3. 3E: Runge-Kutta Usuli (Mashqlar) Matematika

Exercise 9
Use the Runge-midpoint method to estimate (y(0.2)) if (y(t)) is the solution to (y' = sqrt) with (y(0) = 0) . Use a step size of (h = 0.1) .
Qaror
With (f(t, y) = sqrt) , the method produces the following values. I’ll list all the intermediate quantites to make it easy to debug. Everything is rounded off to the fourth decimal place. [oshlanishi y_0 & = 0 f_1 & = 1 y_> & = 0.05 f_> & = 1.0259 y_1 & = 0.1026 f_2 & = 1.0548 y_> & = 0.1553 f_> & = 1.0853 y_2 & = 0.2111 ,.end] The estimate this method gives is (0.2111) .Exercise 10
Use the Runge-trapezoid method to calculate (y(1.4)) if (y(t)) is the solution to the differential equation (y'=frac<1 + y^2><1 + t^2>) with (y(1) = 2) . Use a step size of (h = 0.2) .
Qaror
With (f(t,y) = frac<1 + y^2><1 + t^2>) , the method produces the following values. I’ll list all the intermediate quantites to make it easy to debug. Everything is rounded off to the fourth decimal place. [oshlanishi f_1 & = 2.5 widetilde & = 2.5 widetilde & = 2.9713 y_1 & = 2.5471 f_2 & = 3.0687 widetilde & = 3.1608 widetilde & = 3.7131 y_2 & = 3.2253 ,.end]
The estimate this method gives is (y(1.4) approx 3.2253) .
Exercise 11
Use the Runge-Kutta method with to calculate (y(1.5)) if (y(t)) is the solution to (y'+y=frac<1>) satisfying (y(1) = 1) . Use a step size of (h = 0.5) (so just one step).
Qaror
With (f(t, y) = frac<1> - y) , the method produces the following values. I’ll list all the intermediate quantites to make it easy to debug. Everything is rounded off to the sixth decimal place. [oshlanishi f_0 & = 0 widetilde>> & = 1 widetilde>> & = -0.2 y_> & = 0.95 f_> & = -0.15 widetilde & = 0.925 widetilde & = -0.258333 y_ <1>& = 0.920139 ,.end] The estimate this method gives is (y(1.5) approx 0.920139) .Exercise 12
Use the second-order Taylor method to estimate (y(2)) if (y(t)) is the solution to (dot=t+frac<1>) with (y(1) = 2) . Use a step size of (h = 0.5) .
Qaror
The second-order Taylor rule is [y_ = y_n + h f(t_n, y_n) + frac<2>left( partial_t f(t_n, y_n) + f(t_n, y_n) partial_y f(t_n, y_n) ight) ,.] Our function is (f(t, y) = t + frac<1>) , with partial derivatives (partial_t f = 1) and (partial_t y = -frac<1>). The step size is (h = 0.5) , and we start with (t_0 = 1) , (y_0 = 2) . After the first step, we get (y_1 = 2.828) at (t_1 = 1.5) , and then we repeat the process and get our approximation, [y(2) approx y_2 = 3.851 ,.]