delta(0) = 0 </math> we have <math>\Delta(s) = 0 </math> for as long as this inequality holds. We have some interval <math>[0, \varepsilon) </math> such that <math>u(X(s)) = U(s) </math> in this interval. Choose the largest <math>\varepsilon </math> such that this is true. Then, by continuity, <math>U(\varepsilon) = u(\mathbf{X}(\varepsilon)) </math>. Provided the ODE still has a solution in some interval after <math>\varepsilon </math>, we can repeat the argument above to find that <math>u(X(s)) = U(s) </math> in a larger interval. Thus, so long as the ODE has a solution, we have <math>u(X(s)) = U(s) </math>.
{{collapse bottom}}

===Fully nonlinear case===
Consider the partial differential equation

{{NumBlk|:|<math>F(x_1,\dots,x_n,u,p_1,\dots,p_n)=0</math>|{{EquationRef|4}}}}

where the variables ''p''<sub>i</sub> are shorthand for the partial derivatives

:<math>p_i = \frac{\partial u}{\partial x_i}.</math>

Let (''x''<sub>i</sub>(''s''),''u''(''s''),''p''<sub>i</sub>(''s'')) be a curve in '''R'''<sup>2n+1</sup>.  Suppose that ''u'' is any solution, and that

:<math>u(s) = u(x_1(s),\dots,x_n(s)).</math>

Along a solution, differentiating ({{EquationNote|4}}) with respect to ''s'' gives{{sfn|John|1991|pp=19-24}}

:<math>\sum_i(F_{x_i} + F_u p_i)\dot{x}_i + \sum_i F_{p_i}\dot{p}_i = 0</math>

:<math>\dot{u} - \sum_i p_i \dot{x}_i = 0</math>

:<math>\sum_i (\dot{x}_i dp_i - \dot{p}_i dx_i)= 0.</math>

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 <math>du - \sum_i p_i \, dx_i = 0</math>.  Manipulating these equations gives

:<math>\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}</math>

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

:<math>\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}}.</math>

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).

:<math>a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0</math>

where <math>a</math> is constant and <math>u</math> is a function of <math>x</math> and <math>t</math>. We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form

:<math> \frac{d}{ds}u(x(s), t(s)) = F(u, x(s), t(s)) ,</math>

where <math>(x(s),t(s))</math> is a characteristic line. First, we find

:<math>\frac{d}{ds}u(x(s), t(s)) = \frac{\partial u}{\partial x} \frac{dx}{ds} + \frac{\partial u}{\partial t} \frac{dt}{ds}</math>

by the chain rule.  Now, if we set <math> \frac{dx}{ds} = a</math> and <math>\frac{dt}{ds} = 1</math> we get

:<math> a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} </math>

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

:<math>\frac{d}{ds}u = a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0.</math>

So, along the characteristic line <math>(x(s), t(s))</math>, the original PDE becomes the ODE <math>u_s = F(u, x(s), t(s)) = 0</math>. That is to say that along the characteristics, the solution is constant. Thus, <math>u(x_s, t_s) = u(x_0, 0)</math> where <math>(x_s, t_s)\,</math> and <math>(x_0, 0)</math> 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:

* <math>\frac{dt}{ds} = 1</math>, letting <math>t(0)=0</math> we know <math>t=s</math>,
* <math>\frac{dx}{ds} = a</math>, letting <math>x(0)=x_0</math> we know <math>x=as+x_0=at+x_0</math>,
* <math>\frac{du}{ds} = 0</math>, letting <math>u(0)=f(x_0)</math> we know <math>u(x(t), t)=f(x_0)=f(x-at)</math>.

In this case, the characteristic lines are straight lines with slope <math>a</math>, and the value of <math>u</math> remains constant along any characteristic line.

== Characteristics of linear differential operators ==
Let ''X'' be a [[differentiable manifold]] and ''P'' a linear [[differential operator]]
:<math>P : C^\infty(X) \to C^\infty(X)</math>
of order ''k''.  In a local coordinate system ''x''<sup>''i''</sup>,
:<math>P = \sum_{|\alpha|\le k} P^{\alpha}(x)\frac{\partial}{\partial x^\alpha}</math>
in which ''α'' denotes a [[multi-index]]. The principal [[Symbol of a differential operator|symbol]] of ''P'', denoted ''σ''<sub>''P''</sub>, is the function on the [[cotangent bundle]] T<sup>∗</sup>''X'' defined in these local coordinates by

:<math>\sigma_P(x,\xi) = \sum_{|\alpha|=k} P^\alpha(x)\xi_\alpha</math>

where the ''ξ''<sub>''i''</sub> are the fiber coordinates on the cotangent bundle induced by the coordinate differentials ''dx''<sup>''i''</sup>.  Although this is defined using a particular coordinate system, the transformation law relating the ''ξ''<sub>''i''</sub> and the ''x''<sup>''i''</sup> ensures that ''σ''<sub>''P''</sub> is a well-defined function on the cotangent bundle.

The function ''σ''<sub>''P''</sub> is [[homogeneous function|homogeneous]] of degree ''k'' in the ''ξ'' variable. The zeros of ''σ''<sub>''P''</sub>, away from the zero section of T<sup>∗</sup>''X'', are the characteristics of ''P''. A hypersurface of ''X'' defined by the equation ''F''(''x'')&nbsp;=&nbsp;''c'' is called a characteristic hypersurface at ''x'' if
:<math>\sigma_P(x,dF(x)) = 0.</math>
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 wave]]s for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to <math>u</math> 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.<ref>{{citation |first=Lokenath |last=Debnath |authorlink=Lokenath Debnath |title=Nonlinear Partial Differential Equations for Scientists and Engineers |location=Boston |publisher=Birkhäuser |edition=2nd |year=2005 |isbn=0-8176-4323-0 |chapter=Conservation Laws and Shock Waves |pages=251–276 }}</ref>

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 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 which [[finite difference]] scheme is best for the problem.

== See also ==
* [[Method of quantum characteristics]]

==Notes==
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{{reflist}}

== References ==
* {{citation|first1=Richard|last1=Courant|author-link1=Richard Courant|first2=David|last2=Hilbert|author-link2=David Hilbert|title=Methods of Mathematical Physics, Volume II|publisher=Wiley-Interscience|year=1962}}
* {{cite journal | last=Demidov | first=S. S. | title=The study of partial differential equations of the first order in the 18th and 19th centuries | journal=Archive for History of Exact Sciences | publisher=Springer Science and Business Media LLC | volume=26 | issue=4 | year=1982 | issn=0003-9519 | doi=10.1007/bf00418753 | pages=325–350}}
* {{citation|first=Lawrence C.|last=Evans|author-link1=Lawrence C. Evans|title=Partial Differential Equations|publisher=American Mathematical Society|publication-place=Providence|year=1998|isbn=0-8218-0772-2}}
* {{cite book | last=John | first=Fritz | title=Partial Differential Equations | publisher=Springer Science & Business Media | publication-place=New York|edition=4th|year=1991|isbn=978-0-387-90609-6|url-access=registration|url=https://archive.org/details/partialdifferent00john_0}}
* {{cite book | last=Zauderer | first=Erich | title=Partial Differential Equations of Applied Mathematics | publisher=Wiley | date=2006 | isbn=978-0-471-69073-3 | doi=10.1002/9781118033302 | doi-access=free}}* {{citation|first1=A. D.|last1=Polyanin|first2=V. F.|last2=Zaitsev|first3=A.|last3=Moussiaux|title=Handbook of First Order Partial Differential Equations|publisher=Taylor & Francis|publication-place=London|year=2002|isbn=0-415-27267-X}}
* {{cite book | last=Pinchover | first=Yehuda | last2=Rubinstein | first2=Jacob | title=An Introduction to Partial Differential Equations | publisher=Cambridge University Press | date=2005 | isbn=978-0-511-80122-8 | doi=10.1017/cbo9780511801228}}
* {{citation|first=A. D.|last=Polyanin|title=Handbook of Linear Partial Differential Equations for Engineers and Scientists|publisher=Chapman & Hall/CRC Press|publication-place=Boca Raton|year=2002|isbn=1-58488-299-9}}
* {{citation|last=Sarra|first=Scott|title=The Method of Characteristics with applications to Conservation Laws|journal=Journal of Online Mathematics and Its Applications|year=2003|url=http://www.scottsarra.org/shock/shock.html}}
* {{citation |last1=Streeter |first1=VL|last2=Wylie |first2=EB|title=Fluid mechanics|publisher=McGraw-Hill Higher Education|edition=International 9th Revised|year=1998}}
* {{cite book | last=Zachmanoglou | first=E. C. | last2=Thoe | first2=Dale W. | title=Introduction to Partial Differential Equations with Applications | publisher=Courier Corporation | publication-place=New York | date=1986 | isbn=0-486-65251-3}}

== External links ==
* [http://www.scottsarra.org/shock/shock.html Prof. Scott Sarra tutorial on Method of Characteristics]
* [https://web.archive.org/web/20211022212853/https://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node5.html Prof. Alan Hood tutorial on Method of Characteristics]

{{Numerical PDE}}


[[Category:Partial differential equations]]
[[Category:Hyperbolic partial differential equations]]