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Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.5. Higher-Order Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial t}+a\frac{\partial^3w}{\partial x^3}=0$,\hfill\break
the linearized Corteveg-de Vries equation.
Solution(s),
Transformation(s),
Integral(s)
:
Particular solutions:\hfill\break
$w_0(x,t)=1$,\\
$w_1(x,t)=x$,\\
$w_2(x,t)=x^2$,\\
$\displaystyle w_3(x,t)=x^3-3!at$,\\
$\displaystyle w_4(x,t)=x^4-4!axt$,\\
$\displaystyle w_5(x,t)=x^5-\frac {5!a}{2!}x^2t$,\\
$\displaystyle w_6(x,t)=x^6-\frac {6!a}{3!}x^3t+\frac {6!a^2}{2!}t^2$,\\
$\displaystyle w_7(x,t)=x^7-\frac {7!a}{4!}x^4t+\frac {7!a^2}{2!}xt^2$,\\
$\displaystyle w_8(x,t)=x^8-\frac {8!a}{5!}x^5t+\frac {8!a^2}{2!2!}x^2t^2$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=\sum_{k=0}^{n-3k\geq0} \frac {(-1)^kn!a^kt^kx^{n-3k}}{(n-3k)!k!}$.\\
Novelty:New solution(s) / integral(s)
Admin's Comment:For other solutions see, for example, 
{\it Handbook of Linear Partial Differential Equations for Engineers and Scientists} by A. D. Polyanin (2002, page 601).
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Mon 12 May 2008 17:57
Edits by author:0

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