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!

Question

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&#039;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&#039;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!

Bianca Chung · Accepted Answer

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            )

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.

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