Group Abstract

Message Boards

WOLFRAM COMMUNITY

5.3K Views

6 Replies

0 Total Likes

View groups...

Follow this post

Share this post:

GROUPS:

Equation Solving Wolfram Language

Need help! : Integrating the function to be solved

hwoarang Polar

Posted 10 years ago

Hello, guys, I think that I am always stuck in the mathmatica grammar. -_-a. Now I am trying to get the time-dependent function H[t], which is correlated to what happens at t'. So I need to integrate the time t' (from -infinity to infinity) to obtain the function with the time t. The code is simple as follows. I think that there are simple problems in Integrate or NSolve... Could you please help me ? Thank you very much in advance. Remove["Global`*"]; \[Lambda] = 800.0; n = 2.48; k = 4.38 ; \[Xi] = \[Lambda]/(4 \[Pi] k); \[Chi] = \[Lambda]/(4 \[Pi] n); v = 5.894; P1 = 0.5; P2 = 0.5; W[t_] = -6 Cos[0.25 t] Exp[-(t/10)^2]; Plot[W[t], {t, -100, 100}, PlotRange -> All] F[t_] = v ((P1/\[Chi] - P2/\[Xi]) Cos[ v/\[Chi] t] - (P2/\[Chi] + P1/\[Xi]) Sin[ v/\[Chi] Abs[t]]) Exp[-Abs[t] v/\[Xi]]; Sol = NSolve[ 8 \[Pi] v (2 P2 H[t] + Integrate[ F[tt - t] H[tt], {tt, -Infinity, Infinity}]) - \[Lambda] W[ t] == 0, H, {t, -30, 30}][[1]]; Plot[H[t] /. Sol, {t, -30, 30}, PlotStyle -> {Blue}, PlotRange -> All]

Hello, guys,

I think that I am always stuck in the mathmatica grammar. -_-a. Now I am trying to get the time-dependent function H[t], which is correlated to what happens at t'. So I need to integrate the time t' (from -infinity to infinity) to obtain the function with the time t. The code is simple as follows. I think that there are simple problems in Integrate or NSolve... Could you please help me ? Thank you very much in advance.

Remove["Global`*"];
\[Lambda] = 800.0; 
n = 2.48; k = 4.38 ; 
\[Xi] = \[Lambda]/(4 \[Pi] k); 
\[Chi] = \[Lambda]/(4 \[Pi] n);
v = 5.894; 
P1 = 0.5; P2 = 0.5;
W[t_] = -6 Cos[0.25 t] Exp[-(t/10)^2];
Plot[W[t], {t, -100, 100}, PlotRange -> All]
F[t_] = v ((P1/\[Chi] - P2/\[Xi]) Cos[
       v/\[Chi] t] - (P2/\[Chi] + P1/\[Xi]) Sin[
       v/\[Chi] Abs[t]]) Exp[-Abs[t] v/\[Xi]];
Sol = NSolve[
    8 \[Pi] v (2 P2 H[t] + 
         Integrate[
          F[tt - t] H[tt], {tt, -Infinity, Infinity}]) - \[Lambda] W[
        t] == 0, H, {t, -30, 30}][[1]];
Plot[H[t] /. Sol, {t, -30, 30}, PlotStyle -> {Blue}, PlotRange -> All]

POSTED BY: hwoarang Polar

6 Replies

Sort By:

hwoarang Polar

Posted 10 years ago

Than you Frank for thoughtful suggestion. :-) Actually, I tried this, but this problem cannot be converged to the differential equation, I think. Rather, if I integrate the previous equation (I mean the equation in NSolve), the simpler equation is obtained as follows. 8 \[Pi] v (Intg[t, H]) - \[Lambda] G[t] == 0 Here, Intg[t,H] and G[t] are already defined before. If we differentiate this function with t, then the previous equation (that I want to solve) is obtained again. If we differentiate higher order, it will be more complex to solve. Maybe, is it good to model/guess H[t] first (using some parameters) and then to use FindFit ? In the meantime, I will try if some transformation (Laplace or Fourier...) can reduce the complexity of this problem...

POSTED BY: hwoarang Polar

Frank Kampas

Frank Kampas, Physicist at Large Consulting

Posted 10 years ago

Are you trying to solve an integral equation? If so, NSolve is not the way to do it. Also, your definition of Intg has H_?NumericQ which will not accept a function as a second argument. In[76]:= Intg1[q_?NumericQ, H_] := NIntegrate[F[T - q] H[T], {T, -Infinity, Infinity}]; In[77]:= Intg1[1, #^2 &] Out[77]= -13.6357

POSTED BY: Frank Kampas

hwoarang Polar

Posted 10 years ago

Thank you, Frank. But, in the integral, H[T] is an unknown function, which should be solved using the following equation. 8 \[Pi] v (2 P2 H[t] + Intg[t, H]) - 10^12 \[Lambda] W[t] == 0 Here, W[q] is a known function (and sorry for typos in previous one, it's better that q needs to be exchanged to t). If H[T] is not known, the integration cannot be done. Maybe, just by guessing H[t] and solving above equation iteratively (or self-consistent ?), then H[t] may be solved. Then, the problem that I have is that how I let the program recognize this. -_____- a

POSTED BY: hwoarang Polar

Frank Kampas

Frank Kampas, Physicist at Large Consulting

Posted 10 years ago

I was illustrating that your definition of Intg would not handle a function as a second argument. However, the main point is that you're trying to solve an integral equation, which NSolve won't do. Can you convert your problem to a differential equation and use NDSolve?

POSTED BY: Frank Kampas

Frank Kampas

Frank Kampas, Physicist at Large Consulting

Posted 10 years ago

Suggest you do the Integrate outside of NSolve and see what you get. That will help you diagnose the problem.

POSTED BY: Frank Kampas

hwoarang Polar

Posted 10 years ago

Thank you, Frank, for the suggestion but I couldn't understand exactly what you suggested. Actually, the difficulty that I think is that the function H[t] which I want to obtain is in the symbolic integration and is even a function of time. Even I am not sure if NSolve is a right command in this case. I have fully tried to search for any leads in archives, but I couldn't find a similar situation to this. Could you help me, please? Any information would be very appreciated. Here is the code which I corrected some typos and unit dimensions. Remove["Global`"]; \[Lambda] = 800000.0; A1 = 0.6; A2 = -1.8; n = 2.48; k = 4.38 ; \[Xi] = \[Lambda]/(4 \[Pi] k); \[Chi] = \[Lambda]/(4 \[Pi] n); v = 5894; P1 = 0.5; P2 = 0.5; G[t_] = -6 10^-4 (Cos[0.25 t] - 0.209611) Exp[-(t/10)^2]; W[t_] = D[G[t], t] Plot[W[t], {t, -100, 100}, PlotRange -> All] F[t_] = v ((P1/\[Chi] - P2/\[Xi]) Cos[ v/\[Chi] t] - (P2/\[Chi] + P1/\[Xi]) Sin[ v/\[Chi] Abs[t]]) Exp[-Abs[t] v/\[Xi]]; Intg[q_?NumericQ, H_?NumericQ] := NIntegrate[ F[T - q] H[T], {T, -Infinity, Infinity}]; ( H is unknown function *) Sol = NSolve[ 8 \[Pi] v (2 P2 H[t] + Intg[q, H]) - 10^12 \[Lambda] W[t] == 0, H][[1]]; Plot[H[t] /. Sol, {t, -30, 30}, PlotStyle -> {Blue}, PlotRange -> All]

Thank you, Frank, for the suggestion but I couldn't understand exactly what you suggested. Actually, the difficulty that I think is that the function H[t] which I want to obtain is in the symbolic integration and is even a function of time. Even I am not sure if NSolve is a right command in this case. I have fully tried to search for any leads in archives, but I couldn't find a similar situation to this. Could you help me, please? Any information would be very appreciated. Here is the code which I corrected some typos and unit dimensions.

Remove["Global`*"];
\[Lambda] = 800000.0; 
A1 = 0.6; A2 = -1.8;  
n = 2.48; k = 4.38 ; 
\[Xi] = \[Lambda]/(4 \[Pi] k); 
\[Chi] = \[Lambda]/(4 \[Pi] n);
v = 5894; 
P1 = 0.5; P2 = 0.5;
G[t_] = -6 10^-4 (Cos[0.25 t] - 0.209611) Exp[-(t/10)^2];
W[t_] = D[G[t], t]
Plot[W[t], {t, -100, 100}, PlotRange -> All]
F[t_] = v ((P1/\[Chi] - P2/\[Xi]) Cos[
       v/\[Chi] t] - (P2/\[Chi] + P1/\[Xi]) Sin[
       v/\[Chi] Abs[t]]) Exp[-Abs[t] v/\[Xi]];
Intg[q_?NumericQ, H_?NumericQ] := 
 NIntegrate[
  F[T - q] H[T], {T, -Infinity, 
   Infinity}]; (* H is unknown function *)
Sol = NSolve[
   8 \[Pi] v (2 P2 H[t] + Intg[q, H]) - 10^12 \[Lambda] W[t] == 0, 
   H][[1]];
Plot[H[t] /. Sol, {t, -30, 30}, PlotStyle -> {Blue}, PlotRange -> All]

POSTED BY: hwoarang Polar

Reply to this discussion

Reply Preview

Attachments

Remove Add a file to this post

Follow this discussion

or Discard

Feedback