I have a question from my book and it says essentially, consider the IVP x ∙ = − x with x ( 0 ) = 1, what is the exact value of x ( 1 ), then using Eulers method with step size1 , estimate x ( 1 ) call this x ∗ ( 1 ) , then repeat for step sizes of 10 − n for n = 1 , 2 , 3 , 4 then finally plot E = | x ∗ ( 1 ) − x ( 1 ) | as a function of step size and then as l n E vs l n t.Now I am having some issues and ill explain,for the first part I get that x ( 1 ) = e − 1 We have f ( x ) = − x and x 0 = 1so I have that by Euler method x 1 = x 0 + f ( x 0 ) twhich would imply that for t = 1 , x 1 = 0and then for the second part it would imply x 1 = 0.9, then 0.99, then 0.999 and finally 0.9999.But this doesn't seem to make any sense to me. Seeing as none of these are close to e − 1 So I am confused in regard to where I am making mistakes, or where everything is going wrong. For the plotting part, I am also stuck because of this. I am looking for any help and advice.

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I have a question from my book and it says essentially, consider the IVP       x          ∙        =  −  x with   x  (  0  )  =  1, what is the exact value of   x  (  1  ), then using Eulers method with step size1 , estimate   x  (  1  ) call this       x          ∗        (  1  ) , then repeat for step sizes of       10          −      n       for   n  =  1  ,  2  ,  3  ,  4 then finally plot   E  =      |        x          ∗        (  1  )  −  x  (  1  )      |   as a function of step size and then as   l  n  E vs   l  n  t.Now I am having some issues and ill explain,for the first part I get that   x  (  1  )  =      e          −      1      We have   f  (  x  )  =  −  x and       x          0        =  1so I have that by Euler method      x          1        =      x          0        +  f  (      x          0        )  twhich would imply that for   t  =  1  ,      x          1        =  0and then for the second part it would imply       x          1        =  0.9, then 0.99, then 0.999 and finally 0.9999.But this doesn&#039;t seem to make any sense to me. Seeing as none of these are close to       e          −      1      So I am confused in regard to where I am making mistakes, or where everything is going wrong. For the plotting part, I am also stuck because of this. I am looking for any help and advice.

Rayan Ali · Accepted Answer

Let   h  =  1      /    m be the step size, Euler&#039;s methode applied to the discretization   (      t    k        )          0      ≤      k      ≤      n       with       t    k    =  k      /    m yields the values       x    0    ,  …  ,      x    m  . With       x    0    =  1 and      x          k      +      1        =      x    k    −  h      x    k    =  (  1  −  h  )      x    k    ,    k  =  0  ,  …  ,  m  −  1This implies, by an easy induction, that      x          k        =            (      1      −      h      )        k    ,    k  =  0  ,  …  ,  mNow the estimate of   x  (  1  )  =      e          −      1       given by Euler&#039;s corresponding to step size   h  =  1      /    m is       x    ∗    (  1  )  =      x    m    =  (  1  −  h      )    m    =            (      1      −              1        m            )        m  , and the corresponding error is  E  =  E  (  m  )  =      |          e              −        1              −                  (        1        −                  1          m                )            m        |    =      e          −      1        −            (      1      −              1        m            )        m  In particular,  E  (  m  )  =            e              −        1                    2      m        +      O        (          1              m        2              )  Or equivalently, in terms of the step size   h:  E  (  h  )  =            e              −        1              2    h  +      O        (          h      2        )  So the plot   (  log  ⁡  h  ,  log  ⁡  E  ) should be very close to the line   y  =  x  −  log  ⁡  2  −  1.

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