# Find an easier substitution in this equation?

Posted 1 year ago
948 Views
|
0 Replies
|
0 Total Likes
|
 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