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plotting - Plot Indefinite integral


I'm having a hard time to find a solution to a numerical limitation on mathematica:


PwD = 0.005*Integrate[(Exp[-(0.25/\[Tau])]/\[Tau])*
  Integrate[Exp[-((Tan[45*Degree]^2 - (z + 30)^2)/(4*\[Tau]))]*
         (1 + 
      2*Sum[Exp[-((n^2*Pi^2*\[Tau])/10000)]*Cos[0.8*n*Pi]*
         Cos[(n*Pi)/2 - (n*Pi*z)/100], {n, 1, 4}]), {z, -50, 
    50}], {\[Tau], 0, t}]


Mathematica graphics


After solving the integral in space {z,-50,50} I need to evaluate the indefinite integral in time {tau,0,t}, so I eventually generate a plot of PwD as function of Log[t]. But it seems that it does not converge.


Can someone provide an alternative way?



Hi everybody thanks for the comments... They sound about right! In the past 2 months I dedicated my time to derive these solutions again - once I had no help from the author. Well, I finally get it! The published expression has a type mistake... The right equation should be:


PwD = (1/200)*Integrate[
    (1/(E^(0.25/\[Tau])*\[Tau]))*Integrate[
      (1 + 2*Sum[(Cos[0.8*n*Pi]*Cos[(n*Pi)/2 - 
              (n*Pi*z)/100])/E^((n^2*Pi^2*\[Tau])/
             10000), {n, 1, 100}])/

       E^((Tan[\[Psi]*Degree]^2*(z + 30)^2)/(4*\[Tau])), 
      {z, -50, 50}], {\[Tau], 0, t}]

Now I'm able to generate the graphics. For different angles \[Psi] this is what I did:


Pw1 = (Exp[-0.25/t]/t)*
   Integrate[
    Exp[-(((Tan[0 Degree]^2)*(z + 30)^2)/(4*t))]*(1 + 
       2*Sum[Exp[-((n^2)*(Pi^2)*t)/(100^2)]*Cos[0.8*n*Pi]*
          Cos[(n*Pi/2) - (n*Pi*z/100)], {n, 1, 100}]), {z, -50, 50}];
Pw2 = (Exp[-0.25/t]/t)*

   Integrate[
    Exp[-(((Tan[30 Degree]^2)*(z + 30)^2)/(4*t))]*(1 + 
       2*Sum[Exp[-((n^2)*(Pi^2)*t)/(100^2)]*Cos[0.8*n*Pi]*
          Cos[(n*Pi/2) - (n*Pi*z/100)], {n, 1, 100}]), {z, -50, 50}];
Pw3 = (Exp[-0.25/t]/t)*
   Integrate[
    Exp[-(((Tan[60 Degree]^2)*(z + 30)^2)/(4*t))]*(1 + 
       2*Sum[Exp[-((n^2)*(Pi^2)*t)/(100^2)]*Cos[0.8*n*Pi]*
          Cos[(n*Pi/2) - (n*Pi*z/100)], {n, 1, 100}]), {z, -50, 50}];


PD1[y_] := 0.005*NIntegrate[Pw1, {t, 0, y}, MaxRecursion -> 20];
PD2[y_] := 0.005*NIntegrate[Pw2, {t, 0, y}, MaxRecursion -> 20];
PD3[y_] := 0.005*NIntegrate[Pw3, {t, 0, y}, MaxRecursion -> 20];

T1 = Table[{y, 
    PD1[y]}, {y, {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 2, 
     2.2, 2.5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 
     18, 19, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 
     90, 100, 110, 120, 130, 140, 150, 200, 250, 300, 350, 400, 450, 
     500, 550, 600, 650, 700, 750, 800, 850, 900, 950, 1000, 1100, 

     1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2100, 2200,
      2300, 2400, 2500, 2600, 2700, 2800, 2900, 3000, 3500, 4000, 
     4500, 5000, 5500, 6000, 6500, 7000, 7500, 8000, 8500, 9000, 9500,
      10000, 11000, 13000, 15000, 17000, 19000, 20000, 22000, 24000, 
     26000, 28000, 30000, 35000, 40000, 45000, 50000, 55000, 60000, 
     65000, 70000, 75000, 80000, 90000, 95000, 100000, 150000, 200000,
      250000, 300000, 350000, 400000, 450000, 500000, 550000, 600000, 
     700000, 800000, 900000, 1000000}}];
T2 = Table[{y, 
    PD2[y]}, {y, {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 2, 

     2.2, 2.5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 
     18, 19, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 
     90, 100, 110, 120, 130, 140, 150, 200, 250, 300, 350, 400, 450, 
     500, 550, 600, 650, 700, 750, 800, 850, 900, 950, 1000, 1100, 
     1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2100, 2200,
      2300, 2400, 2500, 2600, 2700, 2800, 2900, 3000, 3500, 4000, 
     4500, 5000, 5500, 6000, 6500, 7000, 7500, 8000, 8500, 9000, 9500,
      10000, 11000, 13000, 15000, 17000, 19000, 20000, 22000, 24000, 
     26000, 28000, 30000, 35000, 40000, 45000, 50000, 55000, 60000, 
     65000, 70000, 75000, 80000, 90000, 95000, 100000, 150000, 200000,

      250000, 300000, 350000, 400000, 450000, 500000, 550000, 600000, 
     700000, 800000, 900000, 1000000}}];
T3 = Table[{y, 
    PD3[y]}, {y, {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 2, 
     2.2, 2.5, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 
     18, 19, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 
     90, 100, 110, 120, 130, 140, 150, 200, 250, 300, 350, 400, 450, 
     500, 550, 600, 650, 700, 750, 800, 850, 900, 950, 1000, 1100, 
     1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2100, 2200,
      2300, 2400, 2500, 2600, 2700, 2800, 2900, 3000, 3500, 4000, 

     4500, 5000, 5500, 6000, 6500, 7000, 7500, 8000, 8500, 9000, 9500,
      10000, 11000, 13000, 15000, 17000, 19000, 20000, 22000, 24000, 
     26000, 28000, 30000, 35000, 40000, 45000, 50000, 55000, 60000, 
     65000, 70000, 75000, 80000, 90000, 95000, 100000, 150000, 200000,
      250000, 300000, 350000, 400000, 450000, 500000, 550000, 600000, 
     700000, 800000, 900000, 1000000}}];

PwD1 = Interpolation[T1];
PwD2 = Interpolation[T2];
PwD3 = Interpolation[T3];


P1 = LogLogPlot[{PwD1[y], y*PwD1'[y]}, {y, 0.1, 1000000}, 
   PlotRange -> {0.01, 10}, PlotStyle -> {{Black}, {Dashed, Black}}, 
   Frame -> True, FrameLabel -> {tD, "PD e tD*PD'"}, 
   BaseStyle -> {FontSize -> 12}];
P2 = LogLogPlot[{PwD2[y], y*PwD2'[y]}, {y, 0.1, 1000000}, 
   PlotRange -> {0.01, 10}, PlotStyle -> {{Brown}, {Dashed, Brown}}];
P3 = LogLogPlot[{PwD3[y], y*PwD3'[y]}, {y, 0.1, 1000000}, 
   PlotRange -> {0.01, 10}, PlotStyle -> {{Purple}, {Dashed, Purple}}];


Show[P1, P2, P3]

The problem is that it takes a very long time to compute these codes... Since I don't have good programming skills, could anyone propose an alternative way to computing this plot?


thanks



Answer



You could do the indefinite integral and put in the limits afterwards. E.g., on Linux I get a speed-up of about 14 for one integral (I have no time to do more right now ...)


Mathematica 8.0 for Linux x86 (64-bit)
Copyright 1988-2011 Wolfram Research, Inc.

In[1]:= !!i

Print["timing 1: ", 
First @ AbsoluteTiming[
  Pw3 = ExpandAll[(Exp[-0.25/t]/t)*
     Integrate[
      Exp[-((Tan[60*Degree]^2*
           (z + 30)^2)/(4*t))]*
       (1 + 2*Sum[Exp[-(n^2*Pi^2*t)/
            100^2]*Cos[0.8*n*Pi]*
           Cos[n*(Pi/2) - n*Pi*
            (z/100)], {n, 1, 100}]), 

      {z, -50, 50}]]; ]
];
Print["timing 2 : ", 
First @ AbsoluteTiming[
  nPw3 = Expand[(Exp[-0.25/t]/t)*
     ((#1 /. z -> 50) - (#1 /. 
         z -> -50) & )[
      ParallelMap[(ExpandAll[Integrate[#1, z]] & ),
       (TrigToExp[Expand[
          Exp[-((Tan[60*Degree]^2*

            (z + 30)^2)/(4*t))]*
           (1 + 2*Sum[Exp[-(n^2*Pi^2*
            t)/100^2]*Cos[0.8*n*Pi]*
            Cos[n*(Pi/2) - n*Pi*
            (z/100)], {n, 1, 
            100}])]])]] ]; ]
]
Print["check ", Chop[Pw3 - nPw3]]

In[1]:= <
timing 1: 75.121517
Launching kernels...       Mathematica 8.0 for Linux x86 (64-bit) Copyright 1988-2011 Wolfram Research, Inc.                                                                                                            
timing 2 : 5.255553                                                                                                                    
check 0

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