# 微分方程数值解作业 6

微分方程## Problem 1

### Question (a)

The equation to solve is

Use a rectangular grid

and the central difference scheme for first and second order derivatives

where

where

The boundary conditions can be written as

FDEs at the top row (where

Let

then the equation can be written as

where

where

where

and

It is easy to see that

### Question (b)

With

and

To use the conjugate gradient method,

And derive a sufficient condition for

. , covered by the first condition.- The first row of
satifies when the first condition is satisfied. is irreducible, since the directed graph of is equivalent to the grid graph, which is connected.

The test suggests that

Using the same test, we can discover that

is a sufficient condition for

### Question (c)

The symmetric condition requires

### Question (d)

Combining the results from (b) and (c), we can find a sufficient condition for the utilization of the conjugate gradient method is

## Problem 2

### Question (a)

Equation (6.17)

for

The FDEs for

With the boundary condition

Let

we can express the above FDE system as

where

It is obvious that

### Question (b)

Use second ordered forward difference at

and (6.17) at

That is

With

we express the above FDE system as

where

where

It is clear that

## Problem 3

### Question (a)

It is easy to see that the directed graph of

### Question (b)

The Gershgorin-Taussky theorem states that the eigenvalues of a real symmetric matrix lies in the union of intervals

For the given matrix

Therefore

## Problem 4

### Question (a)

From equation (6.15) we can derive