Table 1 A comparison between the solutions obtained by Ege and Misirli²⁸ and the solutions obtained by our study for the space–time fractional nonlinear Klein–Gordon equation using the IBSEF method.

If \(\alpha \ne \beta , a = 1,b = 2, p_{0} = 1, q_{0} = 2,\tau = 1,\) then (4.1.9) solution becomes:

\(V_{{1_{1} }} \left( {x,t} \right) = \frac{{\sqrt 2 + \frac{2\beta \sqrt 2 }{{\alpha \left( { - \frac{\beta }{\alpha } + \frac{1}{{e^{{2\alpha \left( {\frac{m}{\gamma }\left( {x + \frac{1}{{{{\Gamma \gamma }}}}} \right)^{\gamma } + \frac{k}{\gamma }\left( {t + \frac{1}{{{{\Gamma \gamma }}}}} \right)^{\gamma } } \right)}} }}} \right)}}}}{2}.\)

If \(\alpha \ne \beta , a = 1,b = 2, p_{0} = 1, q_{0} = 2, q_{1} = 1,\tau = 1,\) then (4.1.15) solution becomes:

\(V_{{1_{5} }} \left( {x,t} \right) = \frac{{\sqrt 2 + \frac{\sqrt 2 }{{2\left( {\sqrt { - \frac{\beta }{\alpha } + \frac{1}{{e^{{2\alpha \left( {\frac{m}{\gamma }\left( {x + \frac{1}{{{{\Gamma \gamma }}}}} \right)^{\gamma } + \frac{k}{\gamma }\left( {t + \frac{1}{{{{\Gamma \gamma }}}}} \right)^{\gamma } } \right)}} }}} } \right)}}}}{{2 + \frac{1}{{\left( {\sqrt { - \frac{\beta }{\alpha } + \frac{1}{{e^{{2\alpha \left( {\frac{m}{\gamma }\left( {x + \frac{1}{{{{\Gamma \gamma }}}}} \right)^{\gamma } + \frac{k}{\gamma }\left( {t + \frac{1}{{{{\Gamma \gamma }}}}} \right)^{\gamma } } \right)}} }}} } \right)}}}}.\)

If we put \(b=c=1,\) in solution (39) then solution \({u}_{5}(x, t)\) becomes:

\(u_{5} \left( {x, t} \right) = \left( { - 1 + 2\frac{1}{{1 + e^{{\omega x + \frac{{\left( {\sqrt { - 2 + \omega^{2} } } \right)t^{\alpha } }}{{{\Gamma }\left( {1 + \alpha } \right)}}}} }}} \right).\) If we put \(b = c = 1,\) in solution (42) then solution \(u_{5} \left( {x, t} \right)\) becomes:

\(u_{7} \left( {x, t} \right) = \left( { - 1 + 2\frac{1}{{1 + e^{{\omega x - \frac{{\left( {\sqrt { - 2 + \omega^{2} } } \right)t^{\alpha } }}{{{\Gamma }\left( {1 + \alpha } \right)}}}} }}} \right).\)