Airy equation: Difference between revisions
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(The equation discussed here was not the Airy equation) |
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The (homogeneous) '''Airy equation''' is given by | The (homogeneous) '''Airy equation''' is given by | ||
:<math> | :<math> y_{xx} - x y = 0. </math> | ||
This equation can be solved by power series expansion techniques to give the Airy functions. The solutions to a wide class of problems may be expressed in terms of these functions. One such problem is the reduced [[Korteweg-de Vries equation]] | |||
:<math> u_t + u_{xxx} = 0, </math> | |||
which is the linear component of many [[KdV equations|equations of KdV type]]. For applications to nonlinear perturbation problems, it is often important to study the more general inhomogeneous linear component of the KdV equation | |||
:<math> u_t + u_{xxx} = F </math> | :<math> u_t + u_{xxx} = F </math> | ||
Revision as of 19:07, 16 April 2009
The (homogeneous) Airy equation is given by
This equation can be solved by power series expansion techniques to give the Airy functions. The solutions to a wide class of problems may be expressed in terms of these functions. One such problem is the reduced Korteweg-de Vries equation
which is the linear component of many equations of KdV type. For applications to nonlinear perturbation problems, it is often important to study the more general inhomogeneous linear component of the KdV equation
for various forcing terms F. Of course, the inhomogeneous and homogeneous equations are related by Duhamel's formula.
A large number of linear, bilinear, trilinear, and multilinear estimates for this equation are known; see the page on Airy estimates for more details.
