Solve the stiff initial value problem y = -50y + 50 sin(t) + cos(t), 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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PART 3
![Solve the stiff initial value problem
y = -50y + 50 sin(t) + cos(t), 0<t< 2, y(0) = 1, with h=0.05.
The exact solution of this problem is y(t) = sin(t) + e
-50t
(1) Using Euler's method to solve this problem and plot the results to compare with the exact
solution.
(2) Using the classical 4th order Runge-Kutta method (RK4) to solve this problem and plot
the results to compare with the exact solution.
(3) By using the Trapezoidal method, what is the specific formula expressing y;+1 in terms
of yi, ti, and t;+1 for solving this problem?
(4) Using the Trapezoidal method to solve this problem and plot the results to compare with
the exact solution.
(5) For each method (Euler's method, RK4, and Trapezoidal method), based on the interval
of absolute stability, find the restrictions on step size h to obtain qualitative agreement
with the exact solution. Do your observed results in (1),(2) and (4) coincide with the
stability restrictions on step size? [Tip: interval of absolute stability of RK4 is (-2.78, 0);
interval of absolute stability of Trapezoidal method has been derived in Problem 1 (2).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f92a2-1c4d-41a3-af19-0cdf00d27604%2F4257dbc8-6618-45e1-92bd-2cabb2013746%2Fq9nc0ae_processed.png&w=3840&q=75)
Transcribed Image Text:Solve the stiff initial value problem
y = -50y + 50 sin(t) + cos(t), 0<t< 2, y(0) = 1, with h=0.05.
The exact solution of this problem is y(t) = sin(t) + e
-50t
(1) Using Euler's method to solve this problem and plot the results to compare with the exact
solution.
(2) Using the classical 4th order Runge-Kutta method (RK4) to solve this problem and plot
the results to compare with the exact solution.
(3) By using the Trapezoidal method, what is the specific formula expressing y;+1 in terms
of yi, ti, and t;+1 for solving this problem?
(4) Using the Trapezoidal method to solve this problem and plot the results to compare with
the exact solution.
(5) For each method (Euler's method, RK4, and Trapezoidal method), based on the interval
of absolute stability, find the restrictions on step size h to obtain qualitative agreement
with the exact solution. Do your observed results in (1),(2) and (4) coincide with the
stability restrictions on step size? [Tip: interval of absolute stability of RK4 is (-2.78, 0);
interval of absolute stability of Trapezoidal method has been derived in Problem 1 (2).]
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