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| In [[mathematics]], a '''first-order partial differential equation''' is a [[partial differential equation]] that involves only first derivatives of the unknown function of ''n'' variables. The equation takes the form
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| :<math> F(x_1,\ldots,x_n,u,u_{x_1},\ldots u_{x_n}) =0. \,</math>
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| Such equations arise in the construction of characteristic surfaces for [[hyperbolic partial differential equation]]s, in the [[calculus of variations]], in some geometrical problems, and they arise in simple models for gas dynamics whose solution involves the [[method of characteristics]]. If a family of solutions
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| of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.
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| ==Characteristic surfaces for the wave equation==
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| Characteristic surfaces for the [[wave equation]] are level surfaces for solutions of the equation
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| :<math> u_t^2 = c^2 \left(u_x^2 +u_y^2 + u_z^2 \right). \,</math>
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| There is little loss of generality if we set <math>u_t =1</math>: in that case ''u'' satisfies
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| :<math> u_x^2 + u_y^2 + u_z^2= \frac{1}{c^2}. \,</math>
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| In vector notation, let
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| :<math> \vec x = (x,y,z) \quad \hbox{and} \quad \vec p = (u_x, u_y, u_z).\,</math>
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| A family of solutions with planes as level surfaces is given by
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| :<math> u(\vec x) = \vec p \cdot (\vec x - \vec{x_0}), \,</math>
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| where
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| :<math> | \vec p \,| = \frac{1}{c}, \quad \text{and} \quad \vec{x_0} \quad \text{is arbitrary}.\,</math>
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| If ''x'' and ''x''<sub>0</sub> are held fixed, the envelope of these solutions is obtained by finding a point on the sphere of radius 1/''c'' where the value of ''u'' is stationary. This is true if <math> \vec p</math> is parallel to <math>\vec x - \vec{x_0}</math>. Hence the envelope has equation
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| :<math> u(\vec x) = \pm \frac{1}{c} | \vec x -\vec{x_0} \,|.</math>
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| These solutions correspond to spheres whose radius grows or shrinks with velocity ''c''. These are light cones in space-time.
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| The initial value problem for this equation consists in specifying a level surface ''S'' where ''u''=0 for ''t''=0. The solution is obtained by taking the envelope of all the spheres with centers on ''S'', whose radii grow with velocity ''c''. This envelope is obtained by requiring that
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| :<math> \frac{1}{c} | \vec x - \vec{x_0}\, | \quad \hbox{is stationary for} \quad \vec{x_0} \in S. \,</math> | |
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| This condition will be satisfied if <math> | \vec x - \vec{x_0}\, |</math> is normal to ''S''. Thus the envelope corresponds to motion with velocity ''c'' along each normal to ''S''. This is the '''Huygens' construction of wave fronts''': each point on ''S'' emits a spherical wave at time ''t''=0, and the wave front at a later time ''t'' is the envelope of these spherical waves. The normals to ''S'' are the light rays.
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| ==Two-dimensional theory==
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| The notation is relatively simple in two space dimensions, but the main ideas generalize to higher dimensions. A general first-order partial differential equation has the form
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| :<math> F(x,y,u,p,q)=0, \,</math> | |
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| where
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| :<math> p=u_x, \quad q=u_y. \,</math>
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| A '''complete integral''' of this equation is a solution φ(''x'',''y'',''u'') that depends upon two parameters ''a'' and ''b''. (There are ''n'' parameters required in the ''n''-dimensional case.) An envelope of such solutions is obtained by choosing an arbitrary function ''w'', setting ''b''=''w''(''a''), and determining ''A''(''x'',''y'',''u'') by requiring that the total derivative
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| :<math> \frac{d \varphi}{d a} = \varphi_a(x,y,u,A,w(A)) + w'(A)\varphi_b(x,y,u,A,w(A)) =0. \,</math>
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| In that case, a solution <math>u_w</math> is also given by
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| :<math> u_w = \phi(x,y,u,A,w(A)) \,</math>
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| Each choice of the function ''w'' leads to a solution of the PDE. A similar process led to the construction of the light cone as a characteristic surface for the wave equation.
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| If a complete integral is not available, solutions may still be obtained by solving a system of ordinary equations. To obtain this system, first note that the PDE determines a cone (analogous to the light cone) at each point: if the PDE is linear in the derivatives of ''u'' (it is quasi-linear), then the cone degenerates into a line. In the general case, the pairs (''p'',''q'') that satisfy the equation determine a family of planes at a given point:
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| :<math> u - u_0 = p(x-x_0) + q(y-y_0), \,</math>
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| where
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| :<math> F(x_0,y_0,u_0,p,q) =0.\,</math> | |
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| The envelope of these planes is a cone, or a line if the PDE is quasi-linear. The condition for an envelope is
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| :<math> F_p\, dp + F_q \,dq =0, \,</math>
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| where F is evaluated at <math> (x_0, y_0,u_0,p,q)</math>, and ''dp'' and ''dq'' are increments of ''p'' and ''q'' that satisfy ''F''=0. Hence the generator of the cone is a line with direction
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| :<math> dx:dy:du = F_p:F_q:(pF_p + qF_q). \,</math>
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| This direction corresponds to the light rays for the wave equation.
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| To integrate differential equations along these directions, we require increments for ''p'' and ''q'' along the ray. This can be obtained by differentiating the PDE:
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| :<math> F_x +F_u p + F_p p_x + F_q p_y =0, \,</math>
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| :<math> F_y +F_u q + F_p q_x + F_q q_y =0,\,</math>
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| Therefore the ray direction in <math>(x,y,u,p,q)</math> space is
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| :<math> dx:dy:du:dp:dq = F_p:F_q:(pF_p + qF_q):(-F_x-F_u p):(-F_y - F_u q). \,</math>
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| The integration of these equations leads to a ray conoid at each point <math>(x_0,y_0,u_0)</math>. General solutions of the PDE can then be obtained from envelopes of such conoids.
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| == External links ==
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| * [http://www.scottsarra.org/shock/shock.html More detailed information on the Method of Characteristics]
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| == Bibliography ==
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| *R. Courant and D. Hilbert, ''Methods of Mathematical Physics, Vol II'', Wiley (Interscience), New York, 1962.
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| * L.C. Evans, ''Partial Differential Equations'', American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
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| * A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, ''Handbook of First Order Partial Differential Equations'', Taylor & Francis, London, 2002. ISBN 0-415-27267-X
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| * A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
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| * Sarra, Scott ''The Method of Characteristics with applications to Conservation Laws'', Journal of Online Mathematics and its Applications, 2003.
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| [[Category:Partial differential equations]]
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