Modified Euler's Method Calculator

Modified Euler’s method Calculator 

Compute approximate solutions to first-order ordinary differential equations (ODEs) using the Modified Euler's method (or Heun's method) with this calculator.

Also, learn about the complete process by clicking on the “show steps” option below the result to see the calculations done by the tool.

How to use this tool?

Here is how to operate the Modified Euler's method calculator:

1. Input the function with the help of the built-in keyboard. 
2. Choose what you want to divide by, step size or no. of steps.
3. Enter the value of what you chose from either option, in front of h.
4. Input the initial values of t and y.
5. Lastly, enter what is t1 and click Calculate.

What is Modified Euler’s Method?

 The Modified Euler's method, often referred to as Heun's method, is a numerical approach used to approximate solutions to first-order ordinary differential equations. 

Unlike the basic Euler's method, which uses only the initial point of an interval to estimate the solution at the next point, Modified Euler's method refines its prediction by taking into account both the beginning and an initial estimate of the end of the interval.

Formula for the MOdified Euler’s method:

The formula for the Modified Euler's Method (or Heun's Method) to solve the ordinary differential equation dy/dt = f(t,y) is as follows:

• Predictor Step:

y* = yn + hf(tn, yn)

Where:

• y* is the predicted value of y at tn+1. 
• yn is the current value of y.
• ℎ is the step size.
• f(tn, yn) is the value of the function at (tn, yn)

• Corrector Step:

yn+1 = yn + h/2 [f(tn , yn) + f(tn+1 , y*)] 

Where:

• yn+1 is the updated value of y for the next time step.
• tn+1 = tn + h is the next time step.

In essence, you first use Euler's method to predict what the value at tn+1 might be, then you use this prediction to calculate a more accurate slope, and finally, you use this more accurate slope to update the value of y.

How to solve ODEs using modified Euler's method?

• Start with a known point (tn, yn).
• Use the basic Euler's method to predict what the next point might be. This uses the "slope" or rate of change (given by the differential equation) at the starting point to estimate where you might end up after a certain "step size".

Predicted yn+1 = yn + hf(tn , yn) 

• With this predicted point in hand, check the slope or rate of change at this new location.

f(tn+1 , Predicted yn+1)

• Average the rate of change at the beginning and this new rate of change from the predicted point.

Average Slope = ½ (f(tn , yn) + f(tn+1, predicted yn+1)) 

• Use this averaged rate of change to find a more accurate estimate of the next point.

yn+1 = yn + h × Average Slope

Example:

let's solve the following first-order ordinary differential equation as an example:

dy/dt = −2ty, y(0)=1

using the Modified Euler's Method with a step size of h=0.2 up to t=1.0.

Here's the step-by-step solution:

1. Initial Conditions:

t0 = 0

y0 = 1

h=0.2

2. First Iteration (from t=0 to t=0.2):

• Predictor Step:

y* = y0 + hf(t0, y0) = 1+0.2x(−2⋅0⋅1) = 1

• Correcter step:

y1 = y0 + h/2 [f(t0 , y0) + f(t1 , y*)] = 1 + 0.2/2 [(−2⋅0⋅1)+(−2⋅0.2⋅1)] = 0.96

3. Second Iteration (from t=0.2 to t=0.4):

• Predictor Step:

y* = y1 + hf(t1, y1) = 0.96 + 0.2 x (−2⋅0.2⋅0.96) = 0.864

• Corrector Step:

y2 = y1 + h/2 [f(t1 , y1) + f(t2 , y*)] = 0.96+ 0.2 /2 [(−2⋅0.2⋅0.96)+(−2⋅0.4⋅0.864)]=0.8832

Applications:

Here are some examples of its applications:

Physics: It can be used to model the motion of particles, such as in celestial mechanics, or to simulate the behavior of electrical circuits.

Engineering: This method is helpful in solving differential equations that arise in the modeling of mechanical systems, such as the motion of a pendulum or the vibration of a beam.

Biology: It can be used to simulate the population dynamics of a species, the spread of infectious diseases, or the concentration of a drug in the bloodstream over time.