The Central Difference Approximation For Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$

Multiple Choice

The central difference approximation for $c \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , u ( 2 , t ) = 6 , u ( x , 0 ) = 3 x ^ { 2 } / 2$ Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $h = 1 / 2$ and $\frac { \partial u } { \partial t }$ with a forward difference approximation with $k = 1 / 4$ . The resulting equation is

A) $c [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
B) $c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$
C) $c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
D) $c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
E) $c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$