Numerical Methods - Oridnary Differential Equations - 3
12 likes10,906 views
The document discusses numerical methods for solving ordinary differential equations using Runge-Kutta methods of orders 2, 3, and 4. It defines the Runge-Kutta method of order 2, which calculates approximations k1 and k2 to find the solution increment k as their average. It also defines the Runge-Kutta method of order 3, which calculates approximations k1, k2, k3 to find the solution increment k as a weighted average. Finally, it defines the Runge-Kutta method of order 4, which calculates approximations k1, k2, k3, k4 to find the solution increment k as a weighted average.
Numerical Methods - Oridnary Differential Equations - 3
- 1. Numerical Methods Ordinary Differential Equations - 3 Dr. N. B. Vyas Department of Mathematics, Atmiya Institute of Technology & Science, Rajkot (Gujarat) - INDIA niravbvyas@gmail.com Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 2. Ordinary Differential Equations Runge’s Method: (Runge-Kutta Method of 3rd Order) Named after German mathematicians CARL RUNGE (1856-1927) and WILHELM KUTTA (1867-1944) Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 3. Ordinary Differential Equations Runge’s Method: (Runge-Kutta Method of 3rd Order) Named after German mathematicians CARL RUNGE (1856-1927) and WILHELM KUTTA (1867-1944) Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 4. Ordinary Differential Equations Runge’s Method: (Runge-Kutta Method of 3rd Order) Named after German mathematicians CARL RUNGE (1856-1927) and WILHELM KUTTA (1867-1944) Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 5. Ordinary Differential Equations Runge’s Method: (Runge-Kutta Method of 3rd Order) Named after German mathematicians CARL RUNGE (1856-1927) and WILHELM KUTTA (1867-1944) Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k = hf (x0 + h, y0 + k1) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 6. Ordinary Differential Equations Runge’s Method: (Runge-Kutta Method of 3rd Order) Named after German mathematicians CARL RUNGE (1856-1927) and WILHELM KUTTA (1867-1944) Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k = hf (x0 + h, y0 + k1) k3 = hf (x0 + h, y0 + k ) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 7. Ordinary Differential Equations Runge’s Method: (Runge-Kutta Method of 3rd Order) Named after German mathematicians CARL RUNGE (1856-1927) and WILHELM KUTTA (1867-1944) Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k = hf (x0 + h, y0 + k1) k3 = hf (x0 + h, y0 + k ) Finally calculate Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 8. Ordinary Differential Equations Runge’s Method: (Runge-Kutta Method of 3rd Order) Named after German mathematicians CARL RUNGE (1856-1927) and WILHELM KUTTA (1867-1944) Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k = hf (x0 + h, y0 + k1) k3 = hf (x0 + h, y0 + k ) Finally calculate k = 1 6 (k1 + 4k2 + k3) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 9. Ordinary Differential Equations Runge’s Method: (Runge-Kutta Method of 3rd Order) Named after German mathematicians CARL RUNGE (1856-1927) and WILHELM KUTTA (1867-1944) Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k = hf (x0 + h, y0 + k1) k3 = hf (x0 + h, y0 + k ) Finally calculate k = 1 6 (k1 + 4k2 + k3) required approximate value of y = y0 + k Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 10. Runge’s Method Ex. Use Runge’s method to approximate y when x = 1.1 given that y = 1.2 when x = 1 and dy dx = 3x + y2 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 11. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 12. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 x0 = 1, y0 = 1.2 and h = 0.1 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 13. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 x0 = 1, y0 = 1.2 and h = 0.1 k1 = hf(x0, y0) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 14. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 x0 = 1, y0 = 1.2 and h = 0.1 k1 = hf(x0, y0) = 0.444 k2 = hf x0 + h 2 , y0 + k1 2 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 15. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 x0 = 1, y0 = 1.2 and h = 0.1 k1 = hf(x0, y0) = 0.444 k2 = hf x0 + h 2 , y0 + k1 2 = 0.51721 k = hf (x0 + h, y0 + k1) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 16. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 x0 = 1, y0 = 1.2 and h = 0.1 k1 = hf(x0, y0) = 0.444 k2 = hf x0 + h 2 , y0 + k1 2 = 0.51721 k = hf (x0 + h, y0 + k1) = 0.60027 k3 = hf (x0 + h, y0 + k ) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 17. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 x0 = 1, y0 = 1.2 and h = 0.1 k1 = hf(x0, y0) = 0.444 k2 = hf x0 + h 2 , y0 + k1 2 = 0.51721 k = hf (x0 + h, y0 + k1) = 0.60027 k3 = hf (x0 + h, y0 + k ) = 0.65411 Hence k = 1 6(k1 + 4k2 + k3) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 18. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 x0 = 1, y0 = 1.2 and h = 0.1 k1 = hf(x0, y0) = 0.444 k2 = hf x0 + h 2 , y0 + k1 2 = 0.51721 k = hf (x0 + h, y0 + k1) = 0.60027 k3 = hf (x0 + h, y0 + k ) = 0.65411 Hence k = 1 6(k1 + 4k2 + k3) = 0.5278 ∴ approximate value of y = y0 + k Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 19. Runge’s Method Sol. We have dy dx = 3x + y2 , ∴ f(x, y) = 3x + y2 x0 = 1, y0 = 1.2 and h = 0.1 k1 = hf(x0, y0) = 0.444 k2 = hf x0 + h 2 , y0 + k1 2 = 0.51721 k = hf (x0 + h, y0 + k1) = 0.60027 k3 = hf (x0 + h, y0 + k ) = 0.65411 Hence k = 1 6(k1 + 4k2 + k3) = 0.5278 ∴ approximate value of y = y0 + k = 1.7278 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 20. Ordinary Differential Equations Runge-Kutta Method of 2nd Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 21. Ordinary Differential Equations Runge-Kutta Method of 2nd Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 22. Ordinary Differential Equations Runge-Kutta Method of 2nd Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf (x0 + h, y0 + k1) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 23. Ordinary Differential Equations Runge-Kutta Method of 2nd Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf (x0 + h, y0 + k1) Find k = 1 2 (k1 + k2) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 24. Ordinary Differential Equations Runge-Kutta Method of 2nd Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf (x0 + h, y0 + k1) Find k = 1 2 (k1 + k2) ∴ y1 = y0 + k Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 25. Ordinary Differential Equations Runge-Kutta Method of 4th Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 26. Ordinary Differential Equations Runge-Kutta Method of 4th Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 27. Ordinary Differential Equations Runge-Kutta Method of 4th Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 28. Ordinary Differential Equations Runge-Kutta Method of 4th Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k3 = hf x0 + h 2 , y0 + k2 2 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 29. Ordinary Differential Equations Runge-Kutta Method of 4th Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k3 = hf x0 + h 2 , y0 + k2 2 k4 = hf(x0 + h, y0 + k3) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 30. Ordinary Differential Equations Runge-Kutta Method of 4th Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k3 = hf x0 + h 2 , y0 + k2 2 k4 = hf(x0 + h, y0 + k3) Find k = 1 6 (k1 + 2k2 + 2k3 + k4) Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 31. Ordinary Differential Equations Runge-Kutta Method of 4th Order: Consider the differential Equation dy dx = f(x, y), y(x0) = y0 Calculate successively k1 = hf(x0, y0) k2 = hf x0 + h 2 , y0 + k1 2 k3 = hf x0 + h 2 , y0 + k2 2 k4 = hf(x0 + h, y0 + k3) Find k = 1 6 (k1 + 2k2 + 2k3 + k4) ∴ y1 = y0 + k and x1 = x0 + h Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 32. Runge-Kutta Method of 2nd Order Ex. Use Runge-kutta second order method to find the approximate value of y(0.2) given that dy dx = x − y2 and y(0) = 1 and h = 0.1 Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 33. Runge-Kutta Method of 2nd Order Ex. Use 4th order Runge-kutta method to solve dy dx = x2 + y2 , y(0) = 1. Find y(0.2) with h = 0.1. Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3
- 34. Runge-Kutta Method of 2nd Order Ex. Determine y(0.1) and y(0.2) correct to four decimal places from dy dx = 2x + y, y(0) = 1. Use fourth order Runge-Kutta method Dr. N. B. Vyas Numerical Methods Ordinary Differential Equations - 3