Consider the problem of finding the temperature in a semi-infinite rod with zero initial temperature and a fixed constant temperature, $u _ { 0 }$ , at $x = 0$ . The mathematical model for this problem is
A) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( x , 0 ) = 0 , u ( 0 , t ) = u _ { 0 }$
B) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( x , 0 ) = 0 , u ( 0 , t ) = u _ { 0 }$
C) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0 , u ( x , 0 ) = 0 , u ( 0 , t ) = u _ { 0 }$
D) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( x , 0 ) = u _ { 0 } , u ( 0 , t ) = 0$
E) $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0 , u ( x , 0 ) = u _ { 0 } , u ( 0 , t ) = 0$

Question

Quizplus · Accepted Answer

The correct model for heat conduction in a semi-infinite rod with the given conditions is described by the partial differential equation $k \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t }$, along with the initial condition $u ( x , 0 ) = 0$ and the boundary condition $u ( 0 , t ) = u _ { 0 }$. This equation is known as the heat equation, which models the distribution of heat (or variation in temperature) in a given region over time.

Consider the Problem of Finding the Temperature in a Semi-Infinite u0u _ { 0 }u0​

Consider the Problem of Finding the Temperature in a Semi-Infinite $u _ { 0 }$