Solving this first-order differential equation for neutron abundance

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
  • #1
gurbir_s
11
4
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
[2]: https://ui.adsabs.harvard.edu/abs/1966ApJ...146..542P/abstract
 
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  • #2
hello @gurbir_s ,
:welcome: ##\qquad ## !​

It seems ([edit]: :wink: (*) ) 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:
  • #3
BvU said:
hello @gurbir_s ,
:welcome: ##\qquad ## !​

It seems ([edit]: :wink: (*) ) 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.
 

1. What is a first-order differential equation?

A first-order differential equation is an equation that involves a function and its derivative (or rate of change). In this context, it is used to describe the relationship between the neutron abundance and time.

2. Why is it important to solve this equation for neutron abundance?

The neutron abundance is a key factor in understanding the behavior and evolution of nuclear reactions. By solving this equation, we can better predict and control these reactions, which has implications in fields such as nuclear energy and astrophysics.

3. What are the steps involved in solving a first-order differential equation?

The first step is to identify the type of differential equation and determine the appropriate method for solving it. This may involve separation of variables, using an integrating factor, or other techniques. Next, we integrate the equation and apply any initial or boundary conditions to find the specific solution. Finally, we may need to verify the solution and make any necessary adjustments.

4. What factors can affect the accuracy of the solution to this differential equation?

The accuracy of the solution can be affected by the complexity of the equation, the chosen method for solving it, and the accuracy of the initial or boundary conditions. Additionally, any simplifying assumptions made in the process may also impact the accuracy of the solution.

5. How can the solution to this differential equation be applied in real-world situations?

The solution to this differential equation can be used in various applications, such as studying the behavior of nuclear reactions in a controlled environment, predicting the evolution of neutron abundance in stars, or developing more accurate models for nuclear energy production. It can also help in understanding the effects of radiation on biological systems and developing radiation shielding techniques.

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