I think my lack of background in proofs is showing.(adsbygoogle = window.adsbygoogle || []).push({});

From "Partial Differential Equations of Mathematical Physics and Integral Equations" by Ronald B. Guenther and John W. Lee, pp. 107 problem 4-3.10.

Prove:

For the inhomogeneous, initial, boundary value problem

[tex]\left\{\begin{array}{lll}

u_{tt} - c^2u_{xx} = F(x,t), & 0 < x < L, & t > 0, \\

u(x,0) = 0, & u_t(x,0) = 0, & 0 \leq x \leq L, \\

u(0,t) = 0, & u(L,t) = 0, & t \geq 0

\end{array} \right.[/tex]

Let F(x,t) have continuous third-order partial derivatives for [itex]0 \leq x \leq L[/itex] and [itex]t \geq 0[/itex]. Assume that [itex]F(0,t) = F(L,t) = F_{xx}(0,t) = F_{xx}(L,t) = 0 \qquad \forall t \geq 0[/itex]. Then the problem above has a unique solution given by

[tex]u(x,t) = \sum_{n=1}^\infty u_n(t) \sin{(\lambda_n x)}[/tex]

where

[tex]u_n(t) = \frac{1}{c\lambda_n}\int_0^t F_n(\tau) \sin{[c\lambda_n (t - \tau)]}\,d\tau[/tex]

and

[tex]\lambda_n = \frac{n\pi}{L}[/tex]

using the following steps. Fix T > 0 and restrict x and t to [itex]0 \leq x \leq L[/itex] and [itex] 0 \leq t \leq T[/itex].

(a) Write out the expressions for [itex]u_x,u_{xx},u_t,u_{tt}[/itex] assuming that term-by-term differentiation is permissible.

This I can do.

[tex]u_x = \sum_{n=1}^\infty u_n(t) \lambda_n \cos{(\lambda_n x)}[/tex]

[tex]u_{xx} = -\sum_{n=1}^\infty u_n(t) \lambda_n^2 \sin{(\lambda_n x)}[/tex]

[tex]u_t = \sum_{n=1}^\infty \sin{(\lambda_n x)}\int_0^t F(\tau) \cos{[c\lambda_n (t-\tau)]}\,d\tau[/tex]

[tex]u_{tt} = \sum_{n=1}^\infty \sin{(\lambda_n x)}[F(t) - c^2\lambda_n^2 u_n(t)][/tex]

(b) Use the definition of [itex]u_n[/itex] to confirm that

[tex]|\lambda_n^2 u_n(t)| \leq T \lambda_n \frac{||F_n||}{c}[/tex]

[tex]|u_n'(t)| \leq T||F_n||[/tex]

[tex]|u_n''(t)| \leq |F_n(t)| + cT\lambda_n ||F_n||[/tex]

where

[tex]||F_n|| = \max_{0 \leq t \leq T} |F_n(t)|[/tex]

Can do that, too.

(c) Assume that [itex]F(0,t) = F(L,t) = F_{xx}(0,t) = F_{xx}(L,t) = 0 \qquad \forall t \geq 0[/itex] and deduce that

[tex]|F_n(t)| \leq \frac{2}{\lambda_n^3 L}\int_0^L |F_{xxx}(x,t)\,dx[/tex]

This I haven't pulled off yet. I don't quite see how the partial derivatives of u and the derivatives of [itex]u_n[/itex] relate to [itex]F_{xxx}[/itex].

The Fourier sine series expansion of F(x,t) relates F(x,t) with [itex]F_n(t)[/itex], where [itex]F_n(t)[/itex] is the sine coefficient of the Fourier series. This puts the two on the same page, but I couldn't take it anywhere. It seems to me like bringing u(x,t) and un(t) into the proof is going out of the scope of F and Fn.

Honestly, I really can't find much use to the three inequalities in part (b).

Anyway, I'd appreciate any help. Thisishomework due tomorrow, but frankly I don't care if it gets done by then because I have nothing on the line in this class (I have a running bet for $50 that I'll just have to retake it in 2 or 3 years). I'm more concerned with figuring out how the proof works (for 2-3 years down the road).

Sorry for dragging anybody through crap they never wanted to remember.

cookiemonster

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# Inhomogeneous Initial, Boundary Value Theorem Proof

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