Exam 15: Numerical Solutions of Partial Differential Equations

In the previous two problems, using $u _ { i j }$ to denote the value of $u$ at the $i , j$ point, the equations for the values of the unknown function at the interior points are Select all that apply.

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

Laplace's equation is

The wave equation is

The heat equation is

In the previous problem, is the value of $\lambda$ such that the scheme is stable?

In the previous two problems, the values $u _ { i , 1 }$ depend on the values $u _ { i , - 1 }$ . How do you calculate those values?

In the four previous problems, let $c = 1$ . The calculated values of $u _ { i , 1 }$ are

Consider the problem $c \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , u ( 1 , t ) = 2 , u ( x , 0 ) = 2 x ^ { 2 }$ . Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $h = 1 / 3$ and $\frac { \partial u } { \partial t }$ with a forward difference approximation with $k = 1 / 2$ . The resulting equation is

The forward difference approximation of $\frac { \partial u } { \partial t }$ with step size k is

The central difference approximation for $c \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial t } , u ( 0 , t ) = 0 , u ( 2 , t ) = 6 , u ( x , 0 ) = 3 x ^ { 2 } / 2$ Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $h = 1 / 2$ and $\frac { \partial u } { \partial t }$ with a forward difference approximation with $k = 1 / 4$ . The resulting equation is

In the previous five problems, is the value of $\lambda$ such that the numerical scheme is stable?

In the previous two problems, the values $u _ { i , 1 }$ depend on the values $u _ { i , 1 }$ . How do you calculate those values?

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

The central difference approximation for $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with step size $h$ is

In the four previous problems, let $c = 1$ . The calculated values of $u _ { i , 1 }$ are Select all that apply.

The heat equation is

In the previous three problems, the solution at the interior points is Select all that apply.

In the previous two problems, let $c = 1$ . Thesolution for u along the line $t = 0.5$ at the mesh points is Select all that apply.

A Dirichlet problem is a partial differential equation with conditions specifying