Applications of integral bifurcation method together with homogeneous balanced principle on investigating exact solutions of time fractional nonlinear PDEs

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

“…Step 1. Slightly different from the previous hypothetical structure of solutions (Rui 2017(Rui , 2018a; Wu and Rui 2018; Rui 2018b) formed as:…”

“…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:…”

Invariant subspaces and exact solutions for some types of scalar and coupled time-space fractional diffusion equations

“…For this reason, in recent years, both mathematicians and physicists have paid much attention to study the exact and numerical solutions of nonlinear fractional partial differential equations (FPDEs) using various adhoc methods, such as Lie group analysis method [9][10][11][12][13][14][15], Adomian decomposition method [16][17][18], homotopy decomposition method [19], differential transform method [20], function-expansion method [21][22][23] and so on. However, recent investigations have shown that a new analytic method based on the invariant subspace approach provides an effective tool to derive the exact solution of scalar and coupled system of timespace FPDEs.…”