A cosmological boundary value problem

Dear Community,

Recently I have encountered the following system of odes:

where κ and known constants with the boundary conditions , . I would like to construct a proper code by using bvp4c or bvp5c but I am not sure how to handle the boundary conditions.

I would be very grateful for any help you could give me. Have a nice day!

[Jan]

dSdt = S / (3 * G) must fail at G(0) = 0. Then the derivative is not finite and the integration will reply infinite results. This is not a probölemk of Matlab, but a mathematical one.

[James Tursa]

Good point. In fact all three equations have 0/0 at the boundary condition.

[Szilárd Zsóka]

Both of you are right. Numerically I set 0 to 0.01. The problem has an asymptotic power series solution. As then and . Our goal is to find asymptotic solution for the system which cannot be approximated by power series. I have several initial guesses but I am not very good at numerical calculations.

[James Tursa]

Seems like you still have a mathematical problem because asymptotically the rhs has 1/sqrt(t). Sure you can arbitrarily start the problem at 0.01, but does this really have meaning if the actual math doesn't match and tends to infinity? Where are these equations coming from?

[Szilárd Zsóka]

The system is coming from the Einstein-conformally coupled Higgs cosmological model. We want to solve these equation near the initial singularity (the Big Bang). We found all asymptotic power series solution but now we are interested in non-power series solutions.

[Szilárd Zsóka]

By considering the power series solutions and the fact that G can be used as new independent variable I have reformulated the system into the following form:

with and boundary conditions. Can it solved with https://www.mathworks.com/help/matlab/math/solve-bvp-with-singular-term.html? What do you think?