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View Equation

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Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.1. Second-Order Parabolic Equations
Equation(s):$\displaystyle a\frac{\partial w}{\partial t}+\frac{\partial^2w}{\partial x^2}+b\frac{\partial ^2w}{\partial y^2}+c^2w=0$.
Solution(s),
Transformation(s),
Integral(s)
:
Particular solutions:\\
$\displaystyle w_{n,m}(t,x,y)=\sum^{n}_{k=0}\sum^{m-2j\geq0}_{j=0}{\frac{(-1)^{k+j}(k+j)!a^kt^{n-k}b^jy^{m-2j}x^{2k+2j}g(k+j,cx)}{k!j!(2k+2j)!(m-2j)!(n-k)!}}$,\\
$\displaystyle w_{n,m}(t,x,y)=\sum^{n}_{k=0}\sum^{m-2j\geq0}_{j=0}{\frac{(-1)^{k+j}(k+j)!a^kt^{n-k}b^jy^{m-2j}x^{2k+2j+1}g(k+j+1,cx)}{k!j!(2k+2j+1)!(m-2j)!(n-k)!}}$,\\
where $m,\,n=0,\,1,\,2,\,\dots\,$ and\\
$g(0,cx)=\cos(cx)$,\\
$\displaystyle g(1,cx)=\frac {\sin(cx)}{cx}$,\\
$\displaystyle g(2,cx)=\frac {3\sin(cx)}{c^3x^3}-\frac {3\cos(cx)}{c^2x^2}$,\\
$\displaystyle g(3,cx)=\frac {45\sin(cx)}{c^5x^5}-\frac {45\cos(cx)}{c^4x^4}-\frac {15\sin(cx)}{c^3x^3}$,\\
$\dots$,\\
$\displaystyle g(k,cx)=\frac {(2k-1)(2k-3)}{c^2x^2}[g(k-1,cx)-g(k-2,cx)]$.
Remarks:Here $g(k,0)=1$, $k=0,\,1,\,2,\,\dots\,$
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Sat 24 May 2008 20:00
Edits by author:1
Last edit by author:Sun 12 Sep 2010 19:45

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