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Find an easier substitution in this equation?

Jacques Ou
Posted 8 years ago

I use 

/. x -> 0

in eq2, and I can't get the limit. I have to use a much more complicated substitution in eq3

/. {-((A11 E^(
     A3 t - A3 t1 - (-lh + x)^2/(4 Dc t1) - t1 \[Theta]c) (-lh + x))/(
    2 Dc t1^(3/2))) -> 
   0, -((E^(-((-lh + x)^2/(4 Dp t1))) (-lh + x))/(
    2 Dp Sqrt[\[Pi]] t1^(3/2))) -> 
   0, \[Alpha]1 (-lh + x + Dp t1 \[Alpha]1) - (1/2 (-lh + x) + 
      Dp t1 \[Alpha]1)^2/(Dp t1) -> -\[Infinity], 
  E^(\[Alpha]1 (-lh + x + Dp t1 \[Alpha]1)) \[Alpha]1^2 Erfc[(
     1/2 (-lh + x) + Dp t1 \[Alpha]1)/Sqrt[Dp t1]] -> 0}

and I can get the limit. Is there a simpler substitution than eq3?

In[111]:= 
p3[x_, t_] = pinf + (A2 (-1 + E^(-t \[Theta]c)))/\[Theta]c + \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(
\*FractionBox[\(A11\ 
\*SuperscriptBox[\(E\), \(A3\ t - A3\ t1 - 
\*FractionBox[
SuperscriptBox[\((\(-lh\) + x)\), \(2\)], \(4\ Dc\ t1\)] - 
        t1\ \[Theta]c\)]\), 
SqrtBox[\(t1\)]] \[DifferentialD]t1\)\) + \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(\((\(-A8\) + A9 - 
      A15\ 
\*SuperscriptBox[\(E\), \(A3\ \((t - t1)\)\)] - A9\ 
\*SuperscriptBox[\(E\), \(\((\(-t\) + t1)\)\ \[Theta]c\)] - A16\ 
\*SuperscriptBox[\(E\), \(A3\ \((t - t1)\)\)]\ Erf[
\*SqrtBox[\(A5\)]\ 
\*SqrtBox[\(t - t1\)]])\)\ \((
\*FractionBox[
SuperscriptBox[\(E\), \(-
\*FractionBox[
SuperscriptBox[\((\(-lh\) + x)\), \(2\)], \(4\ Dp\ t1\)]\)], \(
\*SqrtBox[\(\[Pi]\)]\ 
\*SqrtBox[\(t1\)]\)] - 
\*SqrtBox[\(Dp\)]\ 
\*SuperscriptBox[\(E\), \(\[Alpha]1\ \((\(-lh\) + x + 
           Dp\ t1\ \[Alpha]1)\)\)]\ \[Alpha]1\ Erfc[
\*FractionBox[\(
\*FractionBox[\(1\), \(2\)]\ \((\(-lh\) + x)\) + Dp\ t1\ \[Alpha]1\), 
SqrtBox[\(Dp\ t1\)]]])\) \[DifferentialD]t1\)\)

Out[111]= pinf + (A2 (-1 + E^(-t \[Theta]c)))/\[Theta]c + \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(
\*FractionBox[\(A11\ 
\*SuperscriptBox[\(E\), \(A3\ t - A3\ t1 - 
\*FractionBox[
SuperscriptBox[\((\(-lh\) + x)\), \(2\)], \(4\ Dc\ t1\)] - 
       t1\ \[Theta]c\)]\), 
SqrtBox[\(t1\)]] \[DifferentialD]t1\)\) + \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(\(\((\(-A8\) + 
      A9 - A15\ 
\*SuperscriptBox[\(E\), \(A3\ \((t - t1)\)\)] - A9\ 
\*SuperscriptBox[\(E\), \(\((\(-t\) + t1)\)\ \[Theta]c\)] - A16\ 
\*SuperscriptBox[\(E\), \(A3\ \((t - t1)\)\)]\ Erf[
\*SqrtBox[\(A5\)]\ 
\*SqrtBox[\(t - t1\)]])\)\ \((
\*FractionBox[
SuperscriptBox[\(E\), \(-
\*FractionBox[
SuperscriptBox[\((\(-lh\) + x)\), \(2\)], \(4\ Dp\ t1\)]\)], \(
\*SqrtBox[\(\[Pi]\)]\ 
\*SqrtBox[\(t1\)]\)] - 
\*SqrtBox[\(Dp\)]\ 
\*SuperscriptBox[\(E\), \(\[Alpha]1\ \((\(-lh\) + x + 
           Dp\ t1\ \[Alpha]1)\)\)]\ \[Alpha]1\ Erfc[
\*FractionBox[\(
\*FractionBox[\(1\), \(2\)]\ \((\(-lh\) + x)\) + Dp\ t1\ \[Alpha]1\), 
SqrtBox[\(Dp\ t1\)]]])\)\) \[DifferentialD]t1\)\)

In[112]:= eq1 = D[p3[x, t], x] == 0

Out[112]= \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(\(-
\*FractionBox[\(A11\ 
\*SuperscriptBox[\(E\), \(A3\ t - A3\ t1 - 
\*FractionBox[
SuperscriptBox[\((\(-lh\) + x)\), \(2\)], \(4\ Dc\ t1\)] - 
         t1\ \[Theta]c\)]\ \((\(-lh\) + x)\)\), \(2\ Dc\ 
\*SuperscriptBox[\(t1\), \(3/2\)]\)]\) \[DifferentialD]t1\)\) + \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(\(\((\(-A8\) + 
       A9 - A15\ 
\*SuperscriptBox[\(E\), \(A3\ \((t - t1)\)\)] - A9\ 
\*SuperscriptBox[\(E\), \(\((\(-t\) + t1)\)\ \[Theta]c\)] - A16\ 
\*SuperscriptBox[\(E\), \(A3\ \((t - t1)\)\)]\ Erf[
\*SqrtBox[\(A5\)]\ 
\*SqrtBox[\(t - t1\)]])\)\ \((\(-
\*FractionBox[\(
\*SuperscriptBox[\(E\), \(-
\*FractionBox[
SuperscriptBox[\((\(-lh\) + 
                x)\), \(2\)], \(4\ Dp\ t1\)]\)]\ \((\(-lh\) + 
            x)\)\), \(2\ Dp\ 
\*SqrtBox[\(\[Pi]\)]\ 
\*SuperscriptBox[\(t1\), \(3/2\)]\)]\) + 
\*FractionBox[\(
\*SqrtBox[\(Dp\)]\ 
\*SuperscriptBox[\(E\), \(\[Alpha]1\ \((\(-lh\) + x + 
              Dp\ t1\ \[Alpha]1)\) - 
\*FractionBox[
SuperscriptBox[\((
\*FractionBox[\(1\), \(2\)]\ \((\(-lh\) + x)\) + 
               Dp\ t1\ \[Alpha]1)\), \(2\)], \(Dp\ t1\)]\)]\ \
\[Alpha]1\), \(
\*SqrtBox[\(\[Pi]\)]\ 
\*SqrtBox[\(Dp\ t1\)]\)] - 
\*SqrtBox[\(Dp\)]\ 
\*SuperscriptBox[\(E\), \(\[Alpha]1\ \((\(-lh\) + x + 
            Dp\ t1\ \[Alpha]1)\)\)]\ 
\*SuperscriptBox[\(\[Alpha]1\), \(2\)]\ Erfc[
\*FractionBox[\(
\*FractionBox[\(1\), \(2\)]\ \((\(-lh\) + x)\) + Dp\ t1\ \[Alpha]1\), 
SqrtBox[\(Dp\ t1\)]]])\)\) \[DifferentialD]t1\)\) == 0

In[113]:= eq2 = eq1 /. x -> 0

Out[113]= \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(
\*FractionBox[\(A11\ 
\*SuperscriptBox[\(E\), \(A3\ t - 
\*FractionBox[
SuperscriptBox[\(lh\), \(2\)], \(4\ Dc\ t1\)] - A3\ t1 - 
        t1\ \[Theta]c\)]\ lh\), \(2\ Dc\ 
\*SuperscriptBox[\(t1\), \(3/2\)]\)] \[DifferentialD]t1\)\) + \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(t\)]\(\(\((\(-A8\) + 
       A9 - A15\ 
\*SuperscriptBox[\(E\), \(A3\ \((t - t1)\)\)] - A9\ 
\*SuperscriptBox[\(E\), \(\((\(-t\) + t1)\)\ \[Theta]c\)] - A16\ 
\*SuperscriptBox[\(E\), \(A3\ \((t - t1)\)\)]\ Erf[
\*SqrtBox[\(A5\)]\ 
\*SqrtBox[\(t - t1\)]])\)\ \((
\*FractionBox[\(
\*SuperscriptBox[\(E\), \(-
\*FractionBox[
SuperscriptBox[\(lh\), \(2\)], \(4\ Dp\ t1\)]\)]\ lh\), \(2\ Dp\ 
\*SqrtBox[\(\[Pi]\)]\ 
\*SuperscriptBox[\(t1\), \(3/2\)]\)] + 
\*FractionBox[\(
\*SqrtBox[\(Dp\)]\ 
\*SuperscriptBox[\(E\), \(\[Alpha]1\ \((\(-lh\) + 
              Dp\ t1\ \[Alpha]1)\) - 
\*FractionBox[
SuperscriptBox[\((\(-
\*FractionBox[\(lh\), \(2\)]\) + 
               Dp\ t1\ \[Alpha]1)\), \(2\)], \(Dp\ t1\)]\)]\ \
\[Alpha]1\), \(
\*SqrtBox[\(\[Pi]\)]\ 
\*SqrtBox[\(Dp\ t1\)]\)] - 
\*SqrtBox[\(Dp\)]\ 
\*SuperscriptBox[\(E\), \(\[Alpha]1\ \((\(-lh\) + 
            Dp\ t1\ \[Alpha]1)\)\)]\ 
\*SuperscriptBox[\(\[Alpha]1\), \(2\)]\ Erfc[
\*FractionBox[\(\(-
\*FractionBox[\(lh\), \(2\)]\) + Dp\ t1\ \[Alpha]1\), 
SqrtBox[\(Dp\ t1\)]]])\)\) \[DifferentialD]t1\)\) == 0

In[114]:= eq3 = 
 eq1 /. {-((
     A11 E^(A3 t - A3 t1 - (-lh + x)^2/(4 Dc t1) - 
       t1 \[Theta]c) (-lh + x))/(2 Dc t1^(3/2))) -> 
    0, -((E^(-((-lh + x)^2/(4 Dp t1))) (-lh + x))/(
     2 Dp Sqrt[\[Pi]] t1^(3/2))) -> 
    0, \[Alpha]1 (-lh + x + Dp t1 \[Alpha]1) - (1/2 (-lh + x) + 
       Dp t1 \[Alpha]1)^2/(Dp t1) -> -\[Infinity], 
   E^(\[Alpha]1 (-lh + x + Dp t1 \[Alpha]1)) \[Alpha]1^2 Erfc[(
      1/2 (-lh + x) + Dp t1 \[Alpha]1)/Sqrt[Dp t1]] -> 0}

Out[114]= True
POSTED BY: Jacques Ou
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