变分迭代方法解半无限空间常微分方程

变分迭代方法（VIM）是中国数学家何吉欢发展的一种求非线性方程近似解的方法。具体方法见International Journal of Non-Linear Mechanics 34 (1999) 699—708。

本文以布拉修斯方程（Blasius equation）为例，介绍变分迭代方法解半无限空间常微分方程。

变分迭代方法简要介绍

非线性方程：

\begin{equation} L[u(t)]+N[u(t)]=g(t) \label{nonlinearequations} \end{equation}

其中$L$和$N$分别为线性和非线性算符。

校正泛函：

\begin{equation} u_{n+1}(t)=u_n(t)+\int_{t_0}^t \lambda(s) \{L[u_n(s)]+N[\tilde u_n(s)]-g(s)\}\mathrm ds \label{correctionfunctional} \end{equation}

其中，$\tilde u$ 为限制变分（restricted variation），即 $\delta \tilde u = 0$。

对上式变分为0，得出$\lambda$，然后可得迭代公式：

\begin{equation} u_{n+1}(t)=u_n(t)+\int_{t_0}^t \lambda(s) \{L[u_n(s)]+N[u_n(s)]-g(s)\}\mathrm ds \label{iterationu} \end{equation}

系统的精确解为：

\begin{equation} u(t)=\lim_{n\to \infty }u_n(t) \label{uexact} \end{equation}

解布拉修斯方程（Blasius equation）

参考文献：Applied Mathematics and Computation 188 (2007) 485–491

Blasius equation 形式如下：

\begin{equation} \begin{split} & u'''(x)+\frac{1}{2}u(x)u''(x)=0 \\ & u(0)=0,\quad u'(0)=1,\quad u'(\infty)=0 \end{split} \label{Blasius} \end{equation}

根据 $x=0$ 处边界条件，初值 $u_0(x)$ 写为麦克劳林级数形式

\begin{equation} u_0(x)=1+\frac{A}{2}x^2 \label{u0} \end{equation}

其中 $A=u''(0)$ 是待定常数，根据另一边界条件确定。

下面用数学软件 Mathematica 进行变分迭代方法求解。

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<code>In[64]:= Clear[&quot;Global`*&quot;]
In[65]:= u0[x_] := x + A*x^2/2
In[66]:= u0[x]
Out[66]= x + (A x^2)/2
In[67]:= u1[x_] := u0[x] - 1/2*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(x\)]\(
\*SuperscriptBox[\((s - x)\), \(2\)]*\((D[u0[s], {s, 3}] + 
\*FractionBox[\(1\), \(2\)]*u0[s]*
        D[u0[s], {s, 2}])\) \[DifferentialD]s\)\)
In[68]:= u1[x]
Out[68]= x + (A x^2)/2 + 1/2 (-((A x^4)/24) - (A^2 x^5)/120)
In[71]:= u2[x_] := u1[x] - 1/2*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(x\)]\(
\*SuperscriptBox[\((s - x)\), \(2\)]*\((D[u1[s], {s, 3}] + 
\*FractionBox[\(1\), \(2\)]*u1[s]*
        D[u1[s], {s, 2}])\) \[DifferentialD]s\)\)
In[72]:= u2[x]
Out[72]= x + (A x^2)/2 - (A x^4)/24 - (A^2 x^5)/120
In[82]:= Remove[s]
In[83]:= ?s
In[84]:= u2[x_] := u1[x] - 1/2*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(x\)]\(
\*SuperscriptBox[\((s - x)\), \(2\)]*\((D[u1[s], {s, 3}] + 
\*FractionBox[\(1\), \(2\)]*u1[s]*
        D[u1[s], {s, 2}])\) \[DifferentialD]s\)\)
In[85]:= u2[x]
Out[85]= x + (A x^2)/2 + 1/2 (-((A x^4)/24) - (A^2 x^5)/120) + 
 1/2 ((A x^6)/480 + (11 A^2 x^7)/10080 + (11 A^3 x^8)/80640 - (
    A^2 x^9)/96768 - (A^3 x^10)/259200 - (A^4 x^11)/2851200)
In[86]:= Remove[s]
In[87]:= u3[x_] := u2[x] - 1/2*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(x\)]\(
\*SuperscriptBox[\((s - x)\), \(2\)]*\((D[u2[s], {s, 3}] + 
\*FractionBox[\(1\), \(2\)]*u2[s]*
        D[u2[s], {s, 2}])\) \[DifferentialD]s\)\)
In[88]:= u3[x]
Out[88]= x + (A x^2)/2 + 1/2 (-((A x^4)/24) - (A^2 x^5)/120) + 
 1/2 ((A x^6)/480 + (11 A^2 x^7)/10080 + (11 A^3 x^8)/80640 - (
    A^2 x^9)/96768 - (A^3 x^10)/259200 - (A^4 x^11)/2851200) + 
 1/2 (-((A x^8)/10752) - (19 A^2 x^9)/241920 - (319 A^3 x^10)/
    14515200 + (23 A^2 x^11)/17740800 - (29 A^4 x^11)/14515200 + (
    1157 A^3 x^12)/1277337600 - (A^2 x^13)/52715520 + (
    10033 A^4 x^13)/49816166400 - (967 A^3 x^14)/42268262400 + (
    10033 A^5 x^14)/697426329600 - (1147 A^4 x^15)/126804787200 + (
    17 A^3 x^16)/104044953600 - (5449 A^5 x^16)/3678732288000 + (
    1829 A^4 x^17)/13265731584000 - (5449 A^6 x^17)/62538448896000 + (
    14057 A^5 x^18)/328326856704000 - (A^4 x^19)/3024586801152 + (
    83 A^6 x^19)/14296891392000 - (A^5 x^20)/4236115968000 + (
    83 A^7 x^20)/285937827840000 - (197 A^6 x^21)/3145316106240000 - (
    A^7 x^22)/136572936192000 - (A^8 x^23)/3141177532416000)
</code>

得到第三次迭代的结果，将之作为方程的近似解。

下面确定待定常数 $A$。直接接 $x=\infty$ 代入显然是不行的。我们将 $u'_3(x)$ 展开为对角 Pade ´ approximants， $u'_3(\infty)$ 对角 Pade ´ approximants 分子上最高幂次系数为0。具体过程如下：

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<code>In[89]:= PadeApproximant[D[u3[x], x], {x, 0, 2}]
Out[89]= (1 + (3 A x)/4 + 1/12 (1 - 3 A^2) x^2)/(1 - (A x)/4 + x^2/12)
In[90]:= Solve[1/12 (1 - 3 A^2) == 0, A]
Out[90]= {{A -&gt; -(1/Sqrt[3])}, {A -&gt; 1/Sqrt[3]}}
In[91]:= PadeApproximant[D[u3[x], x], {x, 0, 3}]
Out[91]= (1 + (A (-7 + 30 A^2) x)/(2 (-2 + 15 A^2)) - (3 x^2)/(
 20 (-2 + 15 A^2)) + (3 (-4 A + 15 A^3) x^3)/(
 80 (-2 + 15 A^2)))/(1 - (3 A x)/(2 (-2 + 15 A^2)) + (
 3 (-1 + 10 A^2) x^2)/(20 (-2 + 15 A^2)) + ((-8 A + 15 A^3) x^3)/(
 48 (-2 + 15 A^2)))
In[92]:= Solve[3 (-4 A + 15 A^3) == 0, A]
Out[92]= {{A -&gt; 0}, {A -&gt; -(2/Sqrt[15])}, {A -&gt; 2/Sqrt[15]}}
In[93]:= N[{{A -&gt; 0}, {A -&gt; -(2/Sqrt[15])}, {A -&gt; 2/Sqrt[15]}}]
Out[93]= {{A -&gt; 0.}, {A -&gt; -0.516398}, {A -&gt; 0.516398}}
In[94]:= PadeApproximant[D[u3[x], x], {x, 0, 4}]
Out[94]= (1 + (A (34 - 225 A^2 + 900 A^4) x)/(
   4 (-26 + 60 A^2 + 225 A^4)) - (3 (104 - 873 A^2 + 1840 A^4) x^2)/(
   56 (-26 + 60 A^2 + 225 A^4)) + (3 A (-23 - 66 A^2 + 180 A^4) x^3)/(
   112 (-26 + 60 A^2 + 225 A^4)) + ((-676 + 4080 A^2 - 6615 A^4 + 
      2700 A^6) x^4)/(2240 (-26 + 60 A^2 + 225 A^4)))/(1 - (
   3 A (-46 + 155 A^2) x)/(4 (-26 + 60 A^2 + 225 A^4)) + (
   3 (-104 + 229 A^2 + 330 A^4) x^2)/(56 (-26 + 60 A^2 + 225 A^4)) + (
   A (937 - 3036 A^2 + 1980 A^4) x^3)/(
   336 (-26 + 60 A^2 + 225 A^4)) + ((-676 + 3060 A^2 - 
      5275 A^4) x^4)/(2240 (-26 + 60 A^2 + 225 A^4)))
In[95]:= Solve[(-676 + 4080 A^2 - 6615 A^4 + 2700 A^6) /(
  2240 (-26 + 60 A^2 + 225 A^4)) == 0, A]
Out[95]= {{A -&gt; -(1/6) (-1)^(
    3/4) \[Sqrt](1/
       10 (294 I - (5289 3^(1/6))/(102681 + 8 I Sqrt[92120595])^(
         1/3) - (1763 I 3^(2/3))/(102681 + 8 I Sqrt[92120595])^(
         1/3) + 3^(5/6) (102681 + 8 I Sqrt[92120595])^(1/3) - 
         I (3 (102681 + 8 I Sqrt[92120595]))^(1/3)))}, {A -&gt; 
   1/6 (-1)^(
    3/4) \[Sqrt](1/
       10 (294 I - (5289 3^(1/6))/(102681 + 8 I Sqrt[92120595])^(
         1/3) - (1763 I 3^(2/3))/(102681 + 8 I Sqrt[92120595])^(
         1/3) + 3^(5/6) (102681 + 8 I Sqrt[92120595])^(1/3) - 
         I (3 (102681 + 8 I Sqrt[92120595]))^(1/3)))}, {A -&gt; -(1/(
    6 Sqrt[10]))(-1)^(
     1/4) \[Sqrt](-294 I - (
        5289 3^(1/6))/(102681 + 8 I Sqrt[92120595])^(1/3) + (
        1763 I 3^(2/3))/(102681 + 8 I Sqrt[92120595])^(1/3) + 
        3^(5/6) (102681 + 8 I Sqrt[92120595])^(1/3) + 
        I (3 (102681 + 8 I Sqrt[92120595]))^(1/3))}, {A -&gt; (1/(
   6 Sqrt[10]))(-1)^(
    1/4) \[Sqrt](-294 I - (
       5289 3^(1/6))/(102681 + 8 I Sqrt[92120595])^(1/3) + (
       1763 I 3^(2/3))/(102681 + 8 I Sqrt[92120595])^(1/3) + 
       3^(5/6) (102681 + 8 I Sqrt[92120595])^(1/3) + 
       I (3 (102681 + 8 I Sqrt[92120595]))^(1/3))}, {A -&gt; -(1/
     6) \[Sqrt](1/
       5 (147 + (1763 3^(2/3))/(102681 + 8 I Sqrt[92120595])^(
         1/3) + (3 (102681 + 8 I Sqrt[92120595]))^(1/3)))}, {A -&gt; 
   1/6 \[Sqrt](1/
       5 (147 + (1763 3^(2/3))/(102681 + 8 I Sqrt[92120595])^(
         1/3) + (3 (102681 + 8 I Sqrt[92120595]))^(1/3)))}}
In[96]:= N[%95]
Out[96]= {{A -&gt; 
   0.755308 + 5.55112*10^-17 I}, {A -&gt; -0.755308 - 
    5.55112*10^-17 I}, {A -&gt; -0.522703 + 1.11022*10^-16 I}, {A -&gt; 
   0.522703 - 1.11022*10^-16 I}, {A -&gt; -1.2674 + 0. I}, {A -&gt; 
   1.2674 + 0. I}}
In[97]:= PadeApproximant[D[u3[x], x], {x, 0, 5}]
Out[97]= (1 + (3 A (10052 - 10695 A^2 - 91240 A^4 + 83700 A^6) x)/(
   2 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) + ((29160 + 
      1854 A^2 - 2230045 A^4 + 3990270 A^6 - 167400 A^8) x^2)/(
   12 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) - (
   A (-245484 + 458451 A^2 + 3114160 A^4 + 386880 A^6) x^3)/(
   48 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) + ((64800 + 
      650156 A^2 - 1796719 A^4 + 1925640 A^6 + 200880 A^8) x^4)/(
   448 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) + ((568608 A - 
      632890 A^3 - 6922655 A^5 + 3856350 A^7 - 167400 A^9) x^5)/(
   2240 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)))/(1 - (
   A (-13308 + 135625 A^2 - 663030 A^4 + 27900 A^6) x)/(
   2 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) + ((29160 - 
      77994 A^2 - 1416295 A^4 + 12090 A^6) x^2)/(
   12 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) + (
   5 A (32508 + 12121 A^2 + 135504 A^4 + 24552 A^6) x^3)/(
   48 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) + ((194400 - 
      1619532 A^2 - 13232537 A^4 + 10821540 A^6 - 491040 A^8) x^4)/(
   1344 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) - (
   A (-373032 - 282105 A^2 + 2254510 A^4 + 304575 A^6) x^5)/(
   1440 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)))
In[98]:= Solve[(
  568608 A - 632890 A^3 - 6922655 A^5 + 3856350 A^7 - 167400 A^9)/(
  2240 (8424 + 51770 A^2 - 468375 A^4 + 139500 A^6)) == 0, A]
Out[98]= {{A -&gt; 
   0}, {A -&gt; -\[Sqrt](25709/4464 - (1/
       4464)(\[Sqrt](1/
           5 (2618036029 + (76671493721000 7^(
                2/3))/(363722122031829547 + 
                18 I Sqrt[230095974116695490903297934806259])^(1/3) + 
             248 (7 (363722122031829547 + 
                  18 I Sqrt[230095974116695490903297934806259]))^(
              1/3)))) - 
       1/2 \[Sqrt](2618036029/
           12454560 - (61831849775 7^(
              
              2/3))/(20088 (363722122031829547 + 
                18 I Sqrt[230095974116695490903297934806259])^(
              1/3)) - (1/
           100440)((7 (363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259]))^(
           1/3)) - 
           11653870955069/(2490912 \[Sqrt](1/
                 5 (2618036029 + (76671493721000 7^(
                    2/3))/(363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259)^(
                    1/3) + 
                   248 (7 (363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259))^(
                    1/3))))))}, {A -&gt; \[Sqrt](25709/4464 - (1/
      4464)(\[Sqrt](1/
          5 (2618036029 + (76671493721000 7^(
               2/3))/(363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259])^(1/3) + 
            248 (7 (363722122031829547 + 
                 18 I Sqrt[230095974116695490903297934806259]))^(
             1/3)))) - 
      1/2 \[Sqrt](2618036029/
          12454560 - (61831849775 7^(
             2/3))/(20088 (363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259])^(
             1/3)) - (1/
          100440)((7 (363722122031829547 + 
              18 I Sqrt[230095974116695490903297934806259]))^(1/3)) - 
          
          11653870955069/(2490912 \[Sqrt](1/
                5 (2618036029 + (76671493721000 7^(
                    2/3))/(363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259)^(
                   1/3) + 
                  248 (7 (363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259))^(
                   1/3))))))}, {A -&gt; -\[Sqrt](25709/4464 - (1/
       4464)(\[Sqrt](1/
           5 (2618036029 + (76671493721000 7^(
                2/3))/(363722122031829547 + 
                18 I Sqrt[230095974116695490903297934806259])^(1/3) + 
             248 (7 (363722122031829547 + 
                  18 I Sqrt[230095974116695490903297934806259]))^(
              1/3)))) + 
       1/2 \[Sqrt](2618036029/
           12454560 - (61831849775 7^(
              2/3))/(20088 (363722122031829547 + 
                18 I Sqrt[230095974116695490903297934806259])^(
              1/3)) - (1/
           100440)((7 (363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259]))^(
           1/3)) - 
           11653870955069/(2490912 \[Sqrt](1/
                 5 (2618036029 + (76671493721000 7^(
                    2/3))/(363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259)^(
                    1/3) + 
                   248 (7 (363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259))^(
                    1/3))))))}, {A -&gt; \[Sqrt](25709/4464 - (1/
      4464)(\[Sqrt](1/
          5 (2618036029 + (76671493721000 7^(
               2/3))/(363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259])^(1/3) + 
            248 (7 (363722122031829547 + 
                 18 I Sqrt[230095974116695490903297934806259]))^(
             1/3)))) + 
      1/2 \[Sqrt](2618036029/
          12454560 - (61831849775 7^(
             2/3))/(20088 (363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259])^(
             1/3)) - (1/
          100440)((7 (363722122031829547 + 
              18 I Sqrt[230095974116695490903297934806259]))^(1/3)) - 
          11653870955069/(2490912 \[Sqrt](1/
                5 (2618036029 + (76671493721000 7^(
                    2/3))/(363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259)^(
                   1/3) + 
                  248 (7 (363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259))^(
                   1/3))))))}, {A -&gt; -\[Sqrt](25709/4464 + (1/
       4464)(\[Sqrt](1/
           5 (2618036029 + (76671493721000 7^(
                2/3))/(363722122031829547 + 
                18 I Sqrt[230095974116695490903297934806259])^(1/3) + 
             248 (7 (363722122031829547 + 
                  18 I Sqrt[230095974116695490903297934806259]))^(
              1/3)))) - 
       1/2 \[Sqrt](2618036029/
           12454560 - (61831849775 7^(
              2/3))/(20088 (363722122031829547 + 
                18 I Sqrt[230095974116695490903297934806259])^(
              1/3)) - (1/
           100440)((7 (363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259]))^(
           1/3)) + 
           11653870955069/(2490912 \[Sqrt](1/
                 5 (2618036029 + (76671493721000 7^(
                    2/3))/(363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259)^(
                    1/3) + 
                   248 (7 (363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259))^(
                    1/3))))))}, {A -&gt; \[Sqrt](25709/4464 + (1/
      4464)(\[Sqrt](1/
          5 (2618036029 + (76671493721000 7^(
               2/3))/(363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259])^(1/3) + 
            248 (7 (363722122031829547 + 
                 18 I Sqrt[230095974116695490903297934806259]))^(
             1/3)))) - 
      1/2 \[Sqrt](2618036029/
          
          12454560 - (61831849775 7^(
             2/3))/(20088 (363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259])^(
             1/3)) - (1/
          100440)((7 (363722122031829547 + 
              18 I Sqrt[230095974116695490903297934806259]))^(1/3)) + 
          11653870955069/(2490912 \[Sqrt](1/
                5 (2618036029 + (76671493721000 7^(
                    2/3))/(363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259)^(
                   1/3) + 
                  248 (7 (363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259))^(
                   1/3))))))}, {A -&gt; -\[Sqrt](25709/4464 + (1/
       4464)(\[Sqrt](1/
           5 (2618036029 + (76671493721000 7^(
                2/3))/(363722122031829547 + 
                18 I Sqrt[230095974116695490903297934806259])^(1/3) + 
             248 (7 (363722122031829547 + 
                  18 I Sqrt[230095974116695490903297934806259]))^(
              1/3)))) + 
       1/2 \[Sqrt](2618036029/
           12454560 - (61831849775 7^(
              2/3))/(20088 (363722122031829547 + 
                18 I Sqrt[230095974116695490903297934806259])^(
              1/3)) - (1/
           100440)((7 (363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259]))^(
           1/3)) + 
           11653870955069/(2490912 \[Sqrt](1/
                 5 (2618036029 + (76671493721000 7^(
                    2/3))/(363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259)^(
                    1/3) + 
                   248 (7 (363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259))^(
                    1/3))))))}, {A -&gt; \[Sqrt](25709/4464 + (1/
      4464)(\[Sqrt](1/
          5 (2618036029 + (76671493721000 7^(
               2/3))/(363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259])^(1/3) + 
            248 (7 (363722122031829547 + 
                 18 I Sqrt[230095974116695490903297934806259]))^(
             1/3)))) + 
      1/2 \[Sqrt](2618036029/
          12454560 - (61831849775 7^(
             2/3))/(20088 (363722122031829547 + 
               18 I Sqrt[230095974116695490903297934806259])^(
             1/3)) - (1/
          100440)((7 (363722122031829547 + 
              18 I Sqrt[230095974116695490903297934806259]))^(1/3)) + 
          11653870955069/(2490912 \[Sqrt](1/
                5 (2618036029 + (76671493721000 7^(
                    2/3))/(363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259)^(
                   1/3) + 
                  248 (7 (363722122031829547 + 
                    18 I \[Sqrt]230095974116695490903297934806259))^(
                   1/3))))))}}
In[99]:= N[%98]
Out[99]= {{A -&gt; 0.}, {A -&gt; -3.38967*10^-17 - 0.553576 I}, {A -&gt; 
   3.38967*10^-17 + 0.553576 I}, {A -&gt; -0.510778 - 
    1.266*10^-17 I}, {A -&gt; 
   0.510778 + 1.266*10^-17 I}, {A -&gt; -1.42015 + 
    8.65741*10^-18 I}, {A -&gt; 
   1.42015 - 8.65741*10^-18 I}, {A -&gt; -4.58971 + 
    1.39029*10^-19 I}, {A -&gt; 4.58971 - 1.39029*10^-19 I}}
In[100]:= PadeApproximant[D[u3[x], x], {x, 0, 6}]
Out[100]= (1 + (A (-18051595200 + 28821322908 A^2 - 72301325965 A^4 + 
        200933852520 A^6 + 182269360800 A^8 + 
        4110004800 A^10) x)/(36 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + ((-50965200000 + 306999795960 A^2 - 
        1246427688327 A^4 + 4017653621200 A^6 - 3275035882460 A^8 - 
        260340814800 A^10) x^2)/(264 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + (A (-105305508000 + 1044801068058 A^2 + 
        6182886841371 A^4 - 37770361925990 A^6 + 27055520656080 A^8 + 
        1107594734400 A^10) x^3)/(4752 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + ((-529254000000 + 4825161138600 A^2 - 
        21972531103965 A^4 + 31620708118368 A^6 + 4775060622080 A^8 + 
        68697191640 A^10 - 7535008800 A^12) x^4)/(9504 (320760000 + 
        347363172 A^2 - 22651834825 A^4 + 24154888720 A^6 + 
        6290185200 A^8 + 114166800 A^10)) + ((-1972887840000 A - 
        1045424860128 A^3 + 1096444794215 A^5 - 2683783126420 A^7 + 
        3613332672000 A^9 - 
        191535062400 A^11) x^5)/(126720 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + ((-2940300000000 + 18768273717600 A^2 - 
        54358130936619 A^4 + 86960837890165 A^6 - 
        36933673973230 A^8 + 27902056675200 A^10 + 
        1122664752000 A^12) x^6)/(887040 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)))/(1 - (A (29598955200 - 16316248716 A^2 - 
        743164727735 A^4 + 668642141400 A^6 + 
        44177306400 A^8) x)/(36 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + ((-152895600000 + 1572176402280 A^2 - 
        4098240536733 A^4 - 4296663146570 A^6 + 4885019463420 A^8 + 
        190878296400 A^10) x^2)/(792 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + (A (939089052000 - 8250701529510 A^2 + 
        21802203471069 A^4 - 2425047113450 A^6 + 236317214760 A^8 + 
        7535008800 A^10) x^3)/(4752 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + ((-529254000000 + 2359316500200 A^2 - 
        5043392665137 A^4 - 119138108950 A^6 - 302304295260 A^8 - 
        130381309080 A^10) x^4)/(9504 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + (A (8373546720000 - 41958983537856 A^2 + 
        98505753780525 A^4 - 69048126381080 A^6 + 
        56286197220000 A^8 + 
        2285511307200 A^10) x^5)/(380160 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) + ((-5292540000000 + 24735582321120 A^2 - 
        93246362469639 A^4 + 113275623939876 A^6 + 
        13201774542214 A^8 - 504742921320 A^10 - 
        46717054560 A^12) x^6)/(1596672 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)))
In[101]:= 
Solve[((-2940300000000 + 18768273717600 A^2 - 54358130936619 A^4 + 
       86960837890165 A^6 - 36933673973230 A^8 + 
       27902056675200 A^10 + 
       1122664752000 A^12) )/(887040 (320760000 + 347363172 A^2 - 
        22651834825 A^4 + 24154888720 A^6 + 6290185200 A^8 + 
        114166800 A^10)) == 0, A] // N
Out[101]= {{A -&gt; 0. - 5.12087 I}, {A -&gt; 
   0. + 5.12087 I}, {A -&gt; -0.558742}, {A -&gt; 
   0.558742}, {A -&gt; -0.537983 + 0.290459 I}, {A -&gt; 
   0.537983 - 0.290459 I}, {A -&gt; -0.537983 - 0.290459 I}, {A -&gt; 
   0.537983 + 0.290459 I}, {A -&gt; -0.958373 + 0.771157 I}, {A -&gt; 
   0.958373 - 0.771157 I}, {A -&gt; -0.958373 - 0.771157 I}, {A -&gt; 
   0.958373 + 0.771157 I}}
</code>

显然，$A=0.516398$。原文献Table 1中更高阶结果有误。

变分迭代方法解半无限空间常微分方程

变分迭代方法简要介绍

解布拉修斯方程（Blasius equation）

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