9.1E: The Heat Equation (Exercises)
- Page ID
- 43346
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In some of the exercises we say “perform numerical experiments.” This means that you should perform the computations just described with the formal solution obtained in the exercise.
Q9.1.1
1. Explain Theorem 9.1.3.
2. Explain Theorem 9.1.4.
3. Explain Theorem 9.1.5.
4. Perform numerical experiments with the formal solution obtained in Example 9.1.1.
5. Perform numerical experiments with the formal solution obtained in Example 9.1.2.
6. Perform numerical experiments with the formal solution obtained in Example 9.1.3.
7. Perform numerical experiments with the formal solution obtained in Example 9.1.4.
Q9.1.2
In Exercises 9.2.8-9.2.19 solve the initial-boundary value problem.
8. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=x(1-x),\quad 0\le x\le 1\)
9. \(u_{t}=9u_{xx},\quad 0<x<4,\quad t>0.\)
\(u(0,t)=0,\quad u(4,t)=0,\quad t>0\),
\(u(x,0)=1,\quad 0\le x\le 4\)
10. \(u_{t}=3u_{xx},\quad 0<x<\pi , \quad t>0.\)
\(u(0,t)=0,\quad u(\pi,t)=0,\quad t>0\),
\(u(x,0)=x\sin x,\quad 0\le x\le \pi\)
11. \(u_{t}=9u_{xx},\quad 0<x<2,\quad t>0,\)
\(u(0,t)=0,\quad u(2,t)=0,\quad t>0\),
\(u(x,0)=x^2(2-x),\quad 0\le x\le 2\)
12. \(u_{t}=4u_{xx},\quad 0<x<3,\quad t>0,\)
\(u(0,t)=0,\quad u(3,t)=0,\quad t>0\),
\(u(x,0)=x(9-x^2),\quad 0\le x\le 3\)
13. \(u_{t}=4u_{xx},\quad 0<x<2,\quad t>0,\)
\(u(0,t)=0,\quad u(2,t)=0,\quad t>0\),
\(u(x,0)= \left\{\begin{array}{cl} x,&0\le x\le1,\\2-x,&1\le x\le 2. \end{array}\right.\)
14. \(u_{t}=4u_{xx},\quad 0<x<2,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(2,t)=0,\quad t>0\),
\(u(x,0)=x(x-4),\quad 0\le x\le 2\)
15. \(u_{t}=9u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=x(1-x),\quad 0\le x\le 1\)
26. \(u_{t}=3u_{xx},\quad 0<x<\pi ,\quad t>0,\)
\(u(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0\),
\(u(x,0)=x(\pi-x),\quad 0\le x\le \pi\)
17. \(u_{t}=5u_{xx},\quad 0<x<2,\quad t>0,\)
\(u(0,t)=0,\quad u_x(2,t)=0,\quad t>0\),
\(u(x,0)=x(4-x),\quad 0\le x\le 2\)
18. \(u_{t}=16u_{xx},\quad 0<x<2\pi ,\quad t>0,\)
\(u_x(0,t)=0,\quad u(2\pi,t)=0,\quad t>0\),
\(u(x,0)=4,\quad 0\le x\le 2\pi\)
19. \(u_{t}=3u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=1-x,\quad 0\le x\le 1\)