Table 1 Comparison between Zayed and Al-Joudi’s solutions and solutions obtained in this article

If μ < 0, case 1 yields

\(\begin{aligned} u_{1,1} (\xi ) &= \pm \sqrt {\frac{6\,\mu }{\sigma \,\alpha }} \, \hfill \\ & \quad \times \,\sqrt {\sigma \left[ {1 - \,\left( {\frac{{A\,\sinh (\sqrt { - \mu } \,\xi ) + B\,\cosh (\sqrt { - \mu } \,\xi )}}{{A\,\cosh (\sqrt { - \mu } \,\xi ) + B\,\sinh (\sqrt { - \mu } \,\xi )}}} \right)^{2} } \right]} \hfill \\ \end{aligned}\)

If σ = ± 1 and μ < 0 solution (3.10) becomes

\(\begin{aligned} u(\xi ) & = \pm \sqrt {\frac{6\,\mu }{\sigma \,\alpha }} \hfill \\ & \quad \times \,\,\sqrt {\sigma \left[ {1 - \,\left( {\frac{{A\,\sinh (\sqrt { - \mu } \,\xi ) + B\,\cosh (\sqrt { - \mu } \,\xi )}}{{A\,\cosh (\sqrt { - \mu } \,\xi ) + B\,\sinh (\sqrt { - \mu } \,\xi )}}} \right)^{2} } \right]} \hfill \\ \end{aligned}\)

If λ = 0 and μ < 0, case 2 yields

\(u_{2,1} (\xi ) = \pm \sqrt {\frac{ - 6\,\mu }{\alpha }} \,\left( {\frac{{A\,\sinh (\sqrt { - \mu } \,\xi ) + B\,\cosh (\sqrt { - \mu } \,\xi )}}{{A\,\cosh (\sqrt { - \mu } \,\xi ) + B\,\sinh (\sqrt { - \mu } \,\xi )}}} \right)\)

If σ = ±1 and μ < 0 solution (3.9) becomes

\(u(\xi ) = \pm \sqrt {\frac{ - 6\,\mu }{\alpha }} \,\left( {\frac{{A\,\sinh (\sqrt { - \mu } \,\xi ) + B\,\cosh (\sqrt { - \mu } \,\xi )}}{{A\,\cosh (\sqrt { - \mu } \,\xi ) + B\,\sinh (\sqrt { - \mu } \,\xi )}}} \right)\)

If μ < 0, Case 3 yields

\(\begin{aligned} u_{3,1} (\xi ) = & \pm \sqrt {\frac{3}{2\,\alpha }} \,\left\{ {\sqrt \mu \,\sqrt {\,\left[ {1 - \left( {\frac{\begin{aligned} A\,\sinh (\sqrt { - \mu } \,\xi ) \hfill \\ + B\,\cosh (\sqrt { - \mu } \,\xi ) \hfill \\ \end{aligned} }{\begin{aligned} A\,\cosh (\sqrt { - \mu } \,\xi ) \hfill \\ + B\,\sinh (\sqrt { - \mu } \,\xi ) \hfill \\ \end{aligned} }} \right)^{2} } \right]} } \right. \hfill \\ & \quad + \left. {\sqrt { - \mu } \,\left( {\frac{{A\,\sinh (\sqrt { - \mu } \,\xi ) + B\,\cosh (\sqrt { - \mu } \,\xi )}}{{A\,\cosh (\sqrt { - \mu } \,\xi ) + B\,\sinh (\sqrt { - \mu } \,\xi )}}} \right)} \right\} \hfill \\ \end{aligned}\)

If σ = ±1 and μ < 0 solution (3.11) becomes

\(\begin{aligned} u(\xi ) = \pm \sqrt {\frac{3}{2\,\alpha }} \,\left\{ {\sqrt \mu \,\sqrt {\,\left[ {1 - \left( {\frac{\begin{aligned} A\,\sinh (\sqrt { - \mu } \,\xi ) \hfill \\ + B\,\cosh (\sqrt { - \mu } \,\xi ) \hfill \\ \end{aligned} }{\begin{aligned} A\,\cosh (\sqrt { - \mu } \,\xi ) \hfill \\ + B\,\sinh (\sqrt { - \mu } \,\xi ) \hfill \\ \end{aligned} }} \right)^{2} } \right]} } \right. \hfill \\ + \left. {\sqrt { - \mu } \,\left( {\frac{{A\,\sinh (\sqrt { - \mu } \,\xi ) + B\,\cosh (\sqrt { - \mu } \,\xi )}}{{A\,\cosh (\sqrt { - \mu } \,\xi ) + B\,\sinh (\sqrt { - \mu } \,\xi )}}} \right)} \right\} \hfill \\ \end{aligned}\)

When μ > 0, case 1 yields

\(\begin{aligned} u_{1,5} (\xi ) = \pm \sqrt {\frac{6\,\mu }{\sigma \,\alpha }} \, \hfill \\ \times \,\sqrt {\sigma \left[ {1 - \,\left( {\frac{ - A\,\sin (\sqrt \mu \,\xi ) + B\,\cos (\sqrt \mu \,\xi )}{A\,\cos (\sqrt \mu \,\xi ) + B\,\sin (\sqrt \mu \,\xi )}} \right)^{2} } \right]} \hfill \\ \end{aligned}\)

If σ = ±1 and μ > 0 solution (3.13) becomes

\(\begin{aligned} u(\xi ) = \pm \sqrt {\frac{6\,\mu }{\sigma \,\alpha }} \, \hfill \\ \times \,\sqrt {\sigma \left[ {1 - \,\left( {\frac{{ - A\,\sin \left( {\sqrt \mu \,\xi } \right) + B\,\cos \left( {\sqrt \mu \,\xi } \right)}}{{A\,\cos \left( {\sqrt \mu \,\xi } \right) + B\,\sin \left( {\sqrt \mu \,\xi } \right)}}} \right)^{2} } \right]} \hfill \\ \end{aligned}\)

If λ = 0 and μ > 0, case 2 yields

\(u_{2,2} (\xi ) = \pm \sqrt {\frac{6\,\mu }{\alpha }} \,\left( {\frac{ - A\,\sin (\sqrt \mu \,\xi ) + B\,\cos (\sqrt \mu \,\xi )}{A\,\cos (\sqrt \mu \,\xi ) + B\,\sin (\sqrt \mu \,\xi )}} \right)\)

If σ = ±1 and μ > 0 solution (3.12) becomes

\(u(\xi ) = \pm \sqrt {\frac{6\,\mu }{\alpha }} \,\left( {\frac{ - A\,\sin (\sqrt \mu \,\xi ) + B\,\cos (\sqrt \mu \,\xi )}{A\,\cos (\sqrt \mu \,\xi ) + B\,\sin (\sqrt \mu \,\xi )}} \right)\)

If μ > 0, case 3 yields

\(\begin{aligned} u_{3,2} (\xi ) = \pm \sqrt {\frac{3}{2\,\alpha }} \,\left\{ {\sqrt \mu \,\sqrt {\,\left[ {1 - \left( {\frac{\begin{aligned} - A\,\sin (\sqrt \mu \,\xi ) \hfill \\ + B\,\cos (\sqrt \mu \,\xi ) \hfill \\ \end{aligned} }{\begin{aligned} A\,\cos (\sqrt \mu \,\xi ) \hfill \\ + B\,\sin (\sqrt \mu \,\xi ) \hfill \\ \end{aligned} }} \right)^{2} } \right]} } \right. \hfill \\ + \left. {\sqrt \mu \,\left( {\frac{ - A\,\sin (\sqrt \mu \,\xi ) + B\,\cos (\sqrt \mu \,\xi )}{A\,\cos (\sqrt \mu \,\xi ) + B\,\sin (\sqrt \mu \,\xi )}} \right)} \right\} \hfill \\ \end{aligned}\)

If σ = ±1 and μ > 0 solution (3.14) becomes

\(\begin{aligned} u(\xi ) = \pm \sqrt {\frac{3}{2\,\alpha }} \,\left\{ {\sqrt \mu \,\sqrt {\,\left[ {1 - \left( {\frac{\begin{aligned} - A\,\sin (\sqrt \mu \,\xi ) \hfill \\ + B\,\cos (\sqrt \mu \,\xi ) \hfill \\ \end{aligned} }{\begin{aligned} A\,\cos (\sqrt \mu \,\xi ) \hfill \\ + B\,\sin (\sqrt \mu \,\xi ) \hfill \\ \end{aligned} }} \right)^{2} } \right]} } \right. \hfill \\ + \left. {\sqrt \mu \,\left( {\frac{ - A\,\sin (\sqrt \mu \,\xi ) + B\,\cos (\sqrt \mu \,\xi )}{A\,\cos (\sqrt \mu \,\xi ) + B\,\sin (\sqrt \mu \,\xi )}} \right)} \right\} \hfill \\ \end{aligned}\)