Trying to understand the proof that the numerical method of solving differential equation x i + 1 = x i + τ i F ( t i + τ i 2 , x i + 1 2 ) x i + 1 2 = x i + τ 2 F ( t i , x i )which seems to be the Cauchy-Euler's method.I am stuck with the Taylor approximation of x ( t + τ ) = x ( t ) + τ x ′ ( t ) + 1 2 x ″ ( t ) + O ( τ 3 )Here it is OK for me.I also understand, why x ′ ( t ) = F ( t , x ( t ) ).But why x ″ ( t ) = d d t F ( t , x ( t ) ) = ∂ t F ( t , x ( t ) ) + D x F ( t , x ( t ) ) F ( t , x ( t ) )???I am a little frustrated about this. Normally I am more into computer science and some mathematical concepts can be missing, so sorry for possible trivial or illposed question, thanks for patience :)

Question

Trying to understand the proof that the numerical method of solving differential equation      x          i      +      1        =      x    i    +      τ    i    F  (      t    i    +            τ      i        2    ,      x          i      +              1        2              )      x          i      +              1        2              =      x    i    +      τ    2    F  (      t    i    ,      x    i    )which seems to be the Cauchy-Euler&#039;s method.I am stuck with the Taylor approximation of  x  (  t  +  τ  )  =  x  (  t  )  +  τ      x    ′    (  t  )  +      1    2        x    ″    (  t  )  +  O  (      τ    3    )Here it is OK for me.I also understand, why       x    ′    (  t  )  =  F  (  t  ,  x  (  t  )  ).But why      x    ″    (  t  )  =      d          d      t        F  (  t  ,  x  (  t  )  )  =      ∂    t    F  (  t  ,  x  (  t  )  )  +      D    x    F  (  t  ,  x  (  t  )  )  F  (  t  ,  x  (  t  )  )???I am a little frustrated about this. Normally I am more into computer science and some mathematical concepts can be missing, so sorry for possible trivial or illposed question, thanks for patience :)

Terahertztl · Accepted Answer

This is really a matter of applying the multivariable chain rule, and being clear on what your notation means. Hopefully you know that       ∂    t    F means the function you get when you differentiate   F with respect to its first argument, and   (      ∂    t    F  )  (  t  ,  x  (  t  )  ) means the value of this function when you evaluate it at the point   (  t  ,  x  (  t  )  ), and likewise for the other symbols.So, if you directly apply the chain rule, you get the following:      x    ″    (  t  )  =  (      ∂    t    F  )  (  t  ,  x  (  t  )  )  +  (      D    x    F  )  (  t  ,  x  (  t  )  )  ⋅      x    ′    (  t  )But notice that we also have       x    ′    (  t  )  =  F  (  t  ,  x  (  t  )  ). So, substituting this, we get      x    ″    (  t  )  =  (      ∂    t    F  )  (  t  ,  x  (  t  )  )  +  (      D    x    F  )  (  t  ,  x  (  t  )  )  ⋅  F  (  t  ,  x  (  t  )  )

Cindy Noble · Accepted Answer

Consider   F as a function of   t and   x: then we have                              x          ″                (        t        )                            =                  lim                      h            →            0                                                              x              ′                        (            t            +            h            )            −                          x              ′                        (            t            )                    h                =                  lim                      h            →            0                                                F            (            t            +            h            ,            x            (            t            +            h            )            )            −            F            (            t            ,            x            (            t            )            )                    h                                                  =                  lim                      h            →            0                                                F            (            t            +            h            ,            x            (            t            +            h            )            )            −            F            (            t            ,            x            (            t            +            h            )            )            +            F            (            t            ,            x            (            t            +            h            )            )            −            F            (            t            ,            x            (            t            )            )                    h                                                  =                  lim                      h            →            0                                                F            (            t            +            h            ,            x            (            t            +            h            )            )            −            F            (            t            ,            x            (            t            +            h            )            )                    h                +                  lim                      h            →            0                                                F            (            t            ,            x            (            t            +            h            )            )            −            F            (            t            ,            x            (            t            )            )                    h                                                  ≜                  ∂          t                F        (        t        ,        x        (        t        )        )        +                  D          x                F        (        t        ,        x        (        t        )        )        ⋅                  x          ′                (        t        )        =                  ∂          t                F        (        t        ,        x        (        t        )        )        +                  D          x                F        (        t        ,        x        (        t        )        )        ⋅        F        (        t        ,        x        (        t        )        )        .            The last term in the last line of the above equality is obtained by using the chain rule.

Trying to understand the proof that the numerical method of solving differential equation x_(i+1)=x_i+tau_iF(t_i+tau_i/2,x_(i+1/2))

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