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| In [[mathematics]], the '''method of characteristics''' is a technique for solving [[partial differential equations]]. Typically, it applies to [[first-order partial differential equation|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]].
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| ==Characteristics of first-order partial differential equation==
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| For a first-order PDE ([[partial differential equations]]), 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.
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| 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
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| {{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}}}}
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| Suppose that a solution ''z'' is known, and consider the surface graph ''z'' = ''z''(''x'',''y'') in '''R'''<sup>3</sup>. A [[normal vector]] to this surface is given by
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| :<math>\left(\frac{\partial z}{\partial x}(x,y),\frac{\partial z}{\partial y}(x,y),-1\right).\,</math>
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| As a result,<ref>{{harvnb|John|1991}}</ref> equation ({{EquationNote|1}}) is equivalent to the geometrical statement that the vector field
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| :<math>(a(x,y,z),b(x,y,z),c(x,y,z))\,</math>
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| 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 curve]]s of this vector field. These integral curves are called the characteristic curves of the original partial differential equation.
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| The equations of the characteristic curve may be expressed invariantly by the ''Lagrange-Charpit equations''<ref>{{harvnb|Delgado|1997}}</ref>
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| :<math>\frac{dx}{a(x,y,z)} = \frac{dy}{b(x,y,z)} = \frac{dz}{c(x,y,z)},</math>
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| 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''):
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| :<math>
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| \begin{array}{rcl}
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| \frac{dx}{dt}&=&a(x,y,z)\\
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| \frac{dy}{dt}&=&b(x,y,z)\\
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| \frac{dz}{dt}&=&c(x,y,z).
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| \end{array}
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| </math>
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| These are the '''characteristic equations''' for the original system.<!-- [[Characteristic equations]] redirects to this article -->
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| ===Linear and quasilinear cases===
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| Consider now a PDE of the form
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| :<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>
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| 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 [[Differential equations#Types of differential equations|quasilinear]], ''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.
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| For a linear or quasilinear PDE, the characteristic curves are given parametrically by
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| :<math>(x_1,\dots,x_n,u) = (x_1(s),\dots,x_n(s),u(s))</math>
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| such that the following system of ODEs is satisfied
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| {{NumBlk|:|<math>\frac{dx_i}{ds} = a_i(x_1,\dots,x_n,u)</math>|{{EquationRef|2}}}}
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| {{NumBlk|:|<math>\frac{du}{ds} = c(x_1,\dots,x_n,u).</math>|{{EquationRef|3}}}}
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| Equations ({{EquationNote|2}}) and ({{EquationNote|3}}) give the characteristics of the PDE.
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| ===Fully nonlinear case===
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| Consider the partial differential equation
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| {{NumBlk|:|<math>F(x_1,\dots,x_n,u,p_1,\dots,p_n)=0</math>|{{EquationRef|4}}}}
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| where the variables ''p''<sub>i</sub> are shorthand for the partial derivatives
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| :<math>p_i = \frac{\partial u}{\partial x_i}.</math>
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| 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
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| :<math>u(s) = u(x_1(s),\dots,x_n(s)).</math>
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| Along a solution, differentiating ({{EquationNote|4}}) with respect to ''s'' gives
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| :<math>\sum_i(F_{x_i} + F_u p_i)\dot{x}_i + \sum_i F_{p_i}\dot{p}_i = 0</math>
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| :<math>\dot{u} - \sum_i p_i \dot{x}_i = 0</math>
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| :<math>\sum_i (\dot{x}_i dp_i - \dot{p}_i dx_i)= 0.</math>
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| 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 | |
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| :<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>
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| where λ is a constant. Writing these equations more symmetrically, one obtains the Lagrange-Charpit equations for the characteristic
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| :<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>
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| 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.
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| == Example ==
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| As an example, consider the [[advection equation]] (this example assumes familiarity with PDE notation, and solutions to basic ODEs).
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| :<math>a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0\,</math>
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| 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
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| :<math> \frac{d}{ds}u(x(s), t(s)) = F(u, x(s), t(s)) </math>,
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| where <math>(x(s),t(s))\,</math> is a characteristic line. First, we find
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| :<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>
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| by the chain rule. Now, if we set <math> \frac{dx}{ds} = a</math> and <math>\frac{dt}{ds} = 1</math> we get | |
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| :<math> a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} \,</math>
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| which is the left hand side of the PDE we started with. Thus
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| :<math>\frac{d}{ds}u = a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0.</math>
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| 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. So to determine the general solution, it is enough to find the characteristics by solving the characteristic system of ODEs:
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| * <math>\frac{dt}{ds} = 1</math>, letting <math>t(0)=0\,</math> we know <math>t=s\,</math>,
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| * <math>\frac{dx}{ds} = a</math>, letting <math>x(0)=x_0\,</math> we know <math>x=as+x_0=at+x_0\,</math>,
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| * <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>.
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| 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.
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| == Characteristics of linear differential operators ==
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| Let ''X'' be a [[differentiable manifold]] and ''P'' a linear [[differential operator]]
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| :<math>P : C^\infty(X) \to C^\infty(X)</math>
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| of order ''k''. In a local coordinate system ''x''<sup>''i''</sub>, | |
| :<math>P = \sum_{|\alpha|\le k} P^{\alpha}(x)\frac{\partial}{\partial x^\alpha}</math>
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| 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
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| :<math>\sigma_P(x,\xi) = \sum_{|\alpha|=k} P^\alpha(x)\xi_\alpha</math> | |
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| where the ξ<sub>''i''</sub> are the fiber coordinates on the cotangent bundle induced by the coordinate differentials d''x''<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.
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| 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'') = ''c'' is called a characteristic hypersurface at ''x'' if
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| :<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''.
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| == Qualitative analysis of characteristics ==
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| Characteristics are also a powerful tool for gaining qualitative insight into a PDE.
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| 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.
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| 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.
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| 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.
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| == See also ==
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| * [[Method of quantum characteristics]]
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| ==Notes==
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| <div class="references-small" >
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| <references />
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| </div>
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| == References ==
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| * {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|title=Methods of Mathematical Physics, Volume II|publisher=Wiley-Interscience|year=1962}}
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| *{{citation|jstor=2133111|last=Delgado|first=Manuel|title=The Lagrange-Charpit Method|journal=SIAM Review|volume=39|year=1997|pages=298–304|doi=10.1137/S0036144595293534|issue=2|bibcode = 1997SIAMR..39..298D }}
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| * {{citation|first=Lawrence C.|last=Evans|title=Partial Differential Equations|publisher=American Mathematical Society|publication-place=Providence|year=1998|isbn=0-8218-0772-2}}
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| * {{citation|first=Fritz|last=John|authorlink=Fritz John|title=Partial differential equations|publisher=Springer|edition=4th |year=1991|isbn=978-0-387-90609-6}}
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| * {{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}}
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| * {{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}}
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| * {{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}}.
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| *{{citation|last1=Streeter|first1=VL|last2=Wylie|first2=EB|title=Fluid mechanics|publisher=McGraw-Hill Higher Education|edition=International 9<sup>th</sup> Revised|year=1998}}
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| == External links ==
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| * [http://www.scottsarra.org/shock/shock.html Prof. Scott Sarra tutorial on Method of Characteristics]
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| * [http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node5.html Prof. Alan Hood tutorial on Method of Characteristics]
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| [[Category:Partial differential equations]]
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