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.
 

Related to Solving this first-order differential equation for neutron abundance

1. What is a first-order differential equation in the context of neutron abundance?

A first-order differential equation in the context of neutron abundance is an equation that relates the rate of change of neutron abundance to the current abundance and possibly other variables. It typically takes the form of dy/dt = f(y, t), where y represents neutron abundance and t represents time.

2. What initial conditions are required to solve a first-order differential equation for neutron abundance?

To solve a first-order differential equation for neutron abundance, you need an initial condition that specifies the neutron abundance at a particular time, usually denoted as y(t0) = y0. This initial value allows you to integrate the differential equation and find a specific solution.

3. What methods can be used to solve a first-order differential equation for neutron abundance?

Several methods can be used to solve a first-order differential equation for neutron abundance, including analytical methods like separation of variables and integrating factors, as well as numerical methods such as Euler's method, Runge-Kutta methods, and finite difference methods. The choice of method depends on the form of the differential equation and the desired accuracy.

4. How does neutron abundance affect the behavior of the differential equation?

Neutron abundance affects the behavior of the differential equation by influencing the rate of change of neutron population. For example, in nuclear reactors or astrophysical settings, neutron abundance can impact reaction rates, absorption, and decay processes. The functional form of the differential equation will reflect these dependencies, affecting how neutron abundance evolves over time.

5. Can external factors influence the solution of the differential equation for neutron abundance?

Yes, external factors such as temperature, pressure, presence of other particles, and external radiation fields can influence the solution of the differential equation for neutron abundance. These factors can alter reaction rates, absorption cross-sections, and other parameters within the differential equation, thereby impacting the neutron abundance dynamics.

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