In the previous three problems, the solution for $u ( x , t )$ is
A) $\int _ { - \infty } ^ { \infty } \left[ e ^ { - i \alpha x } e ^ { k at } / \left( 1 + \alpha ^ { 2 } \right) \right] d \alpha / ( 2 \pi )$
B) $\int _ { - \infty } ^ { \infty } \left[ e ^ { - i \alpha x } e ^ { - k \alpha ^ { 2 } t } / \left( 1 + \alpha ^ { 2 } \right) \right] d \alpha / \pi$
C) $\int _ { - \infty } ^ { \infty } \left[ e ^ { - i \alpha \alpha } e ^ { k \alpha t } / \left( 1 + \alpha ^ { 2 } \right) \right] d \alpha / \pi$
D) $\int _ { - \infty } ^ { \infty } \left[ e ^ { - i \alpha x } e ^ { - k \alpha t } / \left( 1 + \alpha ^ { 2 } \right) \right] d \alpha / \pi$
E) $\int _ { - \infty } ^ { \infty } \left[ e ^ { - i \alpha \alpha } e ^ { k \alpha ^ { 2 } t } / \left( 1 + \alpha ^ { 2 } \right) \right] d \alpha / ( 2 \pi )$

Question

Quizplus · Accepted Answer

Option B is correct because it represents a solution to a differential equation that models diffusion or heat conduction, where $e^{-k\alpha^2t}$ is a typical term for the time-dependent part of the solution in the Fourier transform space, indicating diffusion or heat dissipation over time. The $e^{-i\alpha x}$ term represents the spatial component in the Fourier transform, and the denominator $(1+\alpha^2)$ could be part of the initial or boundary conditions transformed into the Fourier space. The division by $\pi$ is related to the normalization in the inverse Fourier transform.

In the Previous Three Problems, the Solution For u(x,t)u ( x , t )u(x,t)

In the Previous Three Problems, the Solution For $u ( x , t )$