Question

Implement the 4th order Runge-Kutta algotithm in MATLAB. Use the script you produced to integrate the following function x(t)--10t + e-t , x(0)--1; t.-0 t, = 1 Vary At and observe the difference in your results. Let At 0.2 sec., 0.1 sec., 0.05 sec. and 0.01 sec. Now integrate the function analytically and compare your result with the results obtained numerically

Answer 1

$\frac{\mathrm{dx} }{\mathrm{d} y} = f(x, y),y(0)= y_o$

The task is to find value of unknown function y at a given point x.

The Runge-Kutta method finds approximate value of y for a given x. Only first order ordinary differential equations can be solved by using the Runge Kutta 4th order method.

Below is the formula used to compute next value y_n+1 from previous value y_n. The value of n are 0, 1, 2, 3, ….(x – x0)/h. Here h is step height and x_n+1 = x₀ + h

$K_1 = hf(x_n, y_n) K_2 = hf(x_n+\frac{h}{2}, y_n+\frac{k_1}{2}) K_3 = hf(x_n+\frac{h}{2}, y_n+\frac{k_2}{2}) K_4 = hf(x_n+h, y_n+k_3) y_n_+_1 = y_n + k_1/6 + k_2/3 + k_3/3 + k_4/6 + O(h^{5})$

The formula basically computes next value y_n+1 using current y_n plus weighted average of four increments.

• k₁ is the increment based on the slope at the beginning of the interval, using y
• k₂ is the increment based on the slope at the midpoint of the interval, using y + hk₁/2.
• k₃ is again the increment based on the slope at the midpoint, using using y + hk₂/2.
• k₄ is the increment based on the slope at the end of the interval, using y + hk₃

_{the code is:-}

// C program to implement Runge Kutta method

#include<stdio.h>

  

// A sample differential equation "dy/dx = (x - y)/2"

float dydx(float x, float y)

{

    return((x - y)/2);

}

  

// Finds value of y for a given x using step size h

// and initial value y0 at x0.

float rungeKutta(float x0, float y0, float x, float h)

{

    // Count number of iterations using step size or

    // step height h

    int n = (int)((x - x0) / h);

  

    float k1, k2, k3, k4, k5;

  

    // Iterate for number of iterations

    float y = y0;

    for (int i=1; i<=n; i++)

    {

        // Apply Runge Kutta Formulas to find

        // next value of y

        k1 = h*dydx(x0, y);

        k2 = h*dydx(x0 + 0.5*h, y + 0.5*k1);

        k3 = h*dydx(x0 + 0.5*h, y + 0.5*k2);

        k4 = h*dydx(x0 + h, y + k3);

  

        // Update next value of y

        y = y + (1.0/6.0)*(k1 + 2*k2 + 2*k3 + k4);;

  

        // Update next value of x

        x0 = x0 + h;

    }

  

    return y;