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Runge kutta method -by Prof.Prashant Goad(R.C.Patel Institute of Technology,Shirpur)
- 1. Runge-Kutta Method Program forRunge-Kutta 4th Order Method Prepared By – Prof.Prashant Goad Prof.Prashant Goad
- 2. Application • Runge kuttaRK4 is 4th order differential integration method is a complex integration • It uses in physics. • It calculate the estimation in 4 steps Example : - • A rocket that is flying through the Earth’s atmosphere. First of all we have the ODE to calculate the acceleration: • Acceleration = (rocket force + force drag) / mass • We know that acceleration is the derivative of velocity (as mentioned earlier) so using RK4 to calculate this should be relatively straight forward Prof.Prashant Goad
- 3. Formulae for R.KMethod • 𝐾1 = ℎ𝑓(𝑥, 𝑦) • 𝐾2 = ℎ𝑓(𝑥 + 1 2 ℎ , 𝑦 + 1 2 𝐾1) • 𝐾 = 𝐾2 Second order R.K Method Prof.Prashant Goad
- 4. Formulae for R.KMethod • 𝐾1 = ℎ𝑓(𝑥, 𝑦) • 𝐾2 = ℎ𝑓(𝑥 + 1 2 ℎ , 𝑦 + 1 2 𝐾1) • 𝐾3 = ℎ𝑓(𝑥 + ℎ, 𝑦 + 2𝐾2 − 𝐾1) • 𝐾 = 1 6 (𝐾1 + 4𝐾2 + 𝐾3) Third order R.K Method Prof.Prashant Goad
- 5. Formulae for R.KMethod • 𝐾1 = ℎ𝑓(𝑥, 𝑦) • 𝐾2 = ℎ𝑓(𝑥 + 1 2 ℎ , 𝑦 + 1 2 𝐾1) • 𝐾3 = ℎ𝑓(𝑥 + 1 2 ℎ, 𝑦 + 1 2 𝐾2) • 𝐾4 = ℎ𝑓(𝑥 + ℎ, 𝑦 + 𝐾3) • 𝐾 = 1 6 (𝐾1 + 2𝐾2 + 2𝐾3 + 𝐾4) Fourth order R.K Method Prof.Prashant Goad
- 6. Apply the FourthOrder Runge Kutta method to find y(0.2) given that y’ =x+y ,y(0)=1 for initial condition x=y=0 with h=0.2 • Solution : - • Here y’ =x+y • So f(x,y) = x+y • Here Initial condition is • 𝑥0 = 0 𝑎𝑛𝑑 𝑦0 = 1 with h = 0.2 • We have to ask to calculate y(0.2) i.e. calculate the value of Y for x = 0.2 Apply the Fourth Order Runge Kutta method to find y(0.2) given that y’ =x+y ,y0=1 for initial condition x0=0 with h=0.2 Prof.Prashant Goad
- 7. • By Fourthorder Ranga kutta method • 𝐾1 = ℎ𝑓(𝑥, 𝑦) • 𝐾1 = 0.2 (𝑥0 + 𝑦0) • 𝐾1 = 0.2 0 + 1 = 0.2 Solution Prof.Prashant Goad
- 8. 𝐾2 = ℎ𝑓(𝑥+ 1 2 ℎ , 𝑦 + 1 2 𝐾1) 𝐾2 = 0.2𝑓 𝑥0 + 1 2 ℎ , 𝑦0 + 1 2 𝐾1 𝐾2 = 0.2𝑓(0 + 1 2 0.2 , 1 + 1 2 0.2) 𝐾2 = 0.2𝑓(0.1 , 1.1) 𝐾2 = 0.2(0.1 + 1.1) 𝐾2 = 0.24 Solution Prof.Prashant Goad
- 9. • 𝐾3 =ℎ𝑓(𝑥0 + 1 2 ℎ, 𝑦0 + 1 2 𝐾2) • 𝐾3 = 0.2𝑓(0 + 1 2 0.2 , 1 + 1 2 0.24) • 𝐾3 = 0.2𝑓(0.1 , 1 + 0.12) • 𝐾3 = 0.2𝑓(0.1 , 1.12) • 𝐾3 = 0.2 0.1 + 1.12 = 0.2(1.22) • 𝐾3 = 0.244 Solution Prof.Prashant Goad
- 10. • 𝐾4 =ℎ𝑓(𝑥0 + ℎ, 𝑦0 + 𝐾3) • 𝐾4 = 0.2 𝑓(0 + 1 2 0.2 , 1 + 0.244) • 𝐾4 = 0.2 𝑓(0.1 , 1.244) • 𝐾4 = 0.2 (0.1 + 1.244) • 𝐾4 = 0.2 (1.344) • 𝐾4 = 0.2688 Solution Prof.Prashant Goad
- 11. • 𝐾 = 1 6 (𝐾1+ 2𝐾2 + 2𝐾3 + 𝐾4) • 𝐾 = 1 6 (0.2 + 2 × 0.24 + 2 × 0.244 + 0.2888) • 𝐾 = 1 6 (0.2 + 0.48 + +0.488 + 0.2888) • 𝐾 = 0.2428 Solution Prof.Prashant Goad
- 12. • So herewe will find • 𝑥1= 𝑥0 + ℎ • 𝑥1= 0 + 0.2 = 0.2 • Here we have to get x1 = 0.2 • So we will find y(0.2) by using • y(0.2) = yo + K = 1+0.2448 = 1.2448 -------------------------- Answer Solution Prof.Prashant Goad
- 13. Prof.Prashant Goad Here ,wehave to ask to find y for x=0.2
- 14. Formulae By using allformulae's we calculated value of “K” with Initial Term X0 and Yo Prof.Prashant Goad
- 15. • By usingall terms we will calculate X1 & Y1 • Here ,We need to check weather value of x now becomes 0.2 that’s we have to ask . • if “ X is not equal to 0.2” then we need to calculate ones again all values of K. i.e. K1,K2,K3,K4 & K. Formulae Prof.Prashant Goad
- 16. • Then onesagain we will check X2 & Y2 ,by using following formulae • X2 = X1 + h if Here X =0.2 Then Calculate the value of Y by using Y2 = Y1 + h Formulae Prof.Prashant Goad
- 17. # Python programto implement Runge Kutta method : # A sample differential equation "dy / dx = (x + y^2)" def dydx(x, y): return (x + y*y) # Finds value of y for a given x using step size h # and initial value y0 at x0. def rungeKutta(x0, y0, x, h): Prof.Prashant Goad
- 18. # Count numberof iterations using step size or # step height h n = (int)((x - x0)/h) # Iterate for number of iterations y = y0 for i in range(1, n + 1): "Apply Runge Kutta Formulas to find next value of y" # Python program to implement Runge Kutta method : Prof.Prashant Goad
- 19. 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) # Python program to implement Runge Kutta method : Prof.Prashant Goad
- 20. # Update nextvalue of y K = (1.0 / 6.0)*(k1 + 2 * k2 + 2 * k3 + k4) y = y + k # Update next value of x x0 = x0 + h # Python program to implement Runge Kutta method : Prof.Prashant Goad
- 21. return y # Drivercode x0 = 0 y = 1 x = 0.2 h = 0.2 print (“The value of y at x is:”, rungeKutta(x0, y, x, h) ) • Output: • The value of y at x is : 1.2735 # Python program to implement Runge Kutta method : Prof.Prashant Goad
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