# 微分方程数值解作业 2

微分方程## Problem 1

To solve (2.4)

We can always obtain the equation at non-boundary positions,

The above FDE approximation has an error of

### Question (a)

The first condition

For the second condition

substitute

we have

And the final equation becomes

The whole FDE system in matrix form

### Question (b)

Therefore

FDE system in matrix form

## Problem 2

### Question (a)

Use central difference to approximate

Substitute back

### Question (b)

Given the differerntial equation

Simply susbtitute

### Question (c)

When

we will be able to seperate

Collect

The above equation is linear, and could be written in matrix form. Denote

Assume the boundary conditions are

Finally, the FDE system is expressed in explicit matrix form

### Question (d)

Therefore, when

## Problem 3

At

And use a forth-ordered forward difference to approximiate

Substitute back to the equation we have one equation