91影视

The radius of the sphere is 1.

The total of the second-order partial derivatives of R, the unknown function, in Cartesian coordinates equals 0, according to Laplace's equation.

Write the Laplace equation in the spherical coordinates.

$$\begin{gathered} {鈭�}^2=\frac{1}{r^2}\frac{鈭�}{鈭�r}\left(r^2\frac{鈭�u}{鈭�r}\right)+\frac{1}{r^2\sin 胃}\frac{鈭�}{鈭�胃}\left(\sin 胃\frac{鈭�u}{鈭�胃}\right)+\frac{1}{r^2{\sin }^2胃}\frac{{鈭�}^{2}u}{鈭�{桅}^2} \\ =0 \end{gathered}$$

Use separation of variables to separate the equation.

$u=R\left(r\right)螛\left(胃\right)唯\left(桅\right)$

Write the general solution as the series of basic functions.

$u=\underset{l=0}{\overset{鈭�}{鈭�}}{c}_l{r}^{l}Pl\left(\cos 胃\right)$

First check the odd-even nature of the function.

$$\begin{gathered} f\left(x\right)={\sin }^2胃+{\cos }^3胃 \\ ={\left(\sin \left(-胃\right)\right)}^2+{\left(\cos \left(-胃\right)\right)}^3 \\ =-\left({\sin }^2胃+{\cos }^3胃\right) \end{gathered}$$

As, $f\left(-x\right)=-f\left(x\right)$the function is odd.

Write the boundary condition as a function of Legendre polynomials.

${\sin }^2胃+{\cos }^3胃=\underset{l=0}{\overset{鈭�}{鈭�}}{c}_l{r}^{l}Pl\left(\cos 胃\right)$

Hence the inside temperature of the sphere with radius 1 is:

$u\left(r,胃,桅\right)=1-\frac{1}{2}r{P}_1\left(\cos 胃\right)+\frac{3}{5}r{P}_1\left(\cos 胃\right)-\frac{2}{3}{r}^2{P}_2\left(\cos 胃\right)+\frac{2}{5}{r}^3{P}_3\left(\cos 胃\right)$