# Improved Euler (Heun's) Method Calculator

## Apply the Heun's method step by step

The calculator will find the approximate solution of the first-order differential equation using the improved Euler (Heun's) method, with steps shown.

Related calculators: Euler's Method Calculator, Modified Euler's Method Calculator

Or $y^{\prime } = f{\left(x,y \right)}$.
Or $x_{0}$.
$y_0=y(t_0)$ or $y_0=y(x_0)$.
Or $x_{1}$.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Our Heun's Method Calculator allows you to handle differential equations using the famous improved Euler's method formula.

## How to Use the Heun's Method Calculator?

• ### Input

Type or paste your differential equation in the specified field. Ensure that it is correctly formatted. Enter the value of $t$ for which you want to approximate $y(t)$. Specify either the number of steps or the step size $h$. Don't forget about the initial condition.

• ### Calculation

Once all values are inputted, click the "Calculate" button. The calculator will process the entered data and output the solution.

• ### Result

The approximate value of the function at the specified point will be displayed, as well as intermediate steps.

## What is the Heun's Method?

The Heun's Method, often referred to as the Improved Euler Method, is a numerical technique used to approximate solutions to ordinary differential equations (ODEs). It is a type of predictor-corrector method that uses two evaluations of the slope at different points in the interval to generate an approximation that is generally more accurate than that given by the standard Euler's method.

Working Principle

The Heun's Method enhances the Euler's method by incorporating an iterative, two-step approach:

• Predictor Step: Using the Euler's method, an initial approximation of the solution at the end of the interval is made.
• Corrector Step: This prediction is refined by computing the average of the function's slope at the beginning and the predicted end of the interval.

Mathematical Formulation

Suppose we are given an ordinary differential equation of the following form:

$$\frac{dy}{dt}=f(t,y)$$

Then the two steps of the Heun's method can be written as follows:

• Initial Estimate (Predictor):

$$\tilde{y}_{n+1}=y_n+h\cdot f\left(t_n,y_n\right)$$
• Refined Estimate (Corrector):

$$y_{n+1}=y_n+\frac{h}{2}\left(f\left(t_n,y_n\right)+f\left(t_{n+1},\tilde{y}_{n+1}\right)\right),$$

where:

• $y_n$ is the approximate solution at the present step.
• $h$ is the specified step size.
• $f\left(t_n,y_n\right)$ is the value of the derivative at the present step.

This formulation emphasizes the two-stage nature of the Heun's method and gives a clear description of the process of its implementation.

Let's understand the method with a simple example. Consider the following differential equation with the given initial condition:

$$\frac{dy}{dt}=-2ty,y(0)=1$$

We want to approximate $y(0.5)$ using the step size $h=0.5$.

Predictor. Use the initial estimate formula:

$$\tilde{y}_{n+1}=y_n+h\cdot f\left(t_n,y_n\right)$$

Substitute in the values:

$$\tilde{y}_1=1+0.5\cdot f(0,1)$$

Given $f(0,1)=-2\cdot0\cdot1=0$,

$$\tilde{y}_1=1+0.5\cdot0=1$$

Corrector. Use the refined estimate formula:

$$y_{n+1}=y_n+\frac{h}{2}\left(f\left(t_n,y_n\right)+f\left(t_{n+1},\tilde{y}_{n+1}\right)\right)$$

Substitute in the values:

$$y_1=1+\frac{0.5}{2}\left(f(0,1)+f(0.5,1)\right)$$

Given $f(0,1)=0$ and $f(0,5;1)=-2\cdot0.5\cdot1=-1$,

$$y_1=y(0.5)=1+\frac{0.5}{2}\left(0+(-1)\right)=0.75$$

Thus, the approximation for $y(0.5)$ using the Heun's method and the given step size is $y(1)=0.75$. The actual value is $y(0.5)=e^{-0.5^2}\approx0.778801$ and the approximation given by the Euler's method is $y(0.5)=1$. As can be seen, the approximation given by the Heun's method is better.

The Heun's method is a simple yet effective way to solve approximate the solution of a differential equation. It first makes a guess using the Euler's method and then improves that guess, increasing the likelihood of getting a more accurate answer to the problem.

## Why Choose Our Heun's Method Calculator?

• ### User-Friendly Design

Our calculator is designed with you in mind. Its intuitive interface ensures you can input data effortlessly and obtain results instantly without the need for knowledge of numerical methods.

• ### High Accuracy

The Heun's method gives better answers than many basic methods.

• ### Fast Сalculations

Our tool quickly solves problems, saving you time, especially with hard equations.

• ### Versatility

Our calculator helps both students with their studies and professionals in their work, covering school, research, and job-related tasks.

### FAQ

#### What is the Heun's Method Calculator?

It's a digital tool designed to help users solve ordinary differential equations using the Heun's method. It gives more accurate solutions than the standard Euler's method.

#### How does the Heun's method differ from the Euler's method?

While both are numerical methods, the Heun's method uses a two-step predictor-corrector approach, making it generally more accurate than the simple Euler's method.

#### Are there any limitations to using the Heun's Method?

Although the Heun's method is more accurate than the Euler's method, even more accurate methods such as Runge-Kutta exist for specific problems. The accuracy of the solution in the Heun's method also depends on the chosen step size.

#### How do I choose the right step size for my calculations?

A smaller step size generally offers better accuracy but requires more calculations. Start with a moderate step size and adjust based on the accuracy of the results.