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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:3.2. Second-Order Hyperbolic Equations
Equation(s):$\displaystyle \frac{\partial^2w}{\partial t^2}=a\frac{\partial ^2w}{\partial
x^2}+b\frac{\partial ^2w}{\partial y^2}+q\frac{\partial
^2w}{\partial z^2}-c^2w$.
Solution(s),
Transformation(s),
Integral(s)
:
Particular solutions:\\
$$
w_{m,n,v}(t,x,y,z)=\sum^{m-2k\geq0}_{k=0}\sum^{n-2j\geq0}_{j=0}\sum^{v-2l\geq0}_{l=0}{\frac{(k+j+l)!t^{2k+2j+2l}a^kx^{m-2k}b^jy^{n-2j}q^lz^{v-2l}g(k+j+l,ct)}{k!j!l!(2k+2j+2l)!(m-2k)!(n-2j)!(v-2l)!}},
$$
$$
w_{m,n,v}(t,x,y,z)=\sum^{m-2k\geq0}_{k=0}\sum^{n-2j\geq0}_{j=0}\sum^{v-2l\geq0}_{l=0}{\frac{(k+j+l)!t^{2k+2j+2l+1}a^kx^{m-2k}b^jy^{n-2j}q^lz^{v-2l}g(k+j+l+1,ct)}{k!j!l!(2k+2j+2l+1)!(m-2k)!(n-2j)!(v-2l)!}},
$$
where $m=0,\,1,\,2,\,\dots\,$;
$n=0,\,1,\,2,\,\dots\,$; $v=0,\,1,\,2,\,\dots\,$; and\\
$g(k,0)=1;\quad k=0,\,1,\,2,\,\dots\,$,\\
$g(0,ct)=\cos(ct)$,\\
$\displaystyle g(1,ct)=\frac {\sin(ct)}{ct}$,\\
$\displaystyle g(2,ct)=\frac {3\sin(ct)}{c^3t^3}-\frac {3\cos(ct)}{c^2t^2}$,\\
$\displaystyle g(3,ct)=\frac {45\sin(ct)}{c^5t^5}-\frac {45\cos(ct)}{c^4t^4}-\frac {15\sin(ct)}{c^3t^3}$,\\
$\dots$,\\
$\displaystyle g(k,ct)=\frac {(2k-1)(2k-3)}{c^2t^2}[g(k-1,ct)-g(k-2,ct)]$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Thu 22 May 2008 19:48
Edits by author:0

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