About Duhamel Principle in Wave Equation

[kakarotyjn]

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

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

 

[gtoguy97]

The sum of the solutions W(x,t;t_j ,Δt_j ) converges to a solution u(x,t) of equation (*). This is because, as Δt_j approaches 0, the terms in the sum approach 0, and the size of the terms decreases to 0. Since the sum of the terms converges, then the limit of the sum also converges to a solution of equation (*).