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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.1. Second-Order Parabolic Equations
Equation(s):$\displaystyle a\frac{\partial w}{\partial t}+\frac{\partial^2w}{\partial x^2}+c^2w=0$.
Solution(s),
Transformation(s),
Integral(s)
:
Solutions:\hfill\break
$\displaystyle w_{n}(t,x)=\sum^{n}_{k=0}{\frac{(-a)^kt^{n-k}x^{2k}g(k,cx)}{(n-k)!(2k)!}},\quad n=0,\,1,\,2,\,...\,$;\\
$\displaystyle w_{n}(t,x)=\sum^{n}_{k=0}{\frac{(-a)^kt^{n-k}x^{2k+1}g(k+1,cx)}{(n-k)!(2k+1)!}},\quad n=0,\,1,\,2,\,...\,$,\\
where\\
$g(0,cx)=\cos(cx)$,\\
$\displaystyle g(1,cx)=\frac {\sin(cx)}{cx}$,\\
$\displaystyle g(2,cx)=\frac {3\sin(cx)}{c^3x^3}-\frac {3\cos(cx)}{c^2x^2}$,\\
$\displaystyle g(3,cx)=\frac {45\sin(cx)}{c^5x^5}-\frac {45\cos(cx)}{c^4x^4}-\frac {15\sin(cx)}{c^3x^3}$,\\
\dots,\\
$\displaystyle g(k,cx)=\frac {(2k-1)(2k-3)}{c^2x^2}[g(k-1,cx)-g(k-2,cx)]$.
Remarks:Here $g(k,0)=1$; $k=0,\,1,\,2,\,...\,$
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 19:34
Edits by author:2
Last edit by author:Sun 12 Sep 2010 19:54

Edit (Only for author/contributor)


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