Similarity solutions of inhomogeneous spherically symmetric spacetime with dust fluid and dark energy

In this work, we investigated an inhomogeneous spherically symmetric star consisting of dust fluid and dark energy components. We obtained similarity solutions for the energy density of the inhomogeneous dust fluid \(\rho (t,r)\), dark energy \(\rho _{DE}(t,r)\), and the mass function inside the star m(t, r). The symmetries of Einstein field equations for this model are derived using the symmetry group analysis method, where the Einstein field equations have been reduced into nonlinear ordinary differential equations (NODEs). Some interesting explicit analytical solutions are obtained for the reduced NODEs. Consequently, a new class of similarity solutions has been obtained for the energy density of the inhomogeneous dust fluid and dark energy as functions of coordinates t and r. These solutions are obtained when the star is composed of both dust fluid and dark energy components, dark energy components only, and dust fluid only. When the star is composed of dust fluid only, we discussed the possibility of black hole formation by assuming that the collapsing star is not initially trapped. Graphical representations of some solutions are given and discussed in the last section.