9.2E: The Wave Equation (Exercises)

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Q9.2.1

In Exercises 9.2.1-9.2.8 solve the initial-boundary value problem.

1. $u_{tt}=9u_{xx},\quad 0<x<1,\quad t>0$,
$u(0,t)=0,\quad u(1,t)=0,\quad t>0$,
$u(x,0)=0,\quad u_t(x,0)= \left\{\begin{array}{cl} x,&0\le x\le{1\over2},\\1-x,&{1\over2}\le x\le1 \end{array}\right.,\quad0\le x\le1$

2. $u_{tt}=9u_{xx},\quad 0<x<1,\quad t>0$,
$u(0,t)=0,\quad u(1,t)=0,\quad t>0$,
$u(x,0)=x(1-x),\quad u_t(x,0)=0,\quad0\le x\le 1$

3. $u_{tt}=9u_{xx},\quad 0<x<2,\quad t>0$,
$u_x(0,t)=0,\quad u(2,t)=0,\quad t>0$,
$u(x,0)=4-x^2,\quad u_t(x,0)=0,\quad0\le x\le2$

4. $u_{tt}=4u_{xx},\quad 0<x<1,\quad t>0$,
$u_x(0,t)=0,\quad u(1,t)=0,\quad t>0$,
$u(x,0)=x^2(1-x),\quad u_t(x,0)=0,\quad0\le x\le 1$

5. $u_{tt}=64u_{xx},\quad 0<x<\pi,\quad t>0$,
$u(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$,
$u(x,0)=x(2\pi-x),\quad u_t(x,0)=0,\quad0\le x\le \pi$

6. $u_{tt}=64u_{xx},\quad 0<x<\pi,\quad t>0$,
$u(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0$,
$u(x,0)=0,\quad u_t(x,0)=x(2\pi-x),\quad0\le x\le \pi$

7. $u_{tt}=5u_{xx},\quad 0<x<2,\quad t>0$,
$u_x(0,t)=0,\quad u_x(2,t)=0,\quad t>0$,
$u(x,0)=2x^2(3-x),\quad u_t(x,0)=0,\quad0\le x\le 2$

8. $u_{tt}=5u_{xx},\quad 0<x<2,\quad t>0$,
$u_x(0,t)=0,\quad u_x(2,t)=0,\quad t>0$,
$u(x,0)=0,\quad u_t(x,0)=2x^2(3-x),\quad0\le x\le 2$

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