Consider the Steady-State Temperature Distribution in a Circular Disc of Radius

Multiple Choice

Consider the steady-state temperature distribution in a circular disc of radius $C$ centere at the origin, with temperature given as a function, $f ( \theta )$ on the boundary $r = c$ and zero on the boundaries $\theta = 0$ and $\theta = \pi$ The mathematical model of this situation is

A) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$
B) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$
C) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$
D) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$
E) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$