12.1E: The Heat Equation (Exercises)

Q12.1.1

1. Explain Definition 12.1.3.

2. Explain Definition 12.1.4.

3. Explain Definition 12.1.5.

4. Perform numerical experiments with the formal solution obtained in Example 12.1.1.

5. Perform numerical experiments with the formal solution obtained in Example 12.1.2.

6. Perform numerical experiments with the formal solution obtained in Example 12.1.3.

7. Perform numerical experiments with the formal solution obtained in Example 12.1.4.

Q12.1.2

In Exercises 12.1.8-12.1.42 solve the initial-boundary value problem. Perform numerical experiments for Exercises 12.1.11, 12.1.17, 12.1.19, 12.1.22, 12.1.26, 12.1.30, 12.1.36, and 12.1.41. To simplify the computation of coefficients in some of these problems, check first to see if u(x, 0) is a polynomial that satisfies the boundary conditions. If it does, apply Theorem 11.3.5; also, see Exercises 11.3.35b, 11.3.42b, and 11.3.50b.

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. \(u_{t}=4u_{xx},\quad 0<x<2,\quad t>0,\)
\(u(0,t)=0,\quad u(2,t)=0,\quad t>0\),
\(u(x,0)= \left\{\begin{array}{cl} x,&0\le x\le1,\\[4pt]2-x,&1\le x\le 2. \end{array}\right.\)

14. \(u_{t}=7u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=x(3x^4-10x^2+7),\quad 0\le x\le 1\)

15. \(u_{t}=5u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=x(x^3-2x^2+1),\quad 0\le x\le 1\)

16. \(u_{t}=2u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=x(3x^4-5x^3+2),\quad 0\le x\le 1\)

17. \(u_{t}=9u_{xx},\quad 0<x<4,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(4,t)=0,\quad t>0\),
\(u(x,0)=x^2,\quad 0\le x\le 4\)

18. \(u_{t}=4u_{xx},\quad 0<x<2,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(2,t)=0,\quad t>0\),
\(u(x,0)=x(x-4),\quad 0\le x\le 2\)

19. \(u_{t}=9u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=x(1-x),\quad 0\le x\le 1\)

20. \(u_{t}=3u_{xx},\quad 0<x<2,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(2,t)=0,\quad t>0\),
\(u(x,0)=2x^2(3-x),\quad 0\le x\le 2\)

21. \(u_{t}=5u_{xx},\quad 0<x<\sqrt{2},\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(\sqrt2,t)=0,\quad t>0\),
\(u(x,0)=3x^2(x^2-4),\quad 0\le x\le \sqrt2\)

22. \(u_{t}=3u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=x^3(3x-4),\quad 0\le x\le 1\)

23. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=x^2(3x^2-8x+6),\quad 0\le x\le 1\)

24. \(u_{t}=u_{xx},\quad 0<x<\pi ,\quad t>0,\)
\(u_x(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0\),
\(u(x,0)=x^2(x-\pi)^2,\quad 0\le x\le \pi\)

25. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=\sin\pi x,\quad 0\le x\le 1\)

26. \(u_{t}=3u_{xx},\quad 0<x<\pi ,\quad t>0,\)
\(u(0,t)=0,\quad u_x(\pi,t)=0,\quad t>0\),
\(u(x,0)=x(\pi-x),\quad 0\le x\le \pi\)

27. \(u_{t}=5u_{xx},\quad 0<x<2,\quad t>0,\)
\(u(0,t)=0,\quad u_x(2,t)=0,\quad t>0\),
\(u(x,0)=x(4-x),\quad 0\le x\le 2\)

28. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=x^2(3-2x),\quad 0\le x\le 1\)

29. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=(x-1)^3+1,\quad 0\le x\le 1\)

30. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=x(x^2-3),\quad 0\le x\le 1\)

31. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=x^3(3x-4),\quad 0\le x\le 1\)

32. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u_x(1,t)=0,\quad t>0\),
\(u(x,0)=x(x^3-2x^2+2),\quad 0\le x\le 1\)

33. \(u_{t}=3u_{xx},\quad 0<x<\pi ,\quad t>0,\)
\(u_x(0,t)=0,\quad u(\pi,t)=0,\quad t>0\),
\(u(x,0)=x^2(\pi-x),\quad 0\le x\le \pi\)

34. \(u_{t}=16u_{xx},\quad 0<x<2\pi ,\quad t>0,\)
\(u_x(0,t)=0,\quad u(2\pi,t)=0,\quad t>0\),
\(u(x,0)=4,\quad 0\le x\le 2\pi\)

35. \(u_{t}=9u_{xx},\quad 0<x<4,\quad t>0,\)
\(u_x(0,t)=0,\quad u(4,t)=0,\quad t>0\),
\(u(x,0)=x^2,\quad 0\le x\le 4\)

36. \(u_{t}=3u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=1-x,\quad 0\le x\le 1\)

37. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=1-x^3,\quad 0\le x\le 1\)

38. \(u_{t}=7u_{xx},\quad 0<x<\pi ,\quad t>0,\)
\(u_x(0,t)=0,\quad u(\pi,t)=0,\quad t>0\),
\(u(x,0)=\pi^2-x^2,\quad 0\le x\le \pi\)

39. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=4x^3+3x^2-7,\quad 0\le x\le 1\)

40. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=2x^3+3x^2-5,\quad 0\le x\le 1\)

41. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=x^4-4x^3+6x^2-3,\quad 0\le x\le 1\)

42. \(u_{t}=u_{xx},\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=0,\quad u(1,t)=0,\quad t>0\),
\(u(x,0)=x^4-2x^3+1,\quad 0\le x\le 1\)

Q12.1.3

Exercises 12.1.43-12.1.46 solve the initial-boundary value problem. Perform numerical experiments for specific values of \(L\) and \(a\). Perform numerical experiments for Exercises 12.1.43-12.1.46.

43. \(u_{t}=a^{2}u_{xx},\quad 0<x<L,\quad t>0,\)
\(u_{x}(0,t)=0,\quad u_{x}(L,t)=0,\quad t>0,\)
\(u(x,0)=\left\{\begin{array}{ll}{1,}&{0\leq x\leq\frac{L}{2}}\\[4pt]{0,}&{\frac{L}{2}<x<L}\end{array} \right.\)

44. \(u_{t}=a^{2}u_{xx},\quad 0<x<L,\quad t>0,\)
\(u(0,t)=0,\quad u(L,t)=0,\quad t>0,\)
\(u(x,0)=\left\{\begin{array}{ll}{1,}&{0\leq x\leq\frac{L}{2}}\\[4pt]{0,}&{\frac{L}{2}<x<L}\end{array} \right.\)

45. \(u_{t}=a^{2}u_{xx},\quad 0<x<L,\quad t>0,\)
\(u_{x}(0,t)=0,\quad u(L,t)=0,\quad t>0,\)
\(u(x,0)=\left\{\begin{array}{ll}{1,}&{0\leq x\leq\frac{L}{2}}\\[4pt]{0,}&{\frac{L}{2}<x<L}\end{array} \right.\)

46. \(u_{t}=a^{2}u_{xx},\quad 0<x<L,\quad t>0,\)
\(u(0,t)=0,\quad u_{x}(L,t)=0,\quad t>0,\)
\(u(x,0)=\left\{\begin{array}{ll}{1,}&{0\leq x\leq\frac{L}{2}}\\[4pt]{0,}&{\frac{L}{2}<x<L}\end{array} \right.\)

Q12.1.4

47. Let \(h\) be continuous on \([0,L]\) and let \(u_0\), \(u_L\), and \(a\) be constants, with \(a>0\). Show that it is always possible to find a function \(q\) that satisfies (a), (b), or (c), but that this isn’t so for (d).

1. \(a^2q''+h=0,\quad q(0)=u_0,\quad q(L)=u_L\)
2. \(a^2q''+h=0,\quad q'(0)=u_0,\quad q(L)=u_L\)
3. \(a^2q''+h=0,\quad q(0)=u_0,\quad q'(L)=u_L\)
4. \(a^2q''+h=0,\quad q'(0)=u_0,\quad q'(L)=u_L\)

Q12.1.5

In Exercises 12.1.48-12.1.53 solve the nonhomogeneous initial-boundary value problem

48. \(u_{t}=9u_{xx}-54x,\quad 0<x<4,\quad t>0,\)
\(u(0,t)=1,\quad u(4,t)=61,\quad t>0\),
\(u(x,0)=2-x+x^3,\quad 0\le x\le 4\)

49. \(u_{t}=u_{xx}-2,\quad 0<x<1,\quad t>0,\)
\(u(0,t)=1,\quad u(1,t)=3,\quad t>0\),
\(u(x,0)=2x^2+1,\quad 0\le x\le 1\)

50. \(u_{t}=3u_{xx}-18x,\quad 0<x<1,\quad t>0,\)
\(u_x(0,t)=-1,\quad u(1,t)=-1,\quad t>0\),
\(u(x,0)=x^3-2x,\quad 0\le x\le 1\)

51. \(u_{t}=9u_{xx}-18,\quad 0<x<4,\quad t>0,\)
\(u_x(0,t)=-1,\quad u(4,t)=10,\quad t>0\),
\(u(x,0)=2x^2-x-2,\quad 0\le x\le 4\)

52. \(u_{t}=u_{xx}+\pi ^{2}\sin\pi x,\quad 0<x<1,\quad t>0,\)
\(u(0,t)=0,\quad u_x(1,t)=-\pi,\quad t>0\),
\(u(x,0)=2\sin\pi x,\quad 0\le x\le 1\)

53. \(u_{t}=u_{xx}-6x,\quad 0<x<L,\quad t>0,\)
\(u(0,t)=3,\quad u_x(1,t)=2,\quad t>0\),
\(u(x,0)=x^3-x^2+x+3,\quad 0\le x\le 1\)

Q12.1.6

54. In this exercise take it as given that the infinite series \(\sum_{n=1}^\infty n^pe^{-qn^2}\) converges for all \(p\) if \(q>0\), and, where appropriate, use the comparison test for absolute convergence of an infinite series.

Let

\[u(x,t)=\sum_{n=1}^\infty \alpha_n e^{-n^2\pi^2 a^2t/L^2}\sin{n\pi x\over L}\nonumber \]

where

\[\alpha_n={2\over L}\int_0^L f(x)\sin{n\pi x\over L}\,dx\nonumber \]

and \(f\) is piecewise smooth on \([0,L]\).

1. Show that \(u\) is defined for \((x,t)\) such that \(t>0\).
2. For fixed \(t>0\), use Theorem 12.1.2 with \(z=x\) to show that \[u_{x}(x,t)=\frac{\pi }{L}\sum_{n=1}^{\infty}n\alpha _{n}e^{-n^{2}\pi ^{2}a^{2}t/L^{2}}\cos\frac{n\pi x}{L},\quad -\infty <x<\infty .\nonumber \]
3. Starting from the result of (a), use Theorem 12.1.2 with \(z=x\) to show that, for a fixed \(t>0\), \[u_{xx}(x,t)=-\frac{\pi ^{2}}{L^{2}}\sum_{n=1}^{\infty }n^{2}\alpha _{n}e^{-n^{2}\pi ^{2}a^{2}t/L^{2}}\sin\frac{n\pi x}{L},\quad -\infty <x<\infty .\nonumber \]
4. For fixed but arbitrary \(x\), use Theorem 12.1.2 with \(z=t\) to show that \[u_{t}(x,t)=-\frac{\pi ^{2}a^{2}}{L^{2}}\sum_{n=1}^{\infty}n^{2}\alpha _{n}e^{-n^{2}\pi ^{2}a^{2}/L^{2}}\sin\frac{n\pi x}{L},\nonumber \] if \(t>t_0>0\), where \(t_0\) is an arbitrary positive number. Then argue that since \(t_0\) is arbitrary, the conclusion holds for all \(t>0\).
5. Conclude from (c) and (d) that \[u_{t}=a^{2}u_{xx},\quad -\infty <x<\infty ,\quad t>0.\nonumber \]

By repeatedly applying the arguments in (a) and (c) , it can be shown that \(u\) can be differentiated term by term any number of times with respect to \(x\) and/or \(t\) if \(t>0\).