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| A '''parabolic partial differential equation''' is a type of second-order [[partial differential equation]] (PDE) of the form
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| : <math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\,</math> | |
| that satisfies the condition
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| :<math>B^2 - AC = 0.\ </math>
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| This definition is analogous to the definition of a planar [[Parabola#Analytic_geometry_equations|parabola]].
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| This form of partial differential equation is used to describe a wide family of problems in science including [[heat diffusion]], [[underwater acoustics|ocean acoustic propagation]], physical or mathematical systems with a time variable, and processes that behave essentially like heat diffusing through a solid.
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| A simple example of a parabolic PDE is the one-dimensional [[heat equation]],
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| :<math>u_t = k u_{xx},\ </math> | |
| where <math>u(t,x)</math> is the temperature at time <math>t</math> and at position <math>x</math>, and <math>k</math> is a constant. The symbol <math>u_t</math> signifies the [[partial derivative]] with respect to the time variable <math>t</math>, and similarly <math>u_{xx}</math> is the second partial derivative with respect to <math>x</math>.
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| This equation says, roughly, that temperature at a given time and point rises or falls at a rate proportional to the difference between the temperature at that point and the average temperature near that point. The quantity <math>u_{xx}</math> measures how far off the temperature is from satisfying the mean value property of [[harmonic functions]].
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| A generalization of the heat equation is
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| :<math>u_t = -Lu,\ </math>
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| where <math>L</math> is a [[second order]] [[elliptic operator]] (implying <math>L</math> must be [[Positive operator|positive]] also; a case where <math>L</math> is non-positive is described below). Such a system can be hidden in an equation of the form
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| :<math>\nabla \cdot (a(x) \nabla u(x)) + b(x)^T \nabla u(x) + cu(x) = f(x)</math>
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| if the matrix-valued function <math>a(x)</math> has a [[Kernel (algebra)|kernel]] of dimension 1.
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| ==Solution==
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| Under broad assumptions, parabolic PDEs as given above have solutions for all ''x,y'' and ''t''>0. An equation of the form <math>u_t = -L(u)</math> is considered parabolic if ''L'' is a (possibly nonlinear) function of ''u'' and its first and second derivatives, with some further conditions on ''L''. With such a nonlinear parabolic differential equation, solutions exist for a short time—but may explode in a [[Mathematical singularity|singularity]] in a finite amount of time. Hence, the difficulty is in determining solutions for all time, or more generally studying the singularities that arise. This is in general quite difficult, as in the [[solution of the Poincaré conjecture]] via [[Ricci flow]].
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| ==Backward parabolic equation==
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| One may occasionally wish to consider PDEs of the form <math>u_t = Lu,\ </math> where <math>L</math> is a positive [[elliptic operator]]. While these problems are no longer necessarily [[well-posed]] (solutions may grow unbounded in finite time, or not even exist), they occur when studying the reflection of singularities of solutions to various other PDEs.<ref>{{citation | first=M. E. | last=Taylor | authorlink = Michael E. Taylor | title=Reflection of singularities of solutions to systems of differential equations | journal=Comm. Pure Appl. Math. | volume=28 | issue=4 | year=1975 | pages=457–478 | doi=10.1002/cpa.3160280403 }}</ref>
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| This class of equations is closely related to standard hyperbolic equations, which can be seen easily by considering the so-called 'backwards heat equation':
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| :<math>\begin{cases} u_{t} = \Delta u & \textrm{on} \ \ \Omega \times (0,T), \\ u=0 & \textrm{on} \ \ \partial\Omega \times (0,T), \\ u = f & \textrm{on} \ \ \Omega \times \left \{ T \right \}. \end{cases} </math>
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| This is essentially the same as the backward hyperbolic equation:
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| :<math>\begin{cases} u_{t} = -\Delta u & \textrm{on} \ \ \Omega \times (0,T), \\ u=0 & \textrm{on} \ \ \partial\Omega \times (0,T), \\ u = f & \textrm{on} \ \ \Omega \times \left \{ 0 \right \}. \end{cases} </math>
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| == Examples ==
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| * [[Heat equation]]
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| * [[Mean curvature flow]]
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| * [[Ricci flow]]
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| == See also == <!-- {{Div col}} -->
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| *[[Hyperbolic partial differential equation]]
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| *[[Elliptic partial differential equation]]
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| *[[Autowave]]
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| <!-- {{Div col end}} --> | |
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| == Notes ==
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| <references/> | |
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| == References ==
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| *{{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | origyear=1998 | url=http://www.ams.org/bull/2000-37-03/.../S0273-0979-00-00868-5.pdf | publisher=[[American Mathematical Society]] | location=Providence, R.I. | edition=2nd | series=Graduate Studies in Mathematics | isbn=978-0-8218-4974-3 | id={{MathSciNet | id = 2597943}} | year=2010 | volume=19}}
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| * {{springer|title=Parabolic partial differential equation|id=p/p071210}}
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| * {{springer|title=Parabolic partial differential equation, numerical methods|id=p/p071220}}
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| {{DEFAULTSORT:Parabolic Partial Differential Equation}}
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| [[Category:Parabolic partial differential equations|*]]
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