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Predictor-Corrector or Modified-Euler method for solving Differential equation

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  • Difficulty Level : Hard
  • Last Updated : 28 Dec, 2021

For a given differential equation \frac{dy}{dx}=f(x, y)  with initial condition y(x_0)=y_0
find the approximate solution using Predictor-Corrector method.
Predictor-Corrector Method : 
The predictor-corrector method is also known as Modified-Euler method
In the Euler method, the tangent is drawn at a point and slope is calculated for a given step size. Thus this method works best with linear functions, but for other cases, there remains a truncation error. To solve this problem the Modified Euler method is introduced. In this method instead of a point, the arithmetic average of the slope over an interval (x_t, x_{t+1})  is used.
Thus in the Predictor-Corrector method for each step the predicted value of y_{t+1}  is calculated first using Euler’s method and then the slopes at the points (x_t, y_t)  and (x_{t+1}, y_{t+1})  is calculated and the arithmetic average of these slopes are added to y_t  to calculate the corrected value of y_{t+1}  .
So, 
 

  • Step – 1 : First the value is predicted for a step(here t+1) : y_{t+1, p} = y_t + h*f(x_t, y_t)
    here h is step size for each increment

 

  • Step – 2 : Then the predicted value is correctedy_{t+1, c} = y_t + h*\frac{f(x_t, y_t)+f(x_{t+1}, y_{t+1, p})}{2}

 

  • Step – 3 : The incrementation is done : x_{t+1}=x_t+h, t=t+1

 

  • Step – 4 : Check for continuation, if x_{t}<x_{n}  then go to step – 1.

 

  • Step – 5 : Terminate the process.

As, in this method, the average slope is used, so the error is reduced significantly. Also, we can repeat the process of correction for convergence. Thus at every step, we are reducing the error thus by improving the value of y.
Examples: 
 

Input : eq = \frac {dy}{dx} = y-2x^2+1  , y(0) = 0.5, step size(h) = 0.2 
To find: y(1) 
Output: y(1) = 2.18147
Explanation: 
\begin{tabular}{|l|l|l|l|l|} \hline x_t & y_t & x_{t+1} & y_{t+1, p} & y_{t+1, c} \\ \hline 0 & 0.5 & 0.2 & 0.8 & 0.82444 \\ \hline 0.2 & 0.82444 & 0.4 & 1.17333 & 1.18543 \\ \hline 0.4 & 1.18543 & 0.6 & 1.55852 & 1.55553 \\ \hline 0.6 & 1.55553 & 0.8 & 1.92263 & 1.9012 \\ \hline 0.8 & 1.9012 & 1 & 2.22544 & 2.18147 \\ \hline \end{tabular}
The final value of y at x = 1 is y=2.18147 
 

Implementation: Here we are considering the differential equation: \frac {dy}{dx} = y-2x^2+1
 

C++




// C++ code for solving the differential equation
// using Predictor-Corrector or Modified-Euler method
// with the given conditions, y(0) = 0.5, step size(h) = 0.2
// to find y(1)
 
#include <bits/stdc++.h>
using namespace std;
 
// consider the differential equation
// for a given x and y, return v
double f(double x, double y)
{
    double v = y - 2 * x * x + 1;
    return v;
}
 
// predicts the next value for a given (x, y)
// and step size h using Euler method
double predict(double x, double y, double h)
{
    // value of next y(predicted) is returned
    double y1p = y + h * f(x, y);
    return y1p;
}
 
// corrects the predicted value
// using Modified Euler method
double correct(double x, double y,
               double x1, double y1,
               double h)
{
    // (x, y) are of previous step
    // and x1 is the increased x for next step
    // and y1 is predicted y for next step
    double e = 0.00001;
    double y1c = y1;
 
    do {
        y1 = y1c;
        y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));
    } while (fabs(y1c - y1) > e);
 
    // every iteration is correcting the value
    // of y using average slope
    return y1c;
}
 
void printFinalValues(double x, double xn,
                      double y, double h)
{
 
    while (x < xn) {
        double x1 = x + h;
        double y1p = predict(x, y, h);
        double y1c = correct(x, y, x1, y1p, h);
        x = x1;
        y = y1c;
    }
 
    // at every iteration first the value
    // of for next step is first predicted
    // and then corrected.
    cout << "The final value of y at x = "
         << x << " is : " << y << endl;
}
 
int main()
{
    // here x and y are the initial
    // given condition, so x=0 and y=0.5
    double x = 0, y = 0.5;
 
    // final value of x for which y is needed
    double xn = 1;
 
    // step size
    double h = 0.2;
 
    printFinalValues(x, xn, y, h);
 
    return 0;
}

Java




// Java code for solving the differential
// equation using Predictor-Corrector
// or Modified-Euler method with the
// given conditions, y(0) = 0.5, step
// size(h) = 0.2 to find y(1)
import java.text.*;
 
class GFG
{
     
// consider the differential equation
// for a given x and y, return v
static double f(double x, double y)
{
    double v = y - 2 * x * x + 1;
    return v;
}
 
// predicts the next value for a given (x, y)
// and step size h using Euler method
static double predict(double x, double y, double h)
{
    // value of next y(predicted) is returned
    double y1p = y + h * f(x, y);
    return y1p;
}
 
// corrects the predicted value
// using Modified Euler method
static double correct(double x, double y,
                    double x1, double y1,
                    double h)
{
    // (x, y) are of previous step
    // and x1 is the increased x for next step
    // and y1 is predicted y for next step
    double e = 0.00001;
    double y1c = y1;
 
    do
    {
        y1 = y1c;
        y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));
    }
    while (Math.abs(y1c - y1) > e);
 
    // every iteration is correcting the value
    // of y using average slope
    return y1c;
}
 
static void printFinalValues(double x, double xn,
                    double y, double h)
{
 
    while (x < xn)
    {
        double x1 = x + h;
        double y1p = predict(x, y, h);
        double y1c = correct(x, y, x1, y1p, h);
        x = x1;
        y = y1c;
    }
 
    // at every iteration first the value
    // of for next step is first predicted
    // and then corrected.
    DecimalFormat df = new DecimalFormat("#.#####");
    System.out.println("The final value of y at x = "+
                        x + " is : "+df.format(y));
}
 
// Driver code
public static void main (String[] args)
{
    // here x and y are the initial
    // given condition, so x=0 and y=0.5
    double x = 0, y = 0.5;
 
    // final value of x for which y is needed
    double xn = 1;
 
    // step size
    double h = 0.2;
 
    printFinalValues(x, xn, y, h);
}
}
 
// This code is contributed by mits

Python3




# Python3 code for solving the differential equation
# using Predictor-Corrector or Modified-Euler method
# with the given conditions, y(0) = 0.5, step size(h) = 0.2
# to find y(1)
 
# consider the differential equation
# for a given x and y, return v
def f(x, y):
    v = y - 2 * x * x + 1;
    return v;
 
# predicts the next value for a given (x, y)
# and step size h using Euler method
def predict(x, y, h):
     
    # value of next y(predicted) is returned
    y1p = y + h * f(x, y);
    return y1p;
 
# corrects the predicted value
# using Modified Euler method
def correct(x, y, x1, y1, h):
     
    # (x, y) are of previous step
    # and x1 is the increased x for next step
    # and y1 is predicted y for next step
    e = 0.00001;
    y1c = y1;
 
    while (abs(y1c - y1) > e + 1):
        y1 = y1c;
        y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));
 
    # every iteration is correcting the value
    # of y using average slope
    return y1c;
 
def printFinalValues(x, xn, y, h):
    while (x < xn):
        x1 = x + h;
        y1p = predict(x, y, h);
        y1c = correct(x, y, x1, y1p, h);
        x = x1;
        y = y1c;
 
    # at every iteration first the value
    # of for next step is first predicted
    # and then corrected.
    print("The final value of y at x =",
                     int(x), "is :", y);
 
# Driver Code
if __name__ == '__main__':
     
    # here x and y are the initial
    # given condition, so x=0 and y=0.5
    x = 0; y = 0.5;
 
    # final value of x for which y is needed
    xn = 1;
 
    # step size
    h = 0.2;
 
    printFinalValues(x, xn, y, h);
 
# This code is contributed by Rajput-Ji

C#




// C# code for solving the differential
// equation using Predictor-Corrector
// or Modified-Euler method with the
// given conditions, y(0) = 0.5, step
// size(h) = 0.2 to find y(1)
using System;
 
class GFG
{
     
// consider the differential equation
// for a given x and y, return v
static double f(double x, double y)
{
    double v = y - 2 * x * x + 1;
    return v;
}
 
// predicts the next value for a given (x, y)
// and step size h using Euler method
static double predict(double x, double y, double h)
{
    // value of next y(predicted) is returned
    double y1p = y + h * f(x, y);
    return y1p;
}
 
// corrects the predicted value
// using Modified Euler method
static double correct(double x, double y,
            double x1, double y1,
            double h)
{
    // (x, y) are of previous step
    // and x1 is the increased x for next step
    // and y1 is predicted y for next step
    double e = 0.00001;
    double y1c = y1;
 
    do
    {
        y1 = y1c;
        y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));
    }
    while (Math.Abs(y1c - y1) > e);
 
    // every iteration is correcting the value
    // of y using average slope
    return y1c;
}
 
static void printFinalValues(double x, double xn,
                    double y, double h)
{
 
    while (x < xn)
    {
        double x1 = x + h;
        double y1p = predict(x, y, h);
        double y1c = correct(x, y, x1, y1p, h);
        x = x1;
        y = y1c;
    }
 
    // at every iteration first the value
    // of for next step is first predicted
    // and then corrected.
    Console.WriteLine("The final value of y at x = "+
                        x + " is : " + Math.Round(y, 5));
}
 
// Driver code
static void Main()
{
    // here x and y are the initial
    // given condition, so x=0 and y=0.5
    double x = 0, y = 0.5;
 
    // final value of x for which y is needed
    double xn = 1;
 
    // step size
    double h = 0.2;
 
    printFinalValues(x, xn, y, h);
}
}
 
// This code is contributed by mits

PHP




<?php
// PHP code for solving the differential equation
// using Predictor-Corrector or Modified-Euler
// method with the given conditions, y(0) = 0.5,
// step size(h) = 0.2 to find y(1)
 
// consider the differential equation
// for a given x and y, return v
function f($x, $y)
{
    $v = $y - 2 * $x * $x + 1;
    return $v;
}
 
// predicts the next value for a given (x, y)
// and step size h using Euler method
function predict($x, $y, $h)
{
    // value of next y(predicted) is returned
    $y1p = $y + $h * f($x, $y);
    return $y1p;
}
 
// corrects the predicted value
// using Modified Euler method
function correct($x, $y, $x1, $y1, $h)
{
     
    // (x, y) are of previous step and
    // x1 is the increased x for next step
    // and y1 is predicted y for next step
    $e = 0.00001;
    $y1c = $y1;
 
    do
    {
        $y1 = $y1c;
        $y1c = $y + 0.5 * $h * (f($x, $y) +
                                f($x1, $y1));
    } while (abs($y1c - $y1) > $e);
 
    // every iteration is correcting the
    // value of y using average slope
    return $y1c;
}
 
function printFinalValues($x, $xn, $y, $h)
{
    while ($x < $xn)
    {
        $x1 = $x + $h;
        $y1p = predict($x, $y, $h);
        $y1c = correct($x, $y, $x1, $y1p, $h);
        $x = $x1;
        $y = $y1c;
    }
 
    // at every iteration first the value
    // of for next step is first predicted
    // and then corrected.
    echo "The final value of y at x = " . $x .
               " is : " . round($y, 5) . "\n";
}
 
    // here x and y are the initial
    // given condition, so x=0 and y=0.5
    $x = 0;
    $y = 0.5;
 
    // final value of x for which y is needed
    $xn = 1;
 
    // step size
    $h = 0.2;
 
    printFinalValues($x, $xn, $y, $h);
 
// This code is contributed by mits
?>

Javascript




<script>
// javascript code for solving the differential
// equation using Predictor-Corrector
// or Modified-Euler method with the
// given conditions, y(0) = 0.5, step
// size(h) = 0.2 to find y(1)
 
    // consider the differential equation
    // for a given x and y, return v
    function f(x , y) {
        var v = y - 2 * x * x + 1;
        return v;
    }
 
    // predicts the next value for a given (x, y)
    // and step size h using Euler method
    function predict(x , y , h) {
        // value of next y(predicted) is returned
        var y1p = y + h * f(x, y);
        return y1p;
    }
 
    // corrects the predicted value
    // using Modified Euler method
    function correct(x , y , x1 , y1 , h) {
        // (x, y) are of previous step
        // and x1 is the increased x for next step
        // and y1 is predicted y for next step
        var e = 0.00001;
        var y1c = y1;
 
        do {
            y1 = y1c;
            y1c = y + 0.5 * h * (f(x, y) + f(x1, y1));
        } while (Math.abs(y1c - y1) > e);
 
        // every iteration is correcting the value
        // of y using average slope
        return y1c;
    }
 
    function printFinalValues(x , xn , y , h) {
 
        while (x < xn) {
            var x1 = x + h;
            var y1p = predict(x, y, h);
            var y1c = correct(x, y, x1, y1p, h);
            x = x1;
            y = y1c;
        }
 
        // at every iteration first the value
        // of for next step is first predicted
        // and then corrected.
 
        document.write("The final value of y at x = " + x + " is : " + y.toFixed(5));
    }
 
    // Driver code
     
        // here x and y are the initial
        // given condition, so x=0 and y=0.5
        var x = 0, y = 0.5;
 
        // final value of x for which y is needed
        var xn = 1;
 
        // step size
        var h = 0.2;
        printFinalValues(x, xn, y, h);
 
// This code is contributed by Rajput-Ji
</script>
Output: 
The final value of y at x = 1 is : 2.18147

 


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