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Use the fourth order Runge-Kutta method to find u(0.2) of the initial value problem u′=−2tu2, u(0) = 1 using h = 0.2. Correct up to three decimal places.
Correct answer is between '0.96,0.963'. Can you explain this answer?
Verified Answer
Use the fourth order Runge-Kutta method to find u(0.2) of the initial ...
Concept:
Runge - Kutta Method of first order:
k1 = h f(x0, y0)
y1 = y0 + k where k = k1 = h f (x0, y0)
y2 = y1 + k where k = h f (x1, y1)
Runge - Kutta Method of Second order:
k1 = h f(x0, y0)
k2 = h f (x0 + h, y0 + k1)
y1 = y0 + k
Runge - Kutta Method of Third order:
k1 = h f (x0, y0)
y1 = y0 + k
Runge - Kutta Method of fourth order:
k1 = h f (x0, y0)
y1 = y0 + k
Calculation:
t0 = 0,u0 = 1
f(u,t) = −2tu2
k1 = hf(t0,u0) = hf(0,1) = 0.2×[−2×0] = 0
= hf(0.1,1) = 0.2×[−2×0.1×12] = −0.04
= hf(0.1,0.98) = 0.2×[−2×0.1×0.982] = −0.038416k4 = hf(t0 + h,u0 + k3) = hf(0.2,0.961584) = 0.2×[−2×0.2×0.9615842] = −0.073972
= −0.0384373u1 = u0 + k = 0.961563
Most Upvoted Answer
Use the fourth order Runge-Kutta method to find u(0.2) of the initial ...
'= 2u, u(0) = 1.
Solution:
We start by defining the necessary variables and functions:
t0 = 0
u0 = 1
h = 0.1
t = 0.2
def f(t, u):
return 2*u
Now we can apply the fourth order Runge-Kutta method:
k1 = h*f(t0, u0)
k2 = h*f(t0 + h/2, u0 + k1/2)
k3 = h*f(t0 + h/2, u0 + k2/2)
k4 = h*f(t0 + h, u0 + k3)
u1 = u0 + 1/6*(k1 + 2*k2 + 2*k3 + k4)
We repeat the process until we reach t = 0.2:
while t0 < />
k1 = h*f(t0, u0)
k2 = h*f(t0 + h/2, u0 + k1/2)
k3 = h*f(t0 + h/2, u0 + k2/2)
k4 = h*f(t0 + h, u0 + k3)
u1 = u0 + 1/6*(k1 + 2*k2 + 2*k3 + k4)
t0 += h
u0 = u1
Finally, we print the result:
print("u(0.2) =", u1)
The output is:
u(0.2) = 1.4918246976412705
Therefore, u(0.2) ≈ 1.4918.
Solution:
We start by defining the necessary variables and functions:
t0 = 0
u0 = 1
h = 0.1
t = 0.2
def f(t, u):
return 2*u
Now we can apply the fourth order Runge-Kutta method:
k1 = h*f(t0, u0)
k2 = h*f(t0 + h/2, u0 + k1/2)
k3 = h*f(t0 + h/2, u0 + k2/2)
k4 = h*f(t0 + h, u0 + k3)
u1 = u0 + 1/6*(k1 + 2*k2 + 2*k3 + k4)
We repeat the process until we reach t = 0.2:
while t0 < />
k1 = h*f(t0, u0)
k2 = h*f(t0 + h/2, u0 + k1/2)
k3 = h*f(t0 + h/2, u0 + k2/2)
k4 = h*f(t0 + h, u0 + k3)
u1 = u0 + 1/6*(k1 + 2*k2 + 2*k3 + k4)
t0 += h
u0 = u1
Finally, we print the result:
print("u(0.2) =", u1)
The output is:
u(0.2) = 1.4918246976412705
Therefore, u(0.2) ≈ 1.4918.
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Use the fourth order Runge-Kutta method to find u(0.2) of the initial value problemu′=−2tu2, u(0) = 1 using h = 0.2. Correct up to three decimal places.Correct answer is between '0.96,0.963'. Can you explain this answer?
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Use the fourth order Runge-Kutta method to find u(0.2) of the initial value problemu′=−2tu2, u(0) = 1 using h = 0.2. Correct up to three decimal places.Correct answer is between '0.96,0.963'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Use the fourth order Runge-Kutta method to find u(0.2) of the initial value problemu′=−2tu2, u(0) = 1 using h = 0.2. Correct up to three decimal places.Correct answer is between '0.96,0.963'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Use the fourth order Runge-Kutta method to find u(0.2) of the initial value problemu′=−2tu2, u(0) = 1 using h = 0.2. Correct up to three decimal places.Correct answer is between '0.96,0.963'. Can you explain this answer?.
Use the fourth order Runge-Kutta method to find u(0.2) of the initial value problemu′=−2tu2, u(0) = 1 using h = 0.2. Correct up to three decimal places.Correct answer is between '0.96,0.963'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Use the fourth order Runge-Kutta method to find u(0.2) of the initial value problemu′=−2tu2, u(0) = 1 using h = 0.2. Correct up to three decimal places.Correct answer is between '0.96,0.963'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Use the fourth order Runge-Kutta method to find u(0.2) of the initial value problemu′=−2tu2, u(0) = 1 using h = 0.2. Correct up to three decimal places.Correct answer is between '0.96,0.963'. Can you explain this answer?.
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