Table 1 Comparison of the modified hybrid method (5.2) and hybrid method [52] for \(L, S_{1}\in ]0,1[\) and \(a=2.0099\), \(q_{1}=0.0189\), \(q_{2}=1.2994\), \(r=10.5923\), \(E=0.9959\), \(k=1.3997\), \(\beta =98.499\), \(\alpha =2.9999\), \(\delta =0.0384\), \(\rho =10.5842\), \(m_{1}=0.6222\), \(m_{2}=0.4422\), \(h\in (0,1)\)

0.48889569

0<L<0.99288325491279

\(0< S_{1}<0.97880134847096\)

0.58889569

0<L<0.983282763587931

\(0< S_{1}<0.95068201684448\)

0.68889569

0<L<0.97635403882826

\(0< S_{1}<0.93072628972275\)

0.78889569

0<L<0.97111704728074

\(0< S_{1}<0.91582972183557\)

0.88889569

0<L<0.96701917628990

\(0< S_{1}<0.90428485868004\)

0.98889569

0<L<0.96372500134675

\(0< S_{1}<0.89507489723917\)