Exact separable solutions of two-dimensional time-fractional nonlinear biological population model under the regularized Prabhakar and ψ-Hilfer fractional-order derivatives

The main aim of this work is to investigate how to compute the exact separable solutions using the invariant subspace method for the fractional-order time derivative of two-dimensional nonlinear partial differential equations involving two space and one-time variables under two different fractional-order derivative definitions, namely the regularized Prabhakar and $\psi$-Hilfer fractional-order derivatives. We also explicitly demonstrate the importance and usefulness of the method of the invariant subspace approach in computing the exact separable solutions for the two-dimensional fractional-order time derivative of the biological population model, which helps more accurately predict how populations will grow or shrink. More specifically, we show systematically how to compute the linear spaces for the above-mentioned model with the help of the invariant subspace approach. Furthermore, the computations of exact separable solutions are investigated for the linear and nonlinear biological population models under the above-mentioned two different time fractional-order derivatives with the help of the computed invariant linear spaces. Additionally, we notice that the computed solutions of the considered equations under the $\psi$-Hilfer fractional derivative are valid under the $\psi$-Riemann–Liouville, $\psi$-Caputo, Hilfer, Katugampola, Caputo–Katugampola, Riemann–Liouville, and Caputo fractional derivatives because the $\psi$-Hilfer fractional derivative is a generalization of those fractional derivatives. Also, note that the computed exact separable solutions to the underlying equation under two fractional-order derivatives are expressed in terms of trigonometric, exponential, and polynomial functions with two or three parameters of Mittag-Leffler functions. In addition, the obtained solutions under different fractional-order derivatives are compared with two-dimensional (2D) graphical representations. Finally, the exact separable solutions are presented for the initial and boundary value problems (IBVPs) of the discussed model under various fractional-order derivatives and their comparison.