Consider the Heat Problem Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$

question 44

Multiple Choice

Consider the heat problem $c \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , u ( 1 , t ) = 3 , u ( x , 0 ) = 3 x ^ { 2 }$ . Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $h = 1 / 3$ and $\frac { \partial u } { \partial t }$ with a forward difference approximation with $k = 1 / 2$ . 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) \(c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k\)
• C) \(c [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k\)
• D) \(c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k\)
• E) \(c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k\)

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