About Duhamel Principle in Wave Equation

  • Thread starter kakarotyjn
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Main Question or Discussion Point

[tex]
\begin{array}{l}
\frac{{\partial ^2 u}}{{\partial t^2 }} - a^2 \frac{{\partial ^2 u}}{{\partial x^2 }} = f(x,t) \\
t = 0:u = 0,\frac{{\partial u}}{{\partial t}} = 0 \\
\end{array}[/tex]......(*)
[tex]
\begin{array}{l}
\frac{{\partial ^2 W}}{{\partial t^2 }} - a^2 \frac{{\partial ^2 W}}{{\partial x^2 }} = 0...(t > t_j ) \\
t = t_j :W = 0,\frac{{\partial W}}{{\partial t}} = f(x,t_j )\Delta t_j \\
\end{array}[/tex].......(**)

let u(x,t) be the solution of equation (*),[tex]W(x,t;t_j ,\Delta t_j )[/tex]be solution of (**)。why [tex]u(x,t) = \mathop {\lim }\limits_{\Delta t_j \to 0} \sum\limits_{j = 1}^l {W(x,t;t_j ,\Delta t_j )} [/tex] exist?
 

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