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

question 24

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) \(U _ { t } = k \alpha ^ { 2 } U - k \alpha u _ { 0 } , U ( \alpha , 0 ) = 0\)

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Definitions:

Temporal Bone

A bone of the skull that surrounds the ear, contributing to the protection of the inner ear structures and supporting facial muscles.

Pinna

The visible part of the ear that is outside the head, responsible for capturing sound vibrations and funneling them into the ear canal.

Optic Chiasma

A structure located at the base of the brain where the optic nerves partially cross, allowing for the integration of visual information from both eyes.

Ganglion Cell Axons

The nerve fibers of ganglion cells in the retina that transmit visual information from the eye to the brain.