I have the following initial value problem on t in [0,T]: y′=y^2(1−elipson y) I wish to use the Trapezoidal method to solve this I.V.P., and I cannot use simply use the pblackictor-corrector method, i.e. improved Euler's method.

Ciara Rose 2022-07-22 Answered
I have the following initial value problem on t [ 0 , T ]:
y = y 2 ( 1 ϵ y )
I wish to use the Trapezoidal method to solve this I.V.P., and I cannot use simply use the pblackictor-corrector method, i.e. improved Euler's method. The trapezoidal method is defined by:
y n + 1 = y n + h 2 ( f n + 1 + f n )
Plugging in the value from the O.D.E., we have after some simplification:
y n + 1 = y n + h 2 ( y n + 1 2 + y n 2 ϵ y n + 1 3 ϵ y n 3 )
This is an implicit method, so I must use a root finding method at each iteration to find the y n + 1 on the right side of the equation. I have read online that Newton's method is usually used for such purposes, but I cannot figure out how to implement it.
y n + 1 = y n f ( y n ) f ( y n )
For example, what values would f ( y n ) and f ( y n ) take in this context? Thank you in advance!
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Answers (1)

Bianca Chung
Answered 2022-07-23 Author has 16 answers
Your whole equation is f. More precisely, y n + 1 is the solution of
0 = f ( y ) = y y n h 2 ( y 2 + y n 2 ϵ ( y 3 + y n 3 ) )
closest to y n .

The derivative is then
f ( y ) = 1 h 2 ( 2 y 3 ϵ y 2 )
so that the Newton iteration is
y + = y f ( y ) f ( y ) = y n h 2 ( y 2 y n 2 ϵ ( 2 y 3 y n 3 ) ) 1 h 2 ( 2 y 3 ϵ y 2 )
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