Using Taylor Series to Improve the Euler Method

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Soil:

By: Alfie Chadwick Date: December 18, 2023 Flower

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Seeds:

Making a Python Library to solve differential Equations

Euler’s Method

Euler’s method is a classic way of approximating first-order differential equations. In short, it uses the derivative of a function and starting condition to estimate the value of the function a short distance from the starting point.

This is commonly written as:

$$\frac{dy}{dx}=f(x,y)$$ $$y(x+h)=y(x)+hf(x,y(x))+ϵ$$ $$\underset{h\to 0}{lim}|ϵ|=0$$

Where $ϵ$ is the error created by the approximation.

Higher Order ODEs

Generalizing Euler’s method to higher order ODEs is pretty easy. All you have to do is think of the ODE as a vector with each entry being the next derivative of the function. You can now write Euler’s Method in terms of this function:

$$\left[\begin{array}{c}y(x+h)\\ y^{\prime }(x+h)\\ y^{″}(x+h)\\ ...\\ y^{n-1}(x+h)\end{array}\right]=\left[\begin{array}{c}y(x)\\ y^{\prime }(x)\\ y^{″}(x)\\ ...\\ y^{n-1}(x)\end{array}\right]+\left[\begin{array}{ccccc}h& 0& 0& ...& 0\\ 0& h& 0& ...& 0\\ 0& 0& h& ...& 0\\ ...& ...& ...& ...& ...\\ 0& 0& 0& ...& h\end{array}\right]\cdot \left[\begin{array}{c}{y}^{\prime }(x)\\ y^{″}(x)\\ y^{‴}(x)\\ ...\\ y^n(x)\end{array}\right]+ϵ$$

Or shifting the $Y^{\prime }$ matrix to make it a bit prettier:

$$\left[\begin{array}{c}y(x+h)\\ y^{\prime }(x+h)\\ y^{″}(x+h)\\ ...\\ y^n(x+h)\end{array}\right]=\left[\begin{array}{ccccc}1& h& 0& ...& 0\\ 0& 1& h& ...& 0\\ 0& 0& 1& ...& 0\\ ...& ...& ...& ...& ...\\ 0& 0& 0& ...& 1\end{array}\right]\cdot \left[\begin{array}{c}y(x)\\ y^{\prime }(x)\\ y^{″}(x)\\ ...\\ y^n(x)\end{array}\right]+ϵ$$

Taylor Series

So everything up to now has been pretty textbook. But when I saw the matrix representation of the Euler method, I couldn’t help but think of another method that combines derivatives and linear algebra, Taylor Series. Taylor Series allows you to express functions as a polynomial of their derivatives at a single point $a$, as defined by this equation:

$$y(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{y^{(n)}(a)}{n!}\cdot (x-a{)}^n$$

The link between the Taylor Series and Euler’s Method becomes clear when we replace $x$ with $x+h$ and $a$ with $x$:

$$y(x+h)=\sum _{n=0}^{\mathrm{\infty }}\frac{y^{(n)}(x)}{n!}\cdot (h{)}^n$$

$$y(x+h)=y(x)+y^{\prime }(x)\cdot h+\frac{y^{″}(x)}{2!}\cdot (h{)}^2+\frac{y^{‴}(x)}{3!}\cdot (h{)}^3...$$

The first two terms of this expansion are the same as Euler’s Method, but the additional terms provide even greater accuracy, minimizing the error in the approximation!

Putting it together

A nice property of Taylor Series is that they have a really simple derivative function:

$$y^{\prime }(a+h)=\sum _{n=0}^{\mathrm{\infty }}\frac{y^{(n+1)}(a)}{n!}\cdot (h{)}^n$$ $$y^{(m)}(a+h)=\sum _{n=0}^{\mathrm{\infty }}\frac{y^{(n+m)}(a)}{n!}\cdot (h{)}^n$$

This means that not only the function can be described as a linear combination of the derivatives at a point, but so too can all derivatives of a function.

Using this, we can go back to the initial matrix representation of the Euler method and include these higher order terms.

$$\left[\begin{array}{c}y(x+h)\\ y^{\prime }(x+h)\\ y^{″}(x+h)\\ ...\\ y^n(x+h)\end{array}\right]=\left[\begin{array}{ccccc}1& \frac{h}{1!}& \frac{h^2}{2!}& ...& \frac{h^n}{n!}\\ 0& 1& \frac{h}{1!}& ...& \frac{h^{n-1}}{(n-1)!}\\ 0& 0& 1& ...& \frac{h^{n-2}}{(n-2)!}\\ ...& ...& ...& ...& ...\\ 0& 0& 0& ...& 1\end{array}\right]\cdot \left[\begin{array}{c}y(x)\\ y^{\prime }(x)\\ y^{″}(x)\\ ...\\ y^n(x)\end{array}\right]+ϵ$$

This should allow us to approximate higher order ODEs with more precision than just using Euler’s method.