Deck 15: Numerical Solutions of Partial Differential Equations

Question 1

In the previous two problems, let $c = 1$ . Thesolutionforu along the line $t = 0.25$ at the mesh points is Select all that apply.

A) $u _ { 31 } = 33 / 8$
B) $u _ { 11 } = 9 / 8$
C) $u _ { 11 } = 11 / 8$
D) $u _ { 21 } = 9 / 4$
E) $u _ { 21 } = 11 / 4$

Answer 1

$u _ { 31 } = 33 / 8$
$u _ { 11 } = 9 / 8$
$u _ { 21 } = 9 / 4$

Question 2

Laplace's equation is

A) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$
B) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } }$
C) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0$
D) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial u } { \partial t } = 0$
E) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0$

Answer 2

$\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$

Question 3

In the previous three problems, the values of $u _ { i , - 1 }$ are

A) $u _ { i , - 1 } = u _ { i , 0 } - k g ( x i )$
B) $u _ { i , - 1 } = u _ { i , 0 } - 2 k g ( x i )$
C) $u _ { i , - 1 } = u _ { i , 1 } + 2 k g ( x i )$
D) $u _ { i , - 1 } = u _ { i , 1 } + k g ( x i )$
E) $u _ { i , - 1 } = u _ { i , 1 } - 2 k g ( x i )$

Answer 3

$u _ { i , - 1 } = u _ { i , 1 } - 2 k g ( x i )$

Question 4

The heat equation is

A) hyperbolic
B) parabolic
C) elliptic
D) none of the above

Question 5

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

A) yes
B) no
C) It is right on the borderline.
D) It cannot be determined from the available data.

Question 6

The five-point approximation of the Laplacian is

A) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y + h ) - 4 u ( x , y ) ]$
B) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y + h ) - 2 u ( x , y ) ]$
C) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y + h ) - 4 u ( x , y ) ] / h$
D) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y + h ) - 4 u ( x , y ) ] / h ^ { 2 }$
E) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y + h ) - 2 u ( x , y ) ] / h ^ { 2 }$

Question 7

Consider the problem $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0 , u ( 0 , y ) = 0 , u ( x , 0 ) = 0 , u ( 1 , y ) = y - y ^ { 2 } , u ( x , 1 ) = x - x ^ { 2 }$ . A finite difference approximation of the solution is desired, using the approximation of the previous problem. Use a mesh size of $h = 1 / 3$ The conditions satisfied by the mesh points on the boundary are Select all that apply.

A) $u = 0$ at (0, 1/3) and (1/3, 0)
B) $u = 0$ at (0, 2/3) and (2/3, 0)
C) $u = 0$ at (1/3, 1/3) and (2/3, 2/3)
D) $u = 2 / 9$ at (1, 1/3) and (1/3, 1)
E) $u = 2 / 3$ at (1, 2/3) and (2/3, 1)

Question 8

Consider the problem $c ^ { 2 } \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = 0 , u ( 1 , t ) = 0 , u ( x , 0 ) = \sin ( \pi x ) , u _ { t } ( x , 0 ) = g ( x )$ . Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $h = 1 / 4$ and $\frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$ with a central difference approximation with $k = 1 / 3$ The resulting equation is

A) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) + u ( x , t - k ) ) / k ^ { 2 }$
B) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - 2 u ( x , t ) + u ( x , t - k ) ) / k ^ { 2 }$
C) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k ^ { 2 }$
D) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$
E) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$

Question 9

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

A) $u _ { i , j + 1 } = \lambda ^ { 2 } { } _ { u i } + 1 , j + \left( 1 + 2 \lambda ^ { 2 } \right) u _ { i j } + \lambda _ { u i - 1 , j - u i j - 1 } ^ { 2 }$
B) $u _ { i , j + 1 } = \lambda ^ { 2 } { } _ { u i } + 1 , j + \left( 1 - 2 \lambda ^ { 2 } \right) u _ { i j } + \lambda _ { u i - 1 , j - u i j - 1 } ^ { 2 }$
C) $u _ { i , j + 1 } = \lambda _ { u i + 1 , j } + ( 1 - \lambda ) u _ { i j } + \lambda _ { u i - 1 , j - u i j - 1 }$
D) $u _ { i , j - 1 } = \lambda ^ { 2 } { } _ { u i } + 1 , j + \left( 1 + 2 \lambda ^ { 2 } \right) u _ { i j } + \lambda _ { u i - 1 , j + u i j - 1 } ^ { 2 }$
E) $u _ { i , j - 1 } = \lambda _ { u i + 1 , j } + ( 1 - 2 \lambda ) u _ { i j } + \lambda _ { u i - 1 , j + u i j - 1 }$

Question 10

Laplace's equation is

A) hyperbolic
B) parabolic
C) elliptic
D) none of the above

Question 11

The wave equation is

A) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$
B) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0$
C) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial u } { \partial y }$
D) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0$
E) $\frac { \partial u } { \partial x } - \frac { \partial u } { \partial t } = 0$

Question 12

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.

A) $- 4 u _ { 11 } + u _ { 21 } + u _ { 12 } = 0$
B) $- 4 u _ { 22 } + u _ { 21 } + u _ { 12 } = - 2 / 9$
C) $- 4 u _ { 12 } + u _ { 11 } + u _ { 22 } = - 2 / 9$
D) $- 4 u _ { 21 } + u _ { 11 } + u _ { 22 } = - 2 / 9$
E) $- 4 u _ { 22 } + u _ { 21 } + u _ { 12 } = - 4 / 9$

Question 13

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

A) Use a central difference approximation in $t$ along the line $t = 0$ .
B) Use a forward difference approximation in $t$ along the line $t = 0$ .
C) Use a backward difference approximation in $t$ along the line $t = 0$ .
D) Use a forward difference approximation in x along the line $t = 0$ .
E) Use a backward difference approximation in x along the line $t = 0$ .

Question 14

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 15

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

A) $( u ( x + h , y ) - 2 u ( x , y ) + u ( x - h , y ) ) / h$
B) $( u ( x + h , y ) - 2 u ( x , y ) + u ( x - h , y ) ) / h ^ { 2 }$
C) $( u ( x , y + h ) - 2 u ( x , y ) + u ( x , y - h ) ) / h$
D) $( u ( x , y + h ) - 2 u ( x , y ) + u ( x , y - h ) ) / h ^ { 2 }$
E) $( u ( x + h , y ) - u ( x - h , y ) ) / ( 2 h )$

Question 16

A Dirichlet problem is a partial differential equation with conditions specifying

A) a linear combination of the values of the unknown function along the boundary and the values of the derivative of the unknown function along the boundary
B) the values of the unknown function along the boundary
C) the values of the derivative of the unknown function along the boundary
D) none of the above

Question 17

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

A) $u _ { 11 } = ( 16 - 7 \sqrt { 2 } + 6 g ( 1 / 4 ) ) / 18$
B) $u _ { 21 } = ( 8 \sqrt { 2 } - 7 + 3 g ( 1 / 2 ) ) / 9$
C) $u _ { 21 } = ( 8 \sqrt { 2 } + 7 + 3 g ( 1 / 2 ) ) / 9$
D) $u _ { 31 } = ( 8 - 7 \sqrt { 2 } / 2 + 3 g ( 3 / 4 ) ) / 9$
E) $u _ { 31 } = ( 8 + 7 \sqrt { 2 } / 2 + 3 g ( 3 / 4 ) ) / 9$

Question 18

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

A) yes
B) no
C) It is in the borderline.
D) It cannot be determined from the available data.

Question 19

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

A) $u _ { 22 } = 1 / 9$
B) $u _ { 22 } = 1 / 6$
C) $u _ { 11 } = 1 / 18$
D) $u _ { 12 } = 1 / 9$
E) $u _ { 21 } = 1 / 9$

Question 20

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

A) $c [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
B) $c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$
C) $c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
D) $c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
E) $c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$

Question 21

The heat equation is

A) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$
B) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial y } = 0$
C) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0$
D) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial u } { \partial t } = 0$
E) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0$

Question 22

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

A) $( u ( x + h , y ) - 2 u ( x , y ) + u ( x - h , y ) ) / h$
B) $( u ( x + h , y ) - 2 u ( x , y ) + u ( x - h , y ) ) / h ^ { 2 }$
C) $( u ( x , y + h ) - 2 u ( x , y ) + u ( x , y - h ) ) / h$
D) $( u ( x , y + h ) - 2 u ( x , y ) + u ( x , y - h ) ) / h ^ { 2 }$
E) $( u ( x + h , y ) - u ( x - h , y ) ) / ( 2 h )$

Question 23

The wave equation is

A) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0$
B) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y } = 0$
C) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial t } = 0$
D) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial u } { \partial t } = 0$
E) $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0$

Question 24

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

A) $u _ { 11 } = - 13 / 24 , u _ { 21 } = - 13 / 24$
B) $u _ { 11 } = - 17 / 24 , u _ { 21 } = - 17 / 24$
C) $u _ { 11 } = - 1 / 2 , u _ { 21 } = - 1 / 2$
D) $u _ { 11 } = - 1 / 24 , u _ { 21 } = - 1 / 24$
E) $u _ { 11 } = - 1 / 4 , u _ { 21 } = - 1 / 4$

Question 25

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

A) Use a forward difference approximation in $t$ along the line $t = 0$ .
B) Use a backward difference approximation in $t$ along the line $t = 0$ .
C) Use a central difference approximation in $$t$$ along the line $t = 0$ .
D) Use a forward difference approximation in $x$ along the line $t = 0$ .
E) Use a backward difference approximation in $x$ along the line $t = 0$ .

Question 26

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

A) $( u ( x + k , t ) - u ( x , t ) ) / k$
B) $( u ( x - k , t ) - u ( x , t ) ) / k ^ { 2 }$
C) $( u ( x , t + k ) - u ( x , t ) ) / k$
D) $( u ( x , t - k ) - u ( x , t ) ) / k$
E) $( u ( x , t + k ) - u ( x , t ) ) / k ^ { 2 }$

Question 27

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

A) $u _ { 22 } = \sqrt { 3 } / 8$
B) $u _ { 22 } = \sqrt { 3 } / 4$
C) $u _ { 11 } = \sqrt { 3 } / 8$
D) $u _ { 12 } = \sqrt { 3 } / 4$
E) $u _ { 21 } = \sqrt { 3 } / 4$

Question 28

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

A) $u _ { i , j - 1 } = \lambda ^ { 2 } u _ { i + 1 , j } + 2 \left( 1 + \lambda ^ { 2 } \right) u _ { i j } + \lambda ^ { 2 } u _ { i - 1 , j } + u _ { i , j - 1 }$
B) $u _ { i , j - 1 } = \lambda u _ { i + 1 , j } + 2 ( 1 - \lambda ) u _ { i j } + \lambda u _ { i - 1 , j } + u _ { i , j - 1 }$
C) $u _ { i , j + 1 } = \lambda ^ { 2 } u _ { i + 1 , j } + 2 \left( 1 + \lambda ^ { 2 } \right) u _ { i j } + \lambda ^ { 2 } u _ { i - 1 , j } - u _ { i , j - 1 }$
D) $u _ { i , j + 1 } = \lambda ^ { 2 } u _ { i + 1 , j } + 2 \left( 1 - \lambda ^ { 2 } \right) u _ { i j } + \lambda ^ { 2 } u _ { i - 1 , j } - u _ { i , j - 1 }$
E) $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - \lambda ) u _ { i j } + \lambda u _ { i - 1 , j } - u _ { i , j - 1 }$

Question 29

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) $u _ { i , j - 1 } = \lambda u _ { i + 1 , j } + ( 1 + 2 \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$
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 30

The wave equation is

A) hyperbolic
B) parabolic
C) elliptic
D) none of the above

Question 31

In the previous three problems, if $g ( x ) = 0$ then the values of $u _ { i , - 1 }$ are

A) $u _ { i , - 1 } = u _ { i , 1 }$
B) $u _ { i _ { . } - 1 } = 0$
C) $u _ { i _ { . } - 1 } = 1$
D) $u _ { i , - 1 } = - 1$
E) none of the above

Question 32

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

A) $c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
B) $c [ u ( x + h , t ) + 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$
C) $c [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
D) $c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$
E) $c [ u ( x + h , t ) - 4 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$

Question 33

Consider the problem $c ^ { 2 } \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } , u ( 0 , t ) = 0 , u ( 1 , t ) = 0 , u ( x , 0 ) = \left\{ \begin{array} { c c c } x & \text { if } & 0 < x < 1 / 2 \\1 - x & \text { if } & 1 / 2 < x < 1\end{array} \right\}$ , $u _ { t } ( x , 0 ) = g ( x )$ . Replace $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $h = 1 / 2$ and $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } }$ with a central difference approximation with $k = 1 / 2$ . The resulting equation is

A) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k ^ { 2 }$
B) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) + u ( x , t ) ) / k$
C) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$
D) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) + u ( x , t - k ) ) / k ^ { 2 }$
E) $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - 2 u ( x , t ) + u ( x , t - k ) ) / k ^ { 2 }$

Question 34

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) $u _ { 11 } = 0$
B) $u _ { 11 } = 20 / 9$
C) $u _ { 11 } = 30 / 9$
D) $u _ { 21 } = 26 / 9$
E) $u _ { 21 } = 32 / 9$

Question 35

A Dirichlet problem is a partial differential equation with conditions specifying

A) the values of the unknown function along the boundary
B) the values of the derivative of the unknown function along the boundary
C) a linear combination of the values of the unknown function along the boundary and the values of the derivative of the unknown function along the boundary
D) none of the above

Question 36

Laplace's equation is

A) hyperbolic
B) parabolic
C) elliptic
D) none of the above

Question 37

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

A) yes
B) no
C) It is right on the borderline.
D) It cannot be determined from the available data.

Question 38

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.

A) $- 4 u _ { 11 } + u _ { 21 } + u _ { 12 } = 0$
B) $- 4 u _ { 22 } + u _ { 21 } + u _ { 12 } = - \sqrt { 3 }$
C) $- 4 u _ { 22 } + u _ { 21 } + u _ { 12 } = - \sqrt { 3 } / 2$
D) $- 4 u _ { 12 } + u _ { 11 } + u _ { 22 } = - \sqrt { 3 } / 2$
E) $- 4 u _ { 21 } + u _ { 11 } + u _ { 22 } = - \sqrt { 3 } / 2$

Question 39

The five point approximation of the Laplacian is

A) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y - h ) - 2 u ( x , y ) ] / h$
B) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y - h ) - 4 u ( x , y ) ] / h$
C) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y - h ) - 2 u ( x , y ) ] / h ^ { 2 }$
D) $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y - h ) - 4 u ( x , y ) ] / h ^ { 2 }$
E) $[ u ( x + h , y ) - u ( x , y + h ) + u ( x - h , y ) - u ( x , y - h ) - 4 u ( x , y ) ] / h ^ { 2 }$

Question 40

Consider the problem $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0 , u ( 0 , y ) = 0 , u ( x , 0 ) = 0 , u ( 1 , y ) = \sin ( \pi y ) , u ( x , 1 ) = \sin ( \pi x )$ . A finite difference approximation of the solution is desired, using the approximation of the previous problem. Use a mesh size of $h = 1 / 3$ The conditions satisfied by the mesh points on the boundary are Select all that apply.

A) $u = 1 / 2 \text { at } ( 1,2 / 3 ) \text { and } ( 2 / 3,1 )$
B) $u = \sqrt { 3 } / 2 \text { at } ( 1,1 / 3 ) \text { and } ( 1 / 3,1 )$
C) $u = 0 \text { at } ( 0,1 / 3 ) \text { and } ( 1 / 3,0 )$
D) $u = 0 \text { at } ( 0,2 / 3 ) \text { and } ( 2 / 3,0 )$
E) $u = 0 \text { at } ( 1 / 3,1 / 3 ) \text { and } ( 2 / 3,2 / 3 )$