I am attempting to use Euler's method to express the following system of ODE's... d y i d x = c i ( y i − 1 3 − y i 3 ) + f i ( for i = 1 , … , n )using the discretised equations, which are presented below in matrix notation: y ( t k + 1 ) = C 1 y ( t k ) + C 2 y 3 ( t k ) + fNote f i = 1 when i = 1, but otherwise equals 0 (this considers the initial condition). Whilst I am familiar with Euler's method, I am unsure of the steps to get to the first equation to the second. Can anyone assist me with this? Understanding this is necessary for a project I'm working on, so I would really appreciate any help.

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I am attempting to use Euler&#039;s method to express the following system of ODE&#039;s...            d              y        i                    d      x        =      c    i    (      y          i      −      1        3    −      y          i        3    )  +      f    i      (  for   i  =  1  ,  …  ,  n  )using the discretised equations, which are presented below in matrix notation:  y  (      t          k      +      1        )  =      C    1    y  (      t          k        )  +      C    2        y    3    (      t          k        )  +  fNote       f    i    =  1 when   i  =  1, but otherwise equals 0 (this considers the initial condition). Whilst I am familiar with Euler&#039;s method, I am unsure of the steps to get to the first equation to the second. Can anyone assist me with this? Understanding this is necessary for a project I&#039;m working on, so I would really appreciate any help.

trajeronls · Accepted Answer

Assuming       y    0    =  0 for   i  =  0, and, say, we have two equations:            d              y        1                    d      t        =  −      c    1        y    1    3    +      f    1                d              y        2                    d      t        =      c    2    (      y    1    3    −      y    2    3    )  +      f    2  Write it in vector form:                              y          →                    ˙        =            F      →        (  t  ,            y      →        )So,                              y          →                    ˙        =            F      →        (  t  ,            y      →        )  =      {                            −                      c            1                                y            1            3                    +                      f            1                    (          t          )                                                  c            2                    (                      y            1            3                    −                      y            2            3                    )          +                      f            2                    (          t          )                      }  Given the initial conditions       y    1    (  0  )  =      y          10        ,      y    2    (  0  )  =      y          20      , and a time step   h you can use the Euler Method as:            y      →        (  k  +  1  )  =            y      →        (  k  )  +  h            F      →        (  t  (  k  )  ,            y      →        (  k  )  )You don&#039;t need to use matrices on this system of equations.Edit:If you have an equation that is linear in             y      →       and                     y        →              3  , you can define two vectors                     c        →              1   and                     c        →              2  , so your function is of the form:                              y          →                    ˙        =                    c        →              1    ⋅            y      →        +                    c        →              2    ⋅                    y        →              3  And the time integration scheme will be (admitting constant                     c        →              1   and                     c        →              2  ):            y      →        (  k  +  1  )  =            y      →        (  k  )  +  h      (                            c          →                    1        ⋅                  y        →              (    k    )    +                            c          →                    2        ⋅                            y          →                    3        (    k    )    )  Observe that the ⋅ operator is an element-wise operator. If you don&#039;t have implemented en element-wise operator, you can also use matrix multiplication and define             c      →        =      d    i    a    g    (            c      →        ), so       c   will be a diagonal matrix instead.

George Bray · Accepted Answer

Since the   ith equation only depends on the   i  −  1th equation which is already solved, the equation can be solved sequentially without needing a matrix formulation. Also, I&#039;m not clear as to what you mean by       f    i    =      δ          i      1       as &quot;considering the initial condition&quot;. The initial conditions       y    i    =      y    i    (  0  ) and       f    i   are two separate unrelated things.

I am attempting to use Euler's method to express the following system of ODE's...

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