# Solving this first-order differential equation for neutron abundance

• A
• gurbir_s
In summary: Can I use the Gear method for solving this equation numerically? If yes, could you please provide an example or a reference where I can understand how to use it for this problem? Thank you again for your help!In summary, the time rate of change of neutron abundance ##X_n## is given by the equation $$\frac{dX_n}{dt} = \lambda - (\lambda + \hat\lambda)X_n$$ where ##\lambda## is the neutron production rate per proton and ##\hat\lambda## is the neutron destruction rate per neutron. To calculate ##X_n##, values of ##\lambda## and ##\hat\lambda## at various times are needed. The use of Euler and RK4
gurbir_s
The time rate of change of neutron abundance ##X_n## is given by
$$\frac{dX_n}{dt} = \lambda - (\lambda + \hat\lambda)X_n$$
where ##\lambda## is neutron production rate per proton and ##\hat\lambda## is neutron destruction rate per neutron.
Given the values of ##\lambda## and ##\hat\lambda## at various values of time, I need to calculate ##X_n##.I have also calculated values of ##\lambda 's## at intermediate times. I have tried using Euler method and RK4 method to solve this equation, but the solutions for ##X_n## diverge to inf values.

[Here][2] is the link to the complete research paper "Primordial Helium Abundance and the Primordial Fireball. II" by P.J.E. Peebles.

Any help or idea on how to solve it will be appreciated : ) [1]: Data for ##\lambda 's## https://i.stack.imgur.com/lnW9M.png

hello @gurbir_s ,

It seems (: (*) ) to me you have a differential equation at hand of the so-called 'very stiff' category.
I don't know what tools you have available, but you can try to find an impementation of the Gear method.

(*) the 'primordeal fireball' in the title says it all[edit2]:
A little googling: in https://globaljournals.org/GJSFR_Volume13/2-Numerical-Approach-for-Solving-Stiff.pdf
I find
12. Hindmarsh, A. C. and Gear C.W. (1974), “Ordinary differential equation system solver”, L.L.L. Report UCID -30001, rev. 3, l.l.l. (www.netlib.org/ode/epsode.f)
Good old Fortran !

##\ ##

Last edited:
BvU said:
hello @gurbir_s ,

It seems (: (*) ) to me you have a differential equation at hand of the so-called 'very stiff' category.
I don't know what tools you have available, but you can try to find an impementation of the Gear method.

(*) the 'primordeal fireball' in the title says it all[edit2]:
A little googling: in https://globaljournals.org/GJSFR_Volume13/2-Numerical-Approach-for-Solving-Stiff.pdf
I find Good old Fortran !

##\ ##
Thank you : ) @BvU. I was struggling with this problem from quite a few days.

• Other Physics Topics
Replies
20
Views
2K
• Math Guides, Tutorials and Articles
Replies
1
Views
12K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Engineering and Comp Sci Homework Help
Replies
4
Views
2K