Help Needed with an Integration
Help Needed with an Integration
Hello All,
In Joe khackan’s latest paper – he models the current drawn by a Langmuir probe.
Part of doing this means dealing with some nasty math problems.
One expression from the paper is a nasty double integration.
I am struggling it to solve it. If anyone out there is sharp with Mathematica or MATLAB. I would appreciate an analytical solution (algebra) or some code, or excel file which deals with this. I have made some attempts with software to get at it, and the problem comes when I try to integrate to infinity, or just a really high number... I have not looked at limits yet.
Ideally, the code would take an applied voltage, and spit back an expected current. The current would be the solution to this double integration.
I do not think there is an analytical solution. The paper did reference another math paper dealing with this equation, but it converged too slowly. I got that paper and took this to an expert and they are still working on it.
Here it is:
I need to take this and make it algebra. Fortunately I have estimates of what the solution should be. Roughly speaking:
Applied: Solution:
-11 --- 0.1
-9 --- 0.1
-7 --- 0.1
-5 --- 0.3
-3 --- 0.3
-1 --- 0.4
1 --- 0.6
3 --- 0.8
5 --- 1
7 --- 1.7
9 --- 2.3
11 --- 3.1
13 --- 4.2
15 --- 5.2
17 --- 6.5
19 --- 7.6
21 --- 8.8
23 --- 10.3
25 --- 11.5
27 --- 12.9
29 --- 14.3
Can anyone help?
In Joe khackan’s latest paper – he models the current drawn by a Langmuir probe.
Part of doing this means dealing with some nasty math problems.
One expression from the paper is a nasty double integration.
I am struggling it to solve it. If anyone out there is sharp with Mathematica or MATLAB. I would appreciate an analytical solution (algebra) or some code, or excel file which deals with this. I have made some attempts with software to get at it, and the problem comes when I try to integrate to infinity, or just a really high number... I have not looked at limits yet.
Ideally, the code would take an applied voltage, and spit back an expected current. The current would be the solution to this double integration.
I do not think there is an analytical solution. The paper did reference another math paper dealing with this equation, but it converged too slowly. I got that paper and took this to an expert and they are still working on it.
Here it is:
I need to take this and make it algebra. Fortunately I have estimates of what the solution should be. Roughly speaking:
Applied: Solution:
-11 --- 0.1
-9 --- 0.1
-7 --- 0.1
-5 --- 0.3
-3 --- 0.3
-1 --- 0.4
1 --- 0.6
3 --- 0.8
5 --- 1
7 --- 1.7
9 --- 2.3
11 --- 3.1
13 --- 4.2
15 --- 5.2
17 --- 6.5
19 --- 7.6
21 --- 8.8
23 --- 10.3
25 --- 11.5
27 --- 12.9
29 --- 14.3
Can anyone help?
-
- Posts: 1439
- Joined: Wed Jul 14, 2010 5:27 pm
Re: Help Needed with an Integration
taking a stab n the dark here - and my calculus is very rusty, but looks like the solution might involve changing the bounds of integration
http://math.ucsd.edu/~wgarner/math20b/change_limits.htm
move the terms into the exponent to individual ones? (e^(a+b) to e^a*e^b))?
then move any constant multilipliers, like C, e^(D*E), etc. out to the left of the integral
try changing the bounds of integration, maybe get that cos(theta) out of the exponent? (that reminds me of e^(i(theta))=cos(theta)+i*sin(theta))
that square root term ... pythagorean, or the result of some inner product of some sort?
* you got both R and dR in the integral. perhaps thats a hint.
* you got an e^(in the integral). that looks like the main thing - the base of the integral, so to speak; looks like its an integral of an exponential function. perhaps a rule related to that, also considering the bullet just above.
** that is, maybe you got something like d(x) = f(d^2(x)).
* i'm looking for an integration by parts solution and i don't see it.
**in any case the radical is really messing me up. maybe chaging the bounds of integration, somehow) ( http://math.ucsd.edu/~wgarner/math20b/change_limits.htm )
http://math.ucsd.edu/~wgarner/math20b/change_limits.htm
move the terms into the exponent to individual ones? (e^(a+b) to e^a*e^b))?
then move any constant multilipliers, like C, e^(D*E), etc. out to the left of the integral
try changing the bounds of integration, maybe get that cos(theta) out of the exponent? (that reminds me of e^(i(theta))=cos(theta)+i*sin(theta))
that square root term ... pythagorean, or the result of some inner product of some sort?
* you got both R and dR in the integral. perhaps thats a hint.
* you got an e^(in the integral). that looks like the main thing - the base of the integral, so to speak; looks like its an integral of an exponential function. perhaps a rule related to that, also considering the bullet just above.
** that is, maybe you got something like d(x) = f(d^2(x)).
* i'm looking for an integration by parts solution and i don't see it.
**in any case the radical is really messing me up. maybe chaging the bounds of integration, somehow) ( http://math.ucsd.edu/~wgarner/math20b/change_limits.htm )
-
- Posts: 1439
- Joined: Wed Jul 14, 2010 5:27 pm
Re: Help Needed with an Integration
maybe reverse the order of integration - do theta first, and check out the last two on http://en.wikipedia.org/wiki/List_of_in ... _functions
-
- Posts: 1439
- Joined: Wed Jul 14, 2010 5:27 pm
Re: Help Needed with an Integration
... and get back a modified bessel function. great.
yeah, no, that one beyond me. it would seem matlab would be the way to go. the modern version of "just use a calculator".
there's a free alternative to matlab, "octave": http://www.gnu.org/software/octave/
yeah, no, that one beyond me. it would seem matlab would be the way to go. the modern version of "just use a calculator".
there's a free alternative to matlab, "octave": http://www.gnu.org/software/octave/
Re: Help Needed with an Integration
You might try not integrating to infinity, but only up to the Debye length. That is sufficiently messy looking that I would do it numerically. But I'm pretty lazy when it comes to integrating. lol.
[edit 1] I just noticed the angular dependence. What is the angle wrt? For that matter, is R a distance or just some variable he introduced?
[edit 2] Also, I can one constant that can be eliminated: E. Why not just absorb that into the C constant?
[edit 1] I just noticed the angular dependence. What is the angle wrt? For that matter, is R a distance or just some variable he introduced?
[edit 2] Also, I can one constant that can be eliminated: E. Why not just absorb that into the C constant?
Carter
Re: Help Needed with an Integration
Have you tried Wolfram Alpha? If you sign up for the pro trial, you get additional processing time, which may be able to solve it for you.
http://www.wolframalpha.com/input/?i=integral
I took your equation and turned it into the following, replacing E with G because it thinks any standalone e or E means Euler's number, and theta with T, because I wasn't sure how well it handles Greek:
Just plug that into the "function to integrate" to solve the first integral. Then, assuming it actually gets a result for you, plug the result back into the solver, replacing x with y, and T with x, because the default expects x to be the value of integration. (You can also click the little "variable" text and change it.)
http://www.wolframalpha.com/input/?i=integral
I took your equation and turned it into the following, replacing E with G because it thinks any standalone e or E means Euler's number, and theta with T, because I wasn't sure how well it handles Greek:
Code: Select all
x*(x^2+B*(125-V))^0.5*C*e^(D*(x^2+G+F*(x*cos(T))))
-
- Posts: 1439
- Joined: Wed Jul 14, 2010 5:27 pm
Re: Help Needed with an Integration
x*(x^2+B*(125-V))^0.5*C*e^(D*x^2)*e^(D*G)*e^(D*F*x*cos(T)))
H=C*e^(D*G), I = D*F, J = B*(125-V)
H*x*(x^2+J)^0.5*e^(D*x^2)*e^(I*x*cos(T)))
cleaner.
H=C*e^(D*G), I = D*F, J = B*(125-V)
H*x*(x^2+J)^0.5*e^(D*x^2)*e^(I*x*cos(T)))
cleaner.
-
- Posts: 1439
- Joined: Wed Jul 14, 2010 5:27 pm
Re: Help Needed with an Integration
what happens when you change the limit of integration - R to what the bound would be at a hypothetical voltage (say, Vh)?
R=(125-Vh)^(1/2)*B(1/2)
R^2/B=125-Vh
125-R^2/B=Vh
and then you'd integrate with respect to dVh*d(theta) over 0<Vh<Va or Va<Vh<infinity, depending on whether Va<Vh
also curious if you change the limits from polar to cartesian. i.e. change dRdT to dXdY, thus turning the Rcos(T) term into simply "X".
Edit: dVh/dR = -2R/B = -2(125-Vh)^(1/2)*B^(-1/2)
So to convert the integral to with respect to dVh, one has to "absorb" the right side of the equation, which is the same thing as dividing by the right side times a dR, and multiplying by Vh ( thus multiplying by x/x = 1)....
And then we substitute the R's with what they are in Terms of Vh, and see if we can simplify
Just thinking out loud here.
R=(125-Vh)^(1/2)*B(1/2)
R^2/B=125-Vh
125-R^2/B=Vh
and then you'd integrate with respect to dVh*d(theta) over 0<Vh<Va or Va<Vh<infinity, depending on whether Va<Vh
also curious if you change the limits from polar to cartesian. i.e. change dRdT to dXdY, thus turning the Rcos(T) term into simply "X".
Edit: dVh/dR = -2R/B = -2(125-Vh)^(1/2)*B^(-1/2)
So to convert the integral to with respect to dVh, one has to "absorb" the right side of the equation, which is the same thing as dividing by the right side times a dR, and multiplying by Vh ( thus multiplying by x/x = 1)....
And then we substitute the R's with what they are in Terms of Vh, and see if we can simplify
Just thinking out loud here.
-
- Posts: 1439
- Joined: Wed Jul 14, 2010 5:27 pm
Re: Help Needed with an Integration
When I try to visualize it, R is a radius. It's 3-d, though. The detector is directional in nature. Theta is the angle from the line that represents its direction.
There's two functions here, the e^ part is a "transmission" function, so to speak - the part in the ^() is about how much information is lost as the distance it has to travel increases. And you'll notice one term is r^2 - a function if the surface area. And d is negative, so it's a decreasing one. I'm curious what e^(-d) is. And then there's a term that shows better ( or worse) transmission along the direction of the probe.
The other function - the one not in the e - is related to the data actually being transmitted - what it's value is at that point in space - it's the "original" that gets copied at exponentially decaying strength with each copy. And you'll notice it's close to r^2 - the surface area, and get asymptotically closer as r increases.
There's two functions here, the e^ part is a "transmission" function, so to speak - the part in the ^() is about how much information is lost as the distance it has to travel increases. And you'll notice one term is r^2 - a function if the surface area. And d is negative, so it's a decreasing one. I'm curious what e^(-d) is. And then there's a term that shows better ( or worse) transmission along the direction of the probe.
The other function - the one not in the e - is related to the data actually being transmitted - what it's value is at that point in space - it's the "original" that gets copied at exponentially decaying strength with each copy. And you'll notice it's close to r^2 - the surface area, and get asymptotically closer as r increases.
-
- Posts: 1439
- Joined: Wed Jul 14, 2010 5:27 pm
Re: Help Needed with an Integration
I want to change the limits of integration show that the part in the e^() is constant, and thus I can drop it. But that means moving the dx's up there, which is well, unconventional. And it involves the chain rule somehow.