Method of separation variables combined with homogenous balanced principle for searching exact solutions of nonlinear time-fractional biological population model

Separation variable method combined with integral bifurcation method for solving time-fractional reaction–diffusion models

“…where (21) contains many timefractional nonlinear models such as time-fractional reaction-diffusion models (Sahadevan and Prakash 2016;Odibat and Shaher 2009;Rui 2017), time-fractional biological population models (Wu and Rui 2018), time-fractional dispersive (or diffusion-wave) equations (Rui 2018b), time-fractional heat transfer models (Blasiak 2016), and so forth, which play very important roles in a range of scientific fields, and all of them can be used to accurately describe nonlinear phenomena. In addition, some time-fractional differential equations can be used to describe subjects with phenomenon of memory (Manzoor et al 2017;Tarasov and Tarasova 2018) and real-time problems such as real-time simulation (Tepljakov et al 2012), and real-time control and implementation (Efe 2011;Yang et al 2018).…”

“…In Rui (2018a), using (2.3), two kinds of exact solutions (3.88) and (3.89) of another time-fractional diffusion-convection model have been obtained. In Wu and Rui (2018), under a 0 = 0, a 1 = 1, using (2.3), we obtained three exact solutions (3.67), (3.68), and (3.70) of a time-fractional biological population model in the condition r > 0. In Rui (2018b), using (2.3), five exact solutions (3.28), (3.60), (3.62), (3.68), and (3.69) of time-fractional Hunter-Saxton equation and time-fractional Li-Olver equation have been obtained.…”

“…By substituting (2.4) into (2.1) and using the formula (2.6), Eq. (2.1) can be reduced to a nonlinear ordinary differential equation of v(x) with coefficients t. As in Rui (2018a) and Wu and Rui (2018), using homogenous balanced principle, we make the power of t of all terms to equal in the reduced nonlinear ODE; therefore, we can determine the value of γ in the range of γ > −1. And then, plugging the value of γ into the reduced nonlinear ODE again, and dividing out the variable t, we can obtain a nonlinear ODE with constant coefficients as follows:…”

“…In fact, Jumarie's fractional chain rule has been verified that it is invalid under the Riemann-Liouville definition and Caputo definition, see the Tarasov (2016) , He (2012), and Rui (2017). To avoid this invalid fractional chain rule, based on the method of separation variables and combined with the homogenous balanced principle or integral bifurcation method, we introduced a series of combination methods (Rui 2017(Rui , 2018aWu and Rui 2018;Rui 2018b) for searching exact solutions of nonlinear time-fractional partial differential equations (tfPDEs). In this paper, based our previous works, we will improve a computational method for investigating exact solutions of tfPDEs.…”

“…Investigations of exact solutions or approximate analytical solutions of the FDEs are very important because it can help people to understand the mechanism and complexity of the fractional nonlinear phenomena. Next, as in Rui (2017), Rui (2018a), Wu and Rui (2018) and Rui (2018b), using separation variable method combined with integral bifurcation method, we will investigate exact solutions and their dynamical properties of the following time-fractional reaction-diffusion models:…”