Solution: . Let
The Euler's method is yn + 1 = yn + h⋅f(xn, yn) where xn + 1 = xn + h. Given h = 0.1, x0 = 0, y0 = 1
Using this method we get
Hence, y20 > y0. Therefore, option (c) is correct.
11 Jiyanet When we solve the initial value problem y' = -, V(0) = 1, using...
Question 1: Given the initial-value problem 12-21 0 <1 <1, y(0) = 1, 12+10 with exact solution v(t) = 2t +1 t2 + 1 a. Use Euler's method with h = 0.1 to approximate the solution of y b. Calculate the error bound and compare the actual error at each step to the error bound. c. Use the answers generated in part (a) and linear interpolation to approximate the following values of y, and compare them to the actual value...
3. Consider the initial value problem y(t) = y, y(0) = 1. a. Write down (i.e., write the formula which describes one step, Yn+1 = yn + ...) the second order Taylor method with step size h for this initial value problem. b. Write down the time stepping formula Yn+1 = Yn +... for the modified Euler method 9n+1 := yn + hf(en +3.29 + s(tn, yn)), for this initial value problem. c. What is the difference between the two...
I. Use Euler's method with step size h = 0.1 to numerically solve the initial value problem y,--2ty+y2, y(0) 1 on the interval 0 < t 2. Compare your approximations with the exact solution. I. Use Euler's method with step size h = 0.1 to numerically solve the initial value problem y,--2ty+y2, y(0) 1 on the interval 0
Problem 3. Given the initial conditions, y(0) from t- 0 to 4: and y (0 0, solve the following initial-value problem d2 dt Obtain your solution with (a) Euler's method and (b) the fourth-order RK method. In both cases, use a step size of 0.1. Plot both solutions on the same graph along with the exact solution y- cos(3t). Note: show the hand calculations for t-0.1 and 0.2, for remaining work use the MATLAB files provided in the lectures Problem...
3. Consider the initial value problem y'(t) = y2, y(0) = 1. a. Write down (i.e., write the formula which describes one step, Yn+1 = yn + ...) the second order Taylor method with step size h for this initial value problem. b. Write down the time stepping formula Yn+1 = Yn +... for the modified Euler method h Yn+1 := yn + hf(tn + h 2:9 » Yn + 5 f (tnYn)), for this initial value problem. c. What...
Problem Thre: 125 points) Consider the following initial value problem: dy-2y+ t The y(0) -1 ea dt ical solution of the differential equation is: y(O)(2-2t+3e-2+1)y fr exoc the differential equation numerically over the interval 0 s i s 2.0 and a step size h At 0.5.A Apply the following Runge-Kutta methods for each of the step. (show your calculations) i. [0.0 0.5: Euler method ii. [0.5 1.0]: Heun method. ii. [1.0 1.5): Midpoint method. iv. [1.5 2.0): 4h RK method...
1. Solve the following initial value problem over the interval from x- 0 to 0.5 with a step size h-0.5 where y(0)-1 dy dx Using Heun method with 2 corrector steps. Calculate g for the corrector steps. Using midpoint method a. b. 1. Solve the following initial value problem over the interval from x- 0 to 0.5 with a step size h-0.5 where y(0)-1 dy dx Using Heun method with 2 corrector steps. Calculate g for the corrector steps. Using...
Consider the following initial value problem y = -5y + 5++ 2t, Ostsi, y(0) = 1/3, with h = 0.1. The exact solution of this problem is y(t) = {2 + e-5t. (2) If you use 2-step Adams-Moulton method (AM2) to solve this problem, what is the specific formula expressing Yi+1 in terms of Yi, Yi-1, të+1, ti, and ti-1 for solving this problem? (3) Use 2-step Adams-Moulton method (AM2) to solve this problem and plot the results to compare...
The Program for the code should be matlab 5. [25 pointsl Given the initial value problem with the initial conditions y(0) 2 and y'(0)10, (a) Solve analytically to obtain the exact solution y(x) (b) Solve numerically using the forward Euler, backward Euler, and fourth-order Runge Kutta methods. Please implement all three methods yourselves do not use any built- in integrators (i.e., ode45)). Integrate over 0 3 r < 4, and compare the methods with the exact solution. (For example, using...
9) Using Euler method, solve this with following initial conditions that t = 0 when y = 1, for the range t = 0 to t = 1 with intervals of 0.25 dr + 2x2 +1=0.3 dt 1o) Using second order Taylor Series method, solve with following initial conditions to-0, xo-1 and h-0.24 11) x(1)-2 h-0.02 Solve the following system to find x(1.06) using 2nd, and 3rd and 4th order Runge-Kutta (RK2, RK3 and RK4)method +2x 2 +1-0.3 de sx)-cox(x/2)...