@ecterrab 

Hello, I am sorry for replying so late. 

The desired output should be:

Details can be found in: https://physics.stackexchange.com/questions/93157/variation-of-modified-einstein-hilbert-action where some terms have been ignored. 

Thank you for taking a look into this. 

Just out of curiousity what are the exact ODEs you are trying to solve? 

@acer 

It does do rather interesting things in the negative region..something I had not thought about.

Is there any portion of the code that does deal with stepsize/speed? Not that I want to "speed up" the process just more of a curiousity on how it works. 

The positive result is also interesting, not what I expected. I will keep tinkering with the parameter ranges and see if anything can occur. I appreciate this worksheet, it appears my attempt was far too simplistic. 

Thanks!

@acer 

As mentioned in my comment above I am still have issues with CW suggestion and still recieving errors. This is a much more involed solution then I was expecting although much more elegant than my attempts. I have some questions however, some may be minor in nature. 

This method replaces my looping attempt by plotting the region of which the inequlaity you is satisfied in blue by examining the region alpha=-1.5..0 and beta -2..0 in step sizes of epsilon =e^-5? Is that the correct understanding?

This reason I ask if because I am interested in regions of positive alpha and beta only and switching to alpha=0..1.5 and beta=0..2 and running it has yet to produce a result after approximately 15 minutes real time. So if i wanted to "speed up" could I make the epsilon larger at the cost of precision? 

Thanks

@Christian Wolinski 

Using this substitution I still recieve an error - for some reason I cannot upload worksheet contents only the worksheet link itself. It appears to not be able to actually compute the integration as I recieve the new error in the modified sheet below. 

NestedError_Mod.mw

@Axel Vogt 

In your language I suppose the zero. In general H is less then G and that is why I want to find when H>G but writiing it as you do then I would want to find when f would be zero. 

My H is

Int(alpha^(3/2)*exp(-1/2*erf(1/2*sqrt(-2*alpha)*t)^2/erf(1/2*sqrt(-2*alpha)*b)^2 - 1/2*alpha*t^2)*I/(Pi*erf(1/2*sqrt(-2*alpha)*b)), t = -infinity .. infinity)and G is

and G is

-alpha/(4*Pi*erf(sqrt(-2*alpha)*b/2)^2)

alpha is in (10^(-9),1) and b is in (1,10^3). My procedure is found in LogLogProper.mw 

I am happy with what it returns and it does it quick, I just don't know if it is the best approach. 

If there is a better way to do it and you want to ammend it I appreciate it. Thank you. 

@acer 

I did not believe the end game was entirely prevelant for this question, I was originally just interested in the reason for the numeric integration issue but things spiraled from there. 

I also want to avoid the "do my work for me" reputation on this forum, all though I understand that this may be hard for a new user. 

Final sheet attached showing what I want to do. Nonetheless this is simply what I want to find: the lowest b value for a given alpha such that the inequality H(alpha,b)>G(alpha,b) is satisfied where H is some complicated numerical integral and G is just some expression containing both alpha and b that can be evlauted exactly. 

How I would exactly write a procedure I am not quite sure, maybe I will work on that next but this code seems to run smoothly. My only ammend I would like to do is the once the b value is found a given alpha, when bailing out of the inner loop to the next alpha the inner loop starts at previous b value instead of running over my entire loop again. 

Is there a simple procedure I could impliment for this? I assume there is and if I sat down with it I could probably get it quite easily. 

Thanks. 

LogLogProper.mw

In using both integration methods that were provided by  @acer and @mmcdara it can be seen that the first point in the loop is not the same in the files:

ErrorRespone.mw - acer

timeinttest.mw - mmcdara

Increasing the bound on the integration with z-substitiuion(mmcdara) will bring the value closer to that obtained through the infinite time integral(acer), however when I set z=1 Maple reports an error. An interesting feature as this does not arise in the integral that sticks with the variable t. 

@acer 

I took your procedure and implimented it similar to what I mentioned above and that being in a loop, but I was only able to get it to run - as seen in the file - by doing it as presented. 

Is this bad Maple procedure? Is there something less convoluted? This is also a very quick method and I am glad you provided the quick fix. 

Thanks

ErrorRespone.mw

@mmcdara 

Thank you for this respone, I am slightly embarassed I didn't just change the integral myself, I suppose I was wanting to rely on Maple instead of my own brain power. 

Regarding my title, I am interested in running this integral in a doube loop for various values of alpha and b. Since I am doing this the width of this "gaussian" like function will change drastically so I put the bounds as +/- infinity to get the most accurate asnwer. Here is a rough file of what I am interested doing (implimentating your change). 

timeinttest.mw

Thanks

If you have a worksheet you are working with I could respond and help out from there. 

@maryam sadeghi

The figure is extremely small, can you try to edit your post to include a larger sized photo? 

What exactly are you trying to solve here? You are clearly missing an ODE if you have initial condtions, have you tried anything yet? If you post an worksheet with your progress/attempts it may get more responses. 

As well you are defining a function which depends on the functions itself, was that just suppose to be an equal sign? 

Many questions need answers before a viable solution can be attempted. 

@acer Thank you for this thorough explanation.

I am particuarily new to Maple for more sophistcated computations/procedures/code writing so I appreciate this little lesson. Particularily the digits aspect was something I was completely unaware of.

As I will have to do some more complicated computations that are similar to this example I gave I will be make all my further "codes" off of this template. 

Thank you again for all of your help!