91Ó°ÊÓ

A slab of thickness 10 cm has its two faces at ${\mathbf{10}}^{\mathbf{°}}$and $\mathbf{20}\mathbf{°}$. At t = 0 , the face temperatures are interchanged. Find $\mathit{u}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}$for t > 0.

Show that the Green function (8.28) which is zero on the plane z = 0 is

$\mathit{G}\mathbf{(}\mathbf{r}\mathbf{,}{\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{)}\mathbf{=}\mathbf{−}\frac{\mathbf{1}}{\mathbf{4}\mathbf{π}}{\mathbf{[}{\mathbf{(}\mathbf{x}\mathbf{−}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{(}\mathbf{y}\mathbf{−}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{(}\mathbf{z}\mathbf{−}{\mathbf{z}}^{\mathbf{\text{'}}}\mathbf{)}}^{\mathbf{2}}\mathbf{]}}^{\mathbf{−}\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{4}\mathbf{π}}{\mathbf{[}{\mathbf{(}\mathbf{x}\mathbf{−}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{(}\mathbf{y}\mathbf{−}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{)}}^{\mathbf{2}}\mathbf{+}{\mathbf{(}\mathbf{z}\mathbf{+}{\mathbf{z}}^{\mathbf{\text{'}}}\mathbf{)}}^{\mathbf{2}}\mathbf{]}}^{\mathbf{−}\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{.}$

Hence write a triple integral for the solution of (8.22) for z > 0 which is zero for z = 0 .

Find the steady-state temperature distribution inside a sphere of radius 1 when the surface temperatures are as given in Problems 1 to 10.

$\left\{\begin{array}{l}\begin{array}{cc}\mathbf{100}\mathbf{°}\mathbf{,}& \mathbf{0}\mathbf{<}\mathbf{θ}\mathbf{<}\mathbf{π}\mathbf{/}\mathbf{3}\mathbf{,}\end{array}\\ \begin{array}{cc}\mathbf{0}\mathbf{°}\mathbf{,}& \mathbf{\text{otherwise.}}\end{array}\end{array}\mathbf{Hint}\mathbf{:}\mathbf{See}\mathbf{\text{ }}\mathbf{Problem}\mathbf{\text{ }}\mathbf{9}\mathbf{.8}\mathbf{\text{ }}\mathbf{of}\mathbf{\text{ }}\mathbf{Chapter}\mathbf{\text{ }}\mathbf{12}\mathbf{.}\right.$

Question: Do Problem 6 in polar coordinates to find the eigenfunctions and energy eigenvalues of a particle in a circular box $\mathbf{r}\mathbf{<}\mathbf{a}$.

A string of length l has initial displacement ${\mathbf{\text{y}}}_{\mathbf{\text{0}}}\mathbf{=}\mathbf{\text{x}}\mathbf{(}\mathbf{\text{l}}\mathbf{−}\mathbf{\text{x}}\mathbf{)}$.Find the displacement as a function of x and t.

Find the steady-state temperature distribution inside a sphere of radius 1 when the surface temperatures are as given in Problems 1 to 10 .

$\mathbf{3}\mathbf{²õ¾\pm ²Ô賦´Ç²õθ²õ¾\pm ²ÔÏ•}$Hint: See equation (7.10) and Chapter 12, equation (10.6).

Show that our results can be extended to find the following solution of (8.22) which satisfies given nonzero boundary conditions:

$\mathit{u}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{=}∭G\mathbf{(}\mathbf{r}\mathbf{,}{\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{)}\mathit{f}\mathbf{\left(}{\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{\right)}\mathit{d}{\mathit{Ï„}}^{\mathbf{\text{'}}}\mathbf{+}âˆ\neg u\mathbf{\left(}{\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{\right)}\frac{\mathbf{∂}\mathbf{G}\mathbf{(}\mathbf{r}\mathbf{,}{\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{)}}{\mathbf{∂}{\mathbf{n}}^{\mathbf{\text{'}}}}\mathit{d}{\mathit{σ}}^{\mathbf{\text{'}}}$

Where $\mathit{G}\mathbf{(}\mathbf{r}\mathbf{,}{\mathbf{r}}^{\mathbf{\text{'}}}\mathbf{)}$is the Green function (8.28) which is zero on the surface σ, and $\mathbf{∂}\mathit{G}\mathbf{/}\mathbf{∂}{\mathit{n}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{∇}\mathit{G}\mathbf{â‹\dots }{\mathbf{n}}^{\mathbf{\text{'}}}$is the normal derivative of G (see Chapter 6, Section 6).