## Introduction to Improved Euler Method Calculator:

Improved euler method calculator is an amazing online source that helps you evaluate the ordinary differential equation at its initial value. It is used to **compute the ODEs** to find the approximate value using an improved Euler method formula.

Heun's Method is a lengthy or complicated procedure to solve the Heun's method problem especially when you are doing its calculation by hand. By using our improved euler's method calculator you can easily get a solution just by adding the required input value.

## What is Heun's Method?

Heun's Method is a numerical analysis method that is used to find the **ordinary differential equation** with the initial value problem using Heun's Method formula. It is known as Heun's Method or the improved Euler method.

This method is used to evaluate the accurate solution of a given function. After all, it has less chance of error because it gives the nearest approximate value on a graph.

## Heun's Method Formula:

The **Heun's Method formula** consists of yn value which varies as n=0,1,2,3,.... You can modify this formula as per the desired number of y values you want to find by putting the value of n in it. The heun’s method formula used by the improved euler method calculator is,

$$ y_{n+1} \;=\; y_n + \frac{h}{2}[f(x_n, y_n) + f(x_{n+1}, y_n + hf(x_n, y_n))],\; n\;=\; 0,1,2,... $$

Whereas,

h is the relative size of the interval and for xi value use xi=yi+h as i=0,1,2,...

## How to Find Heun's Method?

To **find an approximate value** of a given ordinary differential equation solution using Heun's Method for a first-order, the heun's method calculator uses the formula of the improved Euler method and get a limit value solution until the interval step size is not achieved.

**Step 1**:

Analyze the given differential equation f(x, y)=dy/dx with the initial condition: y(x0)= y_0, step size: h.

**Step 2**:

Start the iteration process as per the given step size h value

**Step 3**:

Find the value of y1, for this the improved Euler method is used. You just need to evaluate the value and solve the formula for the y1 value.

$$ y_{n+1} \;=\; y_n + \frac{h}{2}[f(x_n, y_n) + f(x_{n+1}, y_n + hf(x_n, y_n))], n \;=\; 0,1,2,... $$

**Step 4**:

For the x1 value add x0 and y0 initial limit points.

$$ x_1 \;=\; x_0 + y_0 $$ (initially) But then it changes the condition

$$ x_2 \;=\; y_1 + h $$

$$ x_3 \;=\; y_2 + h……..x_i \;=\; y_i + h_i $$

**Step 5**:

Repeat this process of iteration until the given sample size of the interval is not achieved.

**Step 6**:

The values yi at each xi give the approximate solution y(x) over the given interval.

## Practical Example of Heun's Method:

**Heun's Method example** with the solution is given below to let you understand the working process of the improved Euler method calculator.

### Example: Find the following with step length 0.1

$$ y(0.5)\; for\; y’ \;=\; x^2 + 3x - 1,\; x_0 \;=\; 0,\; y_0 \;=\; -1 $$

Use the improved Euler method (1st order derivative).

**Solution**:

The given data for the question is,

$$ y’ \;=\; x^2 + 3x - 1 $$

$$ x_0 \;=\; 0,\; y_0 \;=\; -1,\; h \;=\; 0.1,\; x_n \;=\; 0.5 $$

The given differential equation and size of the interval can be written as,

$$ f(x,y) \;=\; x^2 + 3x - 1 $$

$$ h \;=\; 0.1, 0.2, 0.3, 0.4 $$

The improved Euler method formula is,

$$ y_{n+1} \;=\; y_n + \frac{h}{2}[f(x_n, y_n) + f(x_{n+1}, y_n + hf(x_n,\; y_n))],\; n \;=\; 0,1,2,... $$

For the value of y1 the above equation is modified,

$$ y_1 \;=\; y_0 + \frac{1}{2} h[f(x_0,\; y_0) + f(x_0 + h,\; y_0 + hf(x_0, y_0))] $$

Solve the value that are present inside the formula and put it into the above formula to get the value of y1,

$$ f(x_0 + h,\; y_0 + hf(x_0,\; y_0)) \;=\; f(0.1,\; -1.1) \;=\; -0.69 $$

$$ y_1 \;=\; -1 + \frac{0.1}{2} . [-1 - 0.69] \;=\; -1.0845 $$

Now you have the value of y1, for the value of x1 add the x0 from y0,

$$ f(x_0,\; y_0) \;=\; f(0, -1) \;=\; -1 $$

Again for y2 value repeat the process,

$$ y_2 \;=\; y_1 + \frac{1}{2} h[f(x_1, y_1) + f(x_1 + h,\; y_1 + hf(x_1, y_1))] $$

Solve the values that are present inside the formula and put it into the above formula to get the value of y2,

$$ f(x_1 + h,\; y_1 + hf(x_1, y_1)) \;=\; f(0.2, - 1.1535) \;=\; -0.36 $$

$$ y_2 \;=\; -1.0845 + \frac{0.1}{2} . [-0.69 - 0.36] \;=\; -1.137 $$

For the value of x2, add h and y1 value,

$$ f(x_1, y_1) \;=\; g(0.1, -1.0845) \;=\; -0.69 $$

Again for the value of y_3, the formula of the improved modified Euler method become,

$$ y_3 \;=\; y_2 + \frac{1}{2} h[f(x_2, y_2) + f(x_2 +h, y_2 + hf(x_2, y_2))] $$

Put the value in this formula again to get the y3 value,

$$ f(x_2 + h, y_2 + hf(x_2, y_2)) \;=\; f(0.3, -1.173) \;=\; -0.01 $$

$$ y_3 \;=\; -1.137 + \frac{0.1}{2} . [-0.36 - 0.01] \;=\; -1.1555 $$

Using the add h2 and y2 find the value of x3,

$$ f(x_2, y_2) \;=\; f(0.2, -1.137) \;=\; -0.36 $$

Again for the value of y_4, the formula of the improved modified Euler method become,

$$ y_4 \;=\; y_3 + \frac{1}{2} h[f(x_3, y_3) + f(x_3 + h, y_3 + hf(x_3, y_3))] $$

Put the value in this formula again to get the y4 value,

$$ f(x_3 + h,\; y_3 + hf(x_3, y_3)) \;=\; f(0.4, -1.1565) \;=\; 0.36 $$

$$ y_4 \;=\; -1.1555 + \frac{0.1}{2} . [-0.01 + 0.36] \;=\; -1.138 $$

Using the add h3 and y3 find the value of x4,

$$ f(x_3, y_3) \;=\; f(0.3, -1.1555) \;=\; -0.01 $$

Again for the value of y5, the formula of the improved modified Euler method become

$$ y_5 \;=\; y_4 + \frac{1}{2} h[f(x_4, y_4) + f(x_4 + h, y_4 + hf(x_4, y_4))] $$

Put the value in this formula again to get the y5 value,

$$ f(x_4 + h, y_4 + hf(x_4, y_4)) \;=\; f(0.5, -1.102) \;=\; 0.75 $$

$$ y_5 \;=\; -1.138 + \frac{0.1}{2} . [0.36 + 0.75] \;=\; -1.0825 $$

Using the add h4 and y4 find the value of x5,

$$ f(x_4, y_4) \;=\; f(0.4, -1.138) \;=\; 0.36 $$

Therefore the result of the given improved Euler method is,

$$ y(0.5) \;=\; -1.0825 $$

## How to Use the Improved Euler Method Calculator?

The improved euler's method calculator has a simple layout that allows you to solve Heun's method question. You just need to put your problem and follow some simple steps without any inconvenience during calculation. These steps are:

**Enter the value**of the ordinary differential equation using Heun's method in the input field.- Enter the value of the limit points of OEDs (x0, y0) in the input field.
- Add the size of the interval h in its required input box.
- Add the “an” value which is the average slope of the ordinary differential equation.
- Review your given input value before clicking the calculate button of the Heun's method calculator to get the exact solution of Heun's method problem.
- Click the “Calculate” button for the solution of Heun's method problems.
- If you want to check the calculation of our heun method calculator then use the load example for the calculation to get an idea about its procedure.
- Click the “Recalculate” button for the evaluation of more examples of Heun's method question solution.

## Output from Improved Euler's Method Calculator:

The improved euler method calculator provides you the **solution** for finding the solution of ODEs as per your input values when you click on the calculate button. It may include as:

**In the Result Box**:

When you click on the result button of the improved euler calculator you get the solution of the given equation.

**Steps Box**:

Click on the steps option so that you get the solution to Heun's method questions in a step-by-step method.

## Benefits of Heun's Method Calculator:

The improved eulers method calculator has many useful features that you obtain whenever you use it to solve ordinary differential problems solution. You just need to add the input value to get a solution without imposing a condition of a sign-up.

- The heun method calculator is a
**trustworthy tool**as it always provides you with accurate solutions to the improved Euler method questions. - It is a swift tool that evaluates heun method problems with solutions in less than a minute.
- The improved euler's method calculator is an adaptable tool that helps you solve various types of Heun method problems easily on online platforms.
- It is a handy tool that saves you tons of time which you consume in solving the heun method problem quickly without putting in external effort.
- The improved euler calculator is a free tool that allows you to use it for the calculation of improved Euler method problems.
- Our Calculator is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of improved Euler method problems.
- Improved euler method calculator can operate on all devices desktop, mobile, or laptop to solve ODE equations using Heun method problems.