Table 11 Different values of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation for the case \(S=0.4\), \(\beta =0.2\), \(\lambda =0.2\), \(A=0.3\), \(N=0.2\), \(M=1\), \(\mathit{Pr} =1\), \(\mathit{Ma}=1\) and several values of \(I_{0}\)

\(I_{0}>0\)

β

\(f''(0)\)

θ(1)

\(-\theta '(0)\)

\(I_{0}<0\)

β

\(f''(0)\)

θ(1)

\(-\theta '(0)\)