In the previous problem, using the notation $u _ { i j } = u ( x , t )$ , and letting $c = 1 , \lambda = c k / h ^ { 2 }$ , the equation becomes
A) $u _ { i , j - 1 } = \lambda u _ { i + 1 , j } + ( 1 + 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$
B) $u _ { i , j - 1 } = \lambda u _ { i + 1 , j } + ( 1 - 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$
C) $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 + 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$
D) $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$
E) $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$

Question

Quizplus · Accepted Answer

The correct equation for the explicit finite difference method for the heat equation (or similarly structured partial differential equations) with $c = 1$ and $\lambda = c k / h^2$ typically involves a forward time step ($j + 1$) and a central space difference, leading to the form in option D. This reflects the diffusion process with a time step forward and a weighted average of the neighboring spatial points and the current point, adjusted by $\lambda$.

In the Previous Problem, Using the Notation uij=u(x,t)u _ { i j } = u ( x , t )uij​=u(x,t)

In the Previous Problem, Using the Notation $u _ { i j } = u ( x , t )$