Question

IN PYTHON:

6. Stiff Problem Consider the initial value problem y = -500(y - cost) - sint, to= 0, yo = 2. Recall from class that the problem is stiff. Solve it with the explicit RK45 and implicit Radau in the module scipy.integrate up to time T = 1 and the default tolerances. 1. Plot the solution. 2. Print the number of steps of each method. 3. Print the difference between the methods at the final time T.

Answer 1

CCODE IMAGE

ANSWER .- I have provided the solution of the above Question in both image and the text format so it will be easy for you to copy the code as well as check the connect indentation, > I have provided the image of correct output so you can check the expected output easily. > I have used python modules: » Scipy integrate, solue._ ivp; To solve the initial value problem for given CODE » Numpy.cos, numpy. Sin; to Use Sin and cosine functions. -> mat plotiboly ploto ploti To plot the graph of the results

OUTPUT IMAGE

## Importing import functions from scipy. integrate import solve_ivp from numpy import sin, cos import matplotlib.pyplot as plt ## Defining the ODE Equation def f(t,y): return -500*(y-cos(t))-sin(t) ## solve_ivp using default method RK45 rk45_sol - solve_ivp(f, [0, 1], [2]) ## Solve ivp using 'Radau' method radau_sol = solve_ivp(f, [0, 1], [2], method='Radau') ## Plotting both equations using y and t points plt.plot(rk45_sol.t,rk45_sol.y[@], label-'RK45 Method) plt.plot(radau_sol.t, radau_sol.y[@], label-'Radau Method plt.xlabel('Time (t)) plt.ylabel('Y (y)) plt.title('Plot between y and t') plt.legend() print('No. of steps in RK45 Method: ', len(rk45_sol.t)) print('No. of steps in Radau Method: ', len(radau_sol.t)) print('Difference between RK45 and Radau solution at final point: ',rk45_sol.y[@][-1]-radau_sol.y[@][-1])

No. of steps in RK45 Method: 156 No. of steps in Radau Method: 12 Difference between RK45 and Radau solution at final point: 2.846018177649068e-05 Plot between y and t - RK45 Method - Radau Method Yly) 0. 0 02 0.4 0.6 Time (t) 0.8 10

CODE TEXT

CODE TEXT: ##importing important fonctions From Scipy, integrate import solue - iup from :numpy import sin, cos. import mat plot lib. Dy plot as pit. ## Defining the ODE Ervation. . def f(ty); return - 500*C * -cos tt))-S init ## Solve - ivp using default method RK45 YK 45 -Sol - Solve nivelf, [0,1],[23)

## Solver ip using 'Radau' method Vada u – Sol - Solver ivplf, [0,1], [2], method = 'Radau'). ## plotting . both equations using Y and t points pit. plot (VK 45- Sol. VK45-sol. Y [o], label='RK45 Method') pit. plot (vadov, solit, va davi_Sol. Y[0], la bel = Radau Method') pit. x tabel ('Time (+)'). Plt. y label ('y tyl') Plt. title ('plot between Yand t') Plt. legend I print('No. of Steps in RK45 Method: ; lentok45 - Sol.t)) Plint ('No. of Steps in Radau Method : ', len( yadav _solot)) Print(Difference between RK45 and Radaw. Solution at Firal point: iVK 45-501.46] [0:- YadavSol,YCJELJ)