Consider the Temperature $u ( x , t )$ , in an Infinite Rod

Consider the temperature, $u ( x , t )$ , in an infinite rod $( - \infty < x < \infty )$ , with an initial temperature of $f ( x ) = e ^ { - | x | }$ . The mathematical model for this is

A) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( x , 0 ) = e ^ { - | x | }$
B) $k \frac { \partial u } { \partial x } = \frac { \partial u } { \partial t } , u ( x , 0 ) = e ^ { - | x | }$
C) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( x , 0 ) = e ^ { - | x | }$
D) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( x , 0 ) = e ^ { - | x | }$
E) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0 , u ( x , 0 ) = e ^ { - | x | }$

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