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

Question

Quizplus · Accepted Answer

The correct equation for a finite difference approximation of a differential equation, using the given notation and letting $\lambda = c k / h ^ { 2 }$, is $u _ { i j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - 2 \lambda ) u _ { i j } + \lambda u _ { i - 1 , j }$. This represents a typical discretization for the heat equation or similar parabolic partial differential equations in one spatial dimension, where the future value $u_{ij+1}$ is a weighted sum of the current value and its spatial neighbors, with weights determined by $\lambda$.

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

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