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.

Accepted Answer

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

Question 2

The wave equation is

Accepted Answer

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)  $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0$ 
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)  $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0$ E

Question 3

Laplace's equation is

Accepted Answer

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

Question 4

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.

Accepted Answer

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)  $- 4 u _ { 22 } + u _ { 21 } + u _ { 12 } = - 4 / 9$ 
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 5

The five-point approximation of the Laplacian is

Accepted Answer

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)  $[ u ( x + h , y ) + u ( x , y + h ) + u ( x - h , y ) + u ( x , y + h ) - 2 u ( x , y ) ] / h ^ { 2 }$ 
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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 
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 8

The wave equation is

Accepted Answer

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)  $\frac { \partial u } { \partial x } - \frac { \partial u } { \partial t } = 0$ 
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 9

The five point approximation of the Laplacian is

Accepted Answer

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)  $[ u ( x + h , y ) - u ( x , y + h ) + u ( x - h , y ) - u ( x , y - h ) - 4 u ( x , y ) ] / h ^ { 2 }$ 
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 10

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.

Accepted Answer

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) 
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 11

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

Accepted Answer

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)  $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - u ( x , t ) ) / k$ 
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 12

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

Accepted Answer

A)  $u _ { 22 } = 1 / 9$ 
B)  $u _ { 22 } = 1 / 6$ 
C)  $u _ { 11 } = 1 / 18$ 
D)  $u _ { 12 } = 1 / 9$ 
E)  $u _ { 21 } = 1 / 9$ 
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 13

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

Accepted Answer

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)  $c ^ { 2 } [ u ( x + h , t ) - 2 u ( x , t ) + u ( x - h , t ) ] / h ^ { 2 } = ( u ( x , t + k ) - 2 u ( x , t ) + u ( x , t - k ) ) / k ^ { 2 }$ 
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 14

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.

Accepted Answer

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)  $u = 0 \text { at } ( 1 / 3,1 / 3 ) \text { and } ( 2 / 3,2 / 3 )$ 
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)  $u = 0 \text { at } ( 1 / 3,1 / 3 ) \text { and } ( 2 / 3,2 / 3 )$

Question 15

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

Accepted Answer

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)  $( u ( x + h , y ) - u ( x - h , y ) ) / ( 2 h )$ 
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

Accepted Answer

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

Accepted Answer

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)  $u _ { i , j + 1 } = \lambda u _ { i + 1 , j } + ( 1 - \lambda ) u _ { i , j } + \lambda u _ { i - 1 , j }$ 
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 18

Laplace's equation is

Accepted Answer

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)  $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0$ 
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)  $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0$

Question 19

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

Accepted Answer

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)  $u _ { i j + 1 } = \lambda u _ { i + 1 , j } + ( 1 + 2 \lambda ) u _ { i j } + \lambda u _ { i - 1 , j }$ 
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 20

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

Accepted Answer

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

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

The wave equation is

Laplace's equation is

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.

The five-point approximation of the Laplacian is

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

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

The wave equation is

The five point approximation of the Laplacian is

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

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

A Dirichlet problem is a partial differential equation with conditions specifying

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

Laplace's equation is

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

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

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

Exam 15: Numerical Solutions of Partial Differential Equations

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

The wave equation is

Laplace's equation is

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

The five-point approximation of the Laplacian is

In the previous problem, is the value of λ\lambdaλ such that the scheme is stable?

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

The wave equation is

The five point approximation of the Laplacian is

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

The central difference approximation for ∂u∂x\frac { \partial u } { \partial x }∂x∂u​ with step size hhh is

A Dirichlet problem is a partial differential equation with conditions specifying

In the previous problem, using the notation uij=u(x,t)u _ { i j } = u ( x , \mathrm { t } )uij​=u(x,t) , and letting λ=ck/h2\lambda = c k / h ^ { 2 }λ=ck/h2 , the equation becomes

Laplace's equation is

In the previous problem, using the notation uij=u(x,t)u _ { i j } = u ( x , \mathrm { t } )uij​=u(x,t) , and letting λ=ck/h2\lambda = c k / h ^ { 2 }λ=ck/h2 , the equation becomes

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

Introduction to Differential Equations

First-Order Differential Equations

Modeling With First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

Filters

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

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

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

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

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

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

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