[Partial differential equation Tutorial Exercise reference solution]1.2 several classical equations

1. There is a soft uniform thin line, in the damping medium for small transverse vibration, the unit length chord resistance $F =-ru_t$. The vibration equation is deduced.

Answer: $$\bex \rho u_{tt}=tu_{xx}-ru_t. \eex$$

2. The three-dimensional heat conduction equation has the spherical symmetry form $u (x,y,z,t) =u (r,t) $ ($r =\sqrt{x^2+y^2+z^2}$) solution, trial: $$\bex U_t=a^2\sex{u_{rr}+\frac{2u_r}{r}}. \eex$$

Proof: by $$\bex U_x=u_r\frac{x}{r},\quad u_{xx}=u_{rr}\frac{x^2}{r^2} +u_r\frac{r-x\frac{x}{r}}{r^2} \eex$$ and symmetry known $$\bex \ Lap u=u_{rr}+u_r\frac{3r-r}{r^2} =u_{rr}+\frac{2u_r}{r}. \eex$$

3. If the $n $ Laplace equation $$\bex \frac{\p^2u}{\p x_1^2}+\cdots+\frac{\p^2u}{\p x_n^2}=0 \eex$$ has a spherical symmetry form of the solution $u (x_1,\cdots,x_n) =f ( R) $, where $r =\sqrt{x_1^2+\cdots+x_n^2}$, then $$\bex f (r) =\sedd{\ba{ll} c_1+c_2\dfrac{1}{r^{n-2}},&n\neq 2,\\ C_1+C_2\ ln \dfrac{1}{r},&n=2, \ea} \eex$$ where $C _1,c_2$ is any constant.

Proof: With the 2nd question, Yi $$\bex \lap u=f_{rr}+ (n-1) \frac{f_r}{r}=0. \eex$$ thereupon $$\bex 0=r^{n-1}f_{rr}+ (n-1) r^{n-2} f_r = (r^{n-1}f_r) _r\ra c=r^{n-1} F_r\ra f_r=\frac{c}{r^{n-1}}. \eex$$ if $n =2$, then $$\bex f_r=\frac{c}{r}\ra f=c_1+c_2\ln \frac{1}{r},\quad c_2=-c. \eex$$ if $n =3$, $$\bex F=C_1+C_2\frac {1} {R^{n-2}},\quad c_2=\frac{c}{-n+2}. \eex$$

[Partial differential equation Tutorial Exercise reference solution]1.2 several classical equations