MIT differential equation---2

今天上两节课试试看效果~。

IVP problem: initial value problem

differential equation and the initial value

use Euler's method to solve the equation. The method is originally used to prove the existence problem of a differential equation, but is also used to 

solve the equation.

Euler's method is a numerical method. It's easy to understand and useful because in many cases, differential equations cannot be solved by elementary functions.

There comes the problem that how to derive an explicit formula for numerical methods and is the solution precise?

Well, the numerical solution may be lower or higher than the actual solution. It will be lower if the curve is convex and higher if concave. If it's a straight line, then you get lucky.

(Remainder: second derivative decides whether the curve is convex or concave, and you can just use the RHS to do the job) 

It's easy to think of a better method which is switching to a smaller h, then the approximate curve can follow closer to the actual solution.( sounds not efficient enough as h

gets smaller) 

Euler's error ~ Ch where h is the step-wise increment and C is some coefficient. Therefore, Euler's method is a first-order method 

To improve the accuracy of Euler's error, we can calculate two slopes at each iteration and then, we'll get an improved Euler's method.

If you want further minimize the error, there's RK2 which is a second-order method and RK4(Runge-kutta) which is a fourth-order method. RK4 is also a standard method to approximate the solution. However, it does require more hard steps(calculation of slope in our approximation).

Now, to share a somehow pessimistic view, RK4 method will fail to find solution if the equation is not continuous at some points. For example, y' = y^2( y = 1 / (C - x) ), the method will just fail. There's no way to predict C, which means each possible solution will have its own singularity.  You cannot predict when things begin to go bad in advance.

这学期上了数值分析，对这节课的理解还是很有帮助。 总之，可以通过数值计算的方式逼近常微分方程的解，但是对于有sigularity（不知道中文怎么解释，wiki上没有找到）的方程，数值计算方法很有可能出错。相似的情况也发生在牛顿法对方程求解。