Euler's Method Error: Linear Functions

[evinda]

Hello! (Smile)

Theorem: Let $f \in C([a,b] \times \mathbb{R})$ function that satifies the total Lipschitz condition and let $y \in C^2([a,b])$ the solution of the ODE $\left\{\begin{matrix}
y=f(t,y(t)) &, a \leq t \leq b \\
y(a)=y_0 &
\end{matrix}\right.$
If $y^0, y^1, \dots, y^N$ are the approximations of Euler's method for uniform partition with step $h=\frac{b-a}{N}$ then

$$\max_{0 \leq n \leq N} |y(t^n)-y^n| \leq \frac{M}{2L}(e^{L(b-a)}-1)h$$

where $M=\max_{a \leq t \leq b} |y''(t)|$.

Then there is the following remark:

$$y'=1, 0 \leq t \leq 1 \\ y(0)=0 \\ y(t)=t $$

When the function is linear, there is no error.Can we justify as follows?Suppose that we have a linear function $y(t)=at+b$.
Then $y'(t)=a$ and $y''(t)=0$
So $M=0$ and thus $\max_{0 \leq n \leq N} |y(t^n)-y^n| \leq 0 \Rightarrow \max_{0 \leq n \leq N} |y(t^n)-y^n|=0$.
Also don't we always have a uniform partition at Euler's method? (Thinking)

 

[jvicens]

Hello there! It is great to see someone interested in numerical methods for solving ODEs. Your reasoning is partially correct, but there are a few points that need clarification.

Firstly, it is important to note that the theorem you have stated is for a general function $f$, not just linear functions. So while it may be true that for a linear function $y(t)=at+b$, the error is zero, this does not necessarily hold for all functions.

Secondly, the linear function $y(t)=at+b$ does not satisfy the total Lipschitz condition, as it is not a Lipschitz continuous function. Therefore, the theorem cannot be applied to this case.

Lastly, it is true that Euler's method uses a uniform partition, but this does not necessarily mean that the function being approximated is linear. The error in Euler's method depends on the function itself, not just the partition used.

In conclusion, while it is true that for a linear function $y(t)=at+b$, the error in Euler's method is zero, this does not hold for all functions. The theorem you have stated is a general result for any function satisfying the total Lipschitz condition, and it is not limited to just linear functions. I hope this helps clarify things for you. Happy researching!