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Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.2. Second-Order Hyperbolic Equations
Equation(s):$\displaystyle \frac{\partial^2w}{\partial t^2}=b\frac{\partial ^2w}{\partial x^2}-c^2w
$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Solutions:
$$
w_n(x,t)=\sum^{n-2k\geq0}_{k=0}{\frac{n!b^kg(k,ct)}{(n-2k)!(2k)!}x^{n-2k}t^{2k}},\quad \ n=0,\,1,\ldots\,,
$$
where $g(0,ct)=\cos(ct)$,\\
$\displaystyle g(1,ct)=\frac {\sin(ct)}{ct}$,\\
$\displaystyle g(2,ct)=\frac {3\sin(ct)}{c^3t^3}-\frac {3\cos(ct)}{c^2t^2}$,\\
$\displaystyle g(3,ct)=\frac {45\sin(ct)}{c^5t^5}-\frac {45\cos(ct)}{c^4t^4}-\frac {15\sin(ct)}{c^3t^3}$,\\
\ldots\,,\\
$\displaystyle g(k,ct)=\frac {(2k-1)(2k-3)}{c^2t^2}[g(k-1,ct)-g(k-2,ct)]$.\\
\medskip
Examples:\\
$w_0(x,t)=g(0,ct)=\cos(ct)$,\\
$w_1(x,t)=xg(0,ct)=x\cos(ct)$,\\
$\displaystyle w_2(x,t)=x^2g(0,ct)+bt^2g(1,ct)=x^2\cos(ct)+\frac {bt}c \sin(ct)$,\\
$\displaystyle w_3(x,t)=x^3g(0,ct)+3bxt^2g(1,ct)=x^3\cos(ct)+\frac {3bxt}c \sin(ct)$,\\
$w_4(x,t)=x^4g(0,ct)+6bx^2t^2g(1,ct)+b^2t^4g(2,ct)$\\
$\displaystyle {}\qquad\quad \ =x^4\cos(ct)+\frac{6bx^2t}c\sin(ct)+\frac{3b^2t}{c^3}\sin(ct)+\frac{3b^2t^2}{c^2}\cos(ct)$.\\

2. Solutions:
$$
w_n(x,t)=\sum^{n-2k\geq0}_{k=0}{\frac{n!b^kg(k+1,ct)}{(n-2k)!(2k+1)!}x^{n-2k}t^{2k+1}},\quad \ n=0,\,1,\ldots\,
$$
Examples:\\
$\displaystyle w_0(x,t)=g(1,ct)t=\frac 1c \sin(ct)$,\\
$\displaystyle w_1(x,t)=txg(1,ct)=\frac xc \sin(ct)$,\\
$\displaystyle w_2(x,t)=tx^2g(1,ct)+\frac {bt^3}3 g(2,ct)=\frac {x^2}c \sin(ct)+\frac b{c^3} \sin(ct)-\frac {bt}{c^2} \cos(ct)$,\\
$\displaystyle w_3(x,t)=tx^3g(1,ct)+bxt^3g(2,ct)=\frac {x^3}c \sin(ct)+\frac {3bx}{c^3} \sin(ct)-\frac {3bxt}{c^2} \cos(ct)$,\\
$w_4(x,t)=tx^4g(1,ct)+2bx^2t^3g(2,ct)+\frac 15{b^2t^5}g(3,ct)$\\
$\displaystyle {}\quad\qquad \ =\frac {x^4}c \sin(ct)+\frac {6bx^2}{c^3} \sin(ct)-\frac {6bx^2t}{c^2}
\cos(ct)+\frac {9b^2}{c^5}\sin(ct)-\frac {9b^2t}{c^4} \cos(ct)$
$\displaystyle {}\quad\qquad \ -\frac {3b^2t^2}{c^3} \sin(ct)$.
Remarks:For other solutions see, for example, "Handbook of Linear Partial Differential \hbox{Equations} for Engineers and Scientists" (Chapman \& Hall/CRC, 2002, p. 287) by A.~D.~Polyanin
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Middle(s) name:Germanovich
Country:Latviya
City:Sigulda
Statistic information
Submission date:Sun 16 Mar 2008 12:39
Edits by author:1
Last edit by author:Mon 21 Apr 2008 16:16

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