# Euler methond and the improved Euler method

• nkk2008
In summary, the conversation discusses the confusion and difficulties in using the Euler method for solving differential equations. The method involves finding the slope of the tangent line at each point and using that to approximate the graph of the function. However, it is not exact and there is a potential for error due to the approximation of the tangent line and calculating the next slope at the wrong place. To improve the accuracy, the improved Euler method, also known as Heun method, uses an average of two slopes to calculate the tangent line. There is a similar method called Modified Euler Method, so it is important to be aware of the differences between the two.
nkk2008
I am confused by both of these. I get what it does, and how it does it, but when using it, somehow I ultimately screw up.

Could someone just post a quick description of what to do in conversational English, please? I am pretty sure that would solve this random mental block I have.

Thanks,
Nkk

Euler method: The differential equation, dy/dx= f(x,y) tells you how to find the slope of the tangent line to y= y(x) at each point. Starting at, say, $(x_0,y_0)$, calculate dy/dx and draw the tangent line, $y= f'(x_0,y_0)(x- x_0)+ y_0$.

That will approximate the graph of y= y(x) for a short distance. So follow that tangent from $(x_0,y_0)$ a short distance to $(x_0+ h, f'(x_0,y_0)h+ y_0)$. Now, using that point as "$(x_0,y_0)$" do it again.

Improved Euler's method: The Euler's method isn't exact because the tangent line is only approximate to the actual curve. And there is a "double" problem. Following the tangent line not only gives an point that is a little off, you will be calculating the next slope at the wrong place! Suppose $$\displaystyle f'(x_0,y_0)$$ is positive but decreasing. Following the tangent line not only gives an point that is a little too high (you didn't follow the downard curve of the graph), you will be calculating the next slope at the wrong place! One way to allow for that is to
1) Find $f(x_0,y_0)$ to find the slope and follow the tangent line to $(x_1,y_1)$
2) Find $f(x_1,y_1)$ to approximate the slope of the tangent line at the new point.
3) Use the average of those two slopes as the slope of your tangent line rather than the original.

Euler method is a Taylor expansion of first order (linear approximation) of the solution at every point.
You can try using the chain rule to make a second order approximation to the function:

$$y^{'}=f(x,y)$$
$$y^{''}=f_{x}(x,y)+f_{y}(x,y)y'=f_{x}(x,y)+f_{y}(x,y)f(x,y)$$

So at point $$(x_{0},y_{0})$$ the next point $$(x_{0}+h,y)$$ will satisfy
$$y=y_{0}+f(x_{0},y_{0})h +[f_{x}(x_{0},y_{0})+f_{y}(x_{0},y_{0})f(x_{0},y_{0})]\frac{h^{2}}{2}$$

Just saw in one textbook. Improved Euler method is also known as Heun method. It is a kind of predictor-corrector method.

Beware! There is another method called Modified Euler Method.

## 1. What is the Euler method?

The Euler method is a numerical method used to approximate the solution of a first-order differential equation. It is based on the idea of breaking down the problem into smaller steps and using linear approximations to estimate the solution at each step.

## 2. How does the Euler method work?

The Euler method works by starting at an initial value of the solution and using the derivative of the function at that point to approximate the value of the solution at the next step. This process is repeated until the desired number of steps is reached.

## 3. What are the limitations of the Euler method?

The Euler method is a first-order method, meaning that it has an error that is proportional to the step size. This means that as the step size decreases, the error decreases as well. However, the Euler method can be unstable for certain types of differential equations, and it may require a very small step size to achieve an accurate solution.

## 4. What is the improved Euler method?

The improved Euler method, also known as the Heun's method, is an enhancement of the Euler method that uses a two-step process to estimate the solution at each step. It uses the average of the slopes at the beginning and end of the step to improve the accuracy of the approximation.

## 5. When should the improved Euler method be used instead of the Euler method?

The improved Euler method should be used when a higher level of accuracy is desired, as it generally produces more accurate results than the Euler method. It is also more stable and can handle certain types of differential equations that the Euler method cannot. However, it may require more computational effort due to the two-step process.

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