Traveling wave solutions for density-dependent conformable fractional diffusion–reaction equation by the first integral method and the improved $$\textbf{tan}\left( {{\mathbf{\frac{1}{2}}}{\boldsymbol{\varphi }}\left({\boldsymbol{\upxi}} \right)} \right)$$ tan 1 2 φ ξ -expansion method

Abstract

In this paper, we propose the first integral method (FIM) and the improved $$\text{tan}\left( {\frac{1}{2}{\varphi }\left(\upxi \right)} \right)$$ -expansion method (ITEM) for solving the density-dependent conformable fractional diffusion–reaction equation (CFDRE) which is commonly applied in mathematical biology. We received many new exact soliton solutions for the density-dependent CFDRE which are expressed by exponential function, rational function and hyperbolic function forms. The results show that FIM and ITEM are powerful mathematical tools and efficient techniques for solving the fractional nonlinear partial differential equations.