Difference between revisions of "Category:Hyperbolic"

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This category contains all control problems which are governed by a hyperbolic partial differential equation.  
 
This category contains all control problems which are governed by a hyperbolic partial differential equation.  
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  A second order linear partial differential equation can be written as
 
  A second order linear partial differential equation can be written as
  <math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\quad \textrm{ lower-order terms} = 0</math>.
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  <math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\, \text{lower-order terms} = 0</math>.  
   
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  If <math>A=(a_{ij})_{ij}</math> is indefinite such that <math>n-1</math> eigenvalues have the same sign and the remaining eigenvalue the other sign, the partial differential equation is called hyperbolic.
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  If the matrix <math>A=(a_{ij})_{ij}</math> is indefinite such that <math>n-1</math> eigenvalues have the same sign and the remaining eigenvalue has the other sign, the partial differential equation is called hyperbolic.
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  An example is the wave equation: <math>\frac{\partial^2 u}{\partial t^2}-\Delta u = f</math>,
 
  An example is the wave equation: <math>\frac{\partial^2 u}{\partial t^2}-\Delta u = f</math>,
 
  where <math>\Delta</math> denotes the Laplace operator, <math>u</math> is the unknown, and the function <math>f</math> is given.
 
  where <math>\Delta</math> denotes the Laplace operator, <math>u</math> is the unknown, and the function <math>f</math> is given.
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<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
 
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->

Latest revision as of 16:24, 24 February 2016

This category contains all control problems which are governed by a hyperbolic partial differential equation.

A second order linear partial differential equation can be written as \sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\, \text{lower-order terms} = 0.

If the matrix A=(a_{ij})_{ij} is indefinite such that n-1 eigenvalues have the same sign and the remaining eigenvalue has the other sign, the partial differential equation is called hyperbolic.

An example is the wave equation: \frac{\partial^2 u}{\partial t^2}-\Delta u = f, where \Delta denotes the Laplace operator, u is the unknown, and the function f is given.

Pages in category "Hyperbolic"

This category contains only the following page.