Computational Physics project using Matlab, I need the answer for it, but I would like to understand how we can code it and explain it with good details because I have to write a journal article about it!!

Consider a radioactive decay problem involving two types of nuclei, A and B, with populations N_A(t) and N_B(t). Suppose that type A nuclei decay to form type B nuclei, which then also decay.

This dual decay is described by the following differential equations:

dNA/ dt = NA/ A

dNB/ dt = NA/ A NB/ B

where _A and _B are the decay time constants for each type of nucleus.

First, normalize this set of equations. You will find that the normalization is very simple. Use the Euler method to solve these coupled equations numerically for N_A(t) and N_B(t) as a function of time. To avoid having to assign too many numerical values, use N_A(0) = 100,N_B(0) = 10, and _A = 1 as the unit of time. Then, obtain the analytical solutions for this system of differential equations by using either the paper-and-pencil method or Mathematica.

Compare your analytical solution with the numerical one. Explore what controls the stability and accuracy of your numerical solution. Describe your findings in your journal article with plots to back up your conclusions.

Examine the three different cases for relative _Aand _B.

(a) _A > _B

(b) _A =_B

( (c) _A<_B

Explore the different behavior of these three cases and when possible highlight the physics controlling this behavior. In particular, try to interpret the short and long time behaviors for different values of the ratio _A/_B. Write up your findings in your journal article, using plots of these three cases to back up your findings. Make sure to include the exact solutions also in your plot to show that the numerical solutions are accurate.

Part II

Consider again the same problem as in Part I, but now suppose that nuclei of type A decay into the ones of type B, while nuclei of type B decay into the ones of type A. Strictly speaking, this is not a decay, since it is possible for the type B nuclei to turn back into type A nuclei. A better analogy would be a resonance in which a system can tunnel or move back and forth between two states A and B which have equal energies. The corresponding rate equations are:

dNA/dt= NB/t - NA/t

dNB/dt= NA/t - NB/t

where for simplicity we assume that the two types of decay are characterized by the same constant

. Solve this system of equations numerically for the numbers of nuclei N_A(t) and N_B(t), with initial conditions N_A(0) = 100 and N_B(0) = 10, and take = 1 as the unit of time. Show that your numerical results are consistent with the idea that the system reaches a steady state in which N_A and N_B are constant. In such a steady state, the time derivatives dN_A/dt and dN_B/dt should vanish.