Re: [微積] 請問一下離散的微分方程的解法

[Frobenius (▽．(▽×▽φ)=0)]

※ 引述《igawa (井穿腸 (3A))》之銘言：
: 就是我有一組數據
: du/dt = 10 @ t=0
: du/dt = 22 @ t=1
: du/dt = 73 @ t=2
: du/dt = 68 @ t=3
: du/dt = 17 @ t=4
: du/dt = 11 @ t=5
:       .
:       .
:       .
: 想請問版上先進,可以用哪些方法,把u(t)這個函數解出來
: 謝謝~~!

定義 du/dt = v[t]

RSolve[{v[0] == 10, v[1] == 22, v[2] == 73, v[3] == 68, v[4] == 17,
  v[5] == 11}, v[t], t]

RSolve::deqx: Supplied equations are not difference equations of the given
functions. >>

Solve[{a 0^6 + b 0^5 + c 0^4 + d 0^3 + e 0^2 + f 0 + g == 10,
  a 1^6 + b 1^5 + c 1^4 + d 1^3 + e 1^2 + f 1 + g == 22,
  a 2^6 + b 2^5 + c 2^4 + d 2^3 + e 2^2 + f 2 + g == 73,
  a 3^6 + b 3^5 + c 3^4 + d 3^3 + e 3^2 + f 3 + g == 68,
  a 4^6 + b 4^5 + c 4^4 + d 4^3 + e 4^2 + f 4 + g == 17,
  a 5^6 + b 5^5 + c 5^4 + d 5^3 + e 5^2 + f 5 + g == 11}, {a, b, c, d,
   e, f, g}]

Solve::svars: Equations may not give solutions for all "solve" variables. >>

{{a -> -(4213/7200) - f/120, b -> 4117/480 + f/8,
  c -> -(62441/1440) - (17 f)/24, d -> 7927/96 + (15 f)/8,
  e -> -(126731/3600) - (137 f)/60, g -> 10}}

v[t_] = (-(4213/7200) - f/120) t^6 + (4117/480 + f/
      8) t^5 + (-(62441/1440) - (17 f)/24) t^4 + (7927/96 + (15 f)/
      8) t^3 + (-(126731/3600) - (137 f)/60) t^2 + f t + 10;

Table[v[t], {t, 0, 5, 1}] // FullSimplify

{10, 22, 73, 68, 17, 11}

u[t_] = 1/7 (-(4213/7200) - f/120) t^7 + 1/6 (4117/480 + f/8) t^6 +
  1/5 (-(62441/1440) - (17 f)/24) t^5 +
  1/4 (7927/96 + (15 f)/8) t^4 +
  1/3 (-(126731/3600) - (137 f)/60) t^3 + 1/2 f t^2 + 10 t + h

Table[u[t], {t, 0, 5, 1}] // FullSimplify

{h, (7005059 + 51780 f)/604800 + h, 2256301/37800 + (37 f)/630 + h,
 3073097/22400 + (87 f)/1120 + h,
 845302/4725 + (16 f)/315 + h, (5 (917759 + 660 f))/24192 + h}

          4213     f    7    4117   f   6      62441    17 f   5
       (-(----) - ---) t    (---- + -) t    (-(-----) - ----) t
          7200    120        480    8          1440      24
u[t] = ------------------ + ------------- + --------------------
               7                  6                  5

        7927   15 f   4      126731    137 f   3
       (---- + ----) t    (-(------) - -----) t       2
         96     8             3600      60         f t
     + ---------------- + ---------------------- + ---- + 10 t + h
              4                     3               2

                  4213     f    6    4117   f   5      62441    17 f   4
u'[t] = v[t] = (-(----) - ---) t  + (---- + -) t  + (-(-----) - ----) t
                  7200    120        480    8          1440      24

                7927   15 f   3      126731    137 f   2
             + (---- + ----) t  + (-(------) - -----) t  + f t + 10
                 96     8             3600      60

                      4213     f    5      4117   f   4
u''[t] = v'[t] = 6 (-(----) - ---) t  + 5 (---- + -) t
                      7200    120          480    8

                      62441    17 f   3      7927   15 f   2
               + 4 (-(-----) - ----) t  + 3 (---- + ----) t
                      1440      24            96     8

                      126731    137 f
               + 2 (-(------) - -----) t + f
                       3600      60

u[0] = h                     ; u'[0] = v[0] = 10 ; u''[0] = v'[0] = f

       7005059 + 51780 f                                            1081   f
u[1] = ----------------- + h ; u'[1] = v[1] = 22 ; u''[1] = v'[1] = ---- - -
            604800                                                   25    5

       2256301   37 f                                               7261    f
u[2] = ------- + ---- + h    ; u'[2] = v[2] = 73 ; u''[2] = v'[2] = ---- + --
        37800    630                                                200    10

       3073097   87 f                                                26543    f
u[3] = ------- + ---- + h   ; u'[3] = v[3] = 68 ; u''[3] = v'[3] = - ----- - --
        22400    1120                                                 600    10

       845302   16 f                                                 2638   f
u[4] = ------ + ---- + h    ; u'[4] = v[4] = 17 ; u''[4] = v'[4] = - ---- + -
        4725    315                                                   75    5

       4588795 + 3300 f                                              119
u[5] = ---------------- + h ; u'[5] = v[5] = 11 ; u''[5] = v'[5] = - --- - f
           24192                                                     15

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◆ From: 118.161.243.220
→ Frobenius   :By mathematica 7.0
→ Frobenius   :u'[6] = -(2233/10)- 6f ; u'[7]= -(21901/10)- 42 f
→ Frobenius   :建議令 u''[0] = v'[0] = f = -(3133/60) - z/6
→ Frobenius   : => u'[t] 屬於整數 => u'[6] = 90 + z ;
→ Frobenius   : u'[7]= 3 + 7 z ; u'[8] = -1140 + 28 z ;
→ Frobenius   : u'[9] = -5324 + 84 z ; ... 依此類推
※ 編輯: Frobenius       來自: 118.161.243.220      (04/03 16:20)
→ Frobenius   :f 控制 u'[t] 是否為整數 ; h 控制 u[t] 的初始條件
推 doom8199    :polynomial curve fitting 要注意用高維度來 fit
→ doom8199    :data 會有 over-fitting 的問題
→ doom8199    :一個簡單的方式可以用 s-fold cross validation
→ Frobenius   :請教一下什麼是高維度來fit?over-fitting?s-fold? ^^