Table 2.

Comparison of the solutions to the time fractional nonlinear model.

Bekir and Guner's [51]solution

The obtained solutions

Particularly, if $C_1\ne 0$, $C_2=0$, $\lambda >0$, $\mu =0$ then the solution of hyperbolic form is $u_{\mathrm{1,2}}\left(x,y,t\right)=\pm \sqrt{r}tanh\frac{\lambda }{2}\left(x+iy+\frac{2h}{\text{Γ}\left(1+\alpha \right)}\sqrt{\frac{r}{{\lambda }^2-4\mu }}{t}^{\alpha }\right)$, Also, the trigonometric solution is $u_{\mathrm{1,2}}\left(x,y,t\right)=\pm \sqrt{r}cot\frac{\lambda }{2}\left(x+iy+\frac{2h}{\text{Γ}\left(1+\alpha \right)}\sqrt{\frac{r}{{\lambda }^2-4\mu }}{t}^{\alpha }\right)$.

For special s, if $A_2=0$, and $A_1\ne 0$, the hyperbolic solution is $v=\sqrt{r}\{coth\left(\sqrt{-\lambda }\left(x+iy-\left(\frac{2h\sqrt{r}}{\sqrt{-\lambda }}\right)\frac{t^{\alpha }}{\text{Γ}\left(1+\alpha \right)}\right)\right)$ $\mp csch\left(\sqrt{-\lambda }\left(x+iy-\left(\frac{2h\sqrt{r}}{\sqrt{-\lambda }}\right)\frac{t^{\alpha }}{\text{Γ}\left(1+\alpha \right)}\right)\right)\}$, and the trigonometric solution is $v=\sqrt{-r}\{cot\left(\sqrt{\lambda }\left(x+iy-\left(\frac{2h\sqrt{-r}}{\sqrt{\lambda }}\right)\frac{t^{\alpha }}{\text{Γ}\left(1+\alpha \right)}\right)\right)$ $\pm csc\left(\sqrt{\lambda }\left(x+iy-\left(\frac{2h\sqrt{-r}}{\sqrt{\lambda }}\right)\frac{t^{\alpha }}{\text{Γ}\left(1+\alpha \right)}\right)\right)\}$.