Airy equation: Difference between revisions
(explicit relationship between Airy and linear KdV) |
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The (homogeneous) '''Airy equation''' is given by | The (homogeneous) '''Airy equation''' is given by | ||
:<math> y_{xx} - x y = 0. </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 (or linearized) [[Korteweg-de Vries equation]] | This equation can be solved by power series expansion or Laplace transform 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 (or linearized) [[Korteweg-de Vries equation]] | ||
:<math> u_t + u_{xxx} = 0, </math> | :<math>\, u_t + u_{xxx} = 0, \!</math> | ||
which is the linear component of many [[KdV equations|equations of KdV type]]. The relationship between the two equations is that if y(x) solves the Airy equation, then | which is the linear component of many [[KdV equations|equations of KdV type]]. The relationship between the two equations is that if y(x) solves the Airy equation, then | ||
:<math>u(t,x) := t^{-1/3} y( x / (3t)^{1/3} )</math> | :<math>\, u(t,x) := t^{-1/3} y( x / (3t)^{1/3} )\!</math> | ||
solves the linearized KdV equation (and for the correct choice of y, can in fact be used as the fundamental solution for this equation). | solves the linearized KdV equation (and for the correct choice of y, can in fact be used as the fundamental solution for this equation). | ||
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For applications to nonlinear perturbation problems, it is often important to study the more general inhomogeneous linear component of the KdV equation | 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> | ||
for various forcing terms F. Of course, the inhomogeneous and homogeneous equations are related by [[Duhamel's formula]]. | for various forcing terms F. Of course, the inhomogeneous and homogeneous equations are related by [[Duhamel's formula]]. | ||
Revision as of 14:00, 24 August 2009
The (homogeneous) Airy equation is given by
This equation can be solved by power series expansion or Laplace transform 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 (or linearized) Korteweg-de Vries equation
which is the linear component of many equations of KdV type. The relationship between the two equations is that if y(x) solves the Airy equation, then
solves the linearized KdV equation (and for the correct choice of y, can in fact be used as the fundamental solution for this equation).
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.
