Difference between revisions of "Category:Parabolic"
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<p> | <p> | ||
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} +\ | + | <math>\sum^n_{i,j=1} a_{ij} \frac{\partial^2u}{\partial x_i \partial x_j} +\, \text{lower-order terms} = 0</math>. |
| − | + | </p> | |
| − | If <math>A=(a_{ij})_{ij}</math> is positive or negative semidefinite with exact one eigenvalue zero, the partial differential equation is called parabolic. | + | |
| − | + | <p> | |
| + | If the matrix <math>A=(a_{ij})_{ij}</math> is positive or negative semidefinite with exact one eigenvalue zero, the partial differential equation is called parabolic. | ||
| + | </p> | ||
| + | <p> | ||
An example is the heat equation: <math>\frac{\partial u}{\partial t}-\Delta u = f</math>, | An example is the heat equation: <math>\frac{\partial u}{\partial t}-\Delta u = f</math>, | ||
| − | where <math>\Delta</math> denotes the Laplace operator, <math>u</math> is the unknown, and the function <math>f</math> given. | + | where <math>\Delta</math> denotes the Laplace operator, <math>u</math> is the unknown, and the function <math>f</math> is given. |
</p> | </p> | ||
Latest revision as of 16:21, 24 February 2016
This category contains all control problems which are governed by a parabolic partial differential equation.
A second order linear partial differential equation can be written as
.
If the matrix 
is positive or negative semidefinite with exact one eigenvalue zero, the partial differential equation is called parabolic.
An example is the heat equation: 
,
where 
denotes the Laplace operator, 
is the unknown, and the function 
is given.
Pages in category "Parabolic"
This category contains only the following page.
