Consider the Heat Problem Apply a Fourier Sine Transform $U ( \alpha , t ) = \mathcal { F } _ { s } \{ u ( x , t ) \}$

Multiple Choice

Consider the heat problem $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , 0 < x < \infty , t > 0 , u ( x , 0 ) = 0 , u ( 0 , t ) = u _ { 0 }$ . Apply a Fourier sine transform. The resulting problem for $U ( \alpha , t ) = \mathcal { F } _ { s } \{ u ( x , t ) \}$ is

A) $U _ { t } = - k \alpha U + k \alpha u _ { 0 } , U ( \alpha , 0 ) = 0$
B) $U _ { t } = - k \alpha ^ { 2 } U - k \alpha t _ { 0 } , U ( \alpha , 0 ) = 0$
C) $U _ { t } = - k \alpha ^ { 2 } U + k \alpha t _ { 0 } , U ( \alpha , 0 ) = 0$
D) $U _ { t } = k \alpha ^ { 2 } U + k \alpha u _ { 0 } , U ( \alpha , 0 ) = 0$
E) $U _ { t } = k \alpha ^ { 2 } U - k \alpha u _ { 0 } , U ( \alpha , 0 ) = 0$