Table 3 Represent the solutions of the flow problem utilizing Stehfest’s algorithms with \(\hat{G}_{T} = 5,\tilde{R} = 5,\ddot{\varphi } = 0.04,\hat{G}_{C} ,Sc = 0.7,\dddot M = 10,Pr = 25,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_{1} = 0.5,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_{2} = 0.3,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{K} = 5,\tilde{\alpha } = 0.5,\tilde{\beta } = 0.3.\)

From: Comparative study of heat and mass transfer of generalized MHD Oldroyd-B bio-nano fluid in a permeable medium with ramped conditions

\(\tilde{\alpha }\)

\(\tilde{\beta }\)

\(\tilde{R}\)

\(\ddot{\varphi }\)

Stehfest’s \(\tilde{\Phi }\) (C)

Stehfest’s \(\tilde{\Phi }\) (CF)

Stehfest’s \(\tilde{\Phi }\) (AB)

Stehfest’s \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Theta }\)(C)

Stehfest’s \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Theta }\)(CF)

Stehfest’s \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Theta }\)(AB)

Stehfest’s \(\tilde{U}^{ * }\)(C)

Stehfest’s \(\tilde{U}^{ * }\)(CF)

Stehfest’s \(\tilde{U}^{ * }\)(AB)

0.5

0.3

5

.04

0.11320

0.12310

0.14109

0.02842

0.03561

0.04606

0.06093

0.06291

0.07416

0.9

0.3

5

.04

0.06970

0.07434

0.08292

0.00779

0.01151

0.01455

0.02801

0.03097

0.03550

0.5

0.3

5

.04

-

-

-

-

-

-

0.06093

0.06291

0.07416

0.5

0.5

5

.04

-

-

-

-

-

-

0.06104

0.06300

0.07422

0.5

0.3

3

.04

0.07253

0.08206

0.09785

-

-

-

0.04402

0.04697

0.05735

0.5

0.3

5

.04

0.11320

0.12310

0.14109

-

-

-

0.06093

0.06291

0.07416

0.5

0.3

5

.02

0.17640

0.18555

0.20435

-

-

-

0.08720

0.08724

0.09884

0.5

0.3

5

.04

0.11320

0.12310

0.14109

-

-

-

0.06093

0.06291

0.07416

\(\Pr\)

\(Sc\)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_{1}\)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\lambda } }_{2}\)

Stehfest’s \(\tilde{\Phi }\) (C)

Stehfest’s \(\tilde{\Phi }\) (CF)

Stehfest’s \(\tilde{\Phi }\) (AB)

Stehfest’s \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Theta }\)(C)

Stehfest’s \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Theta }\)(CF)

Stehfest’s \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Theta }\)(AB)

Stehfest’s \(\tilde{U}^{ * }\)(C)

Stehfest’s \(\tilde{U}^{ * }\)(CF)

Stehfest’s \(\tilde{U}^{ * }\)(AB)

10

0.3

0.5

0.3

0.19779

0.20649

0.22506

-

-

-

0.09611

0.09544

0.10694

25

0.3

0.5

0.3

0.11320

0.12310

0.14109

-

-

-

0.06093

0.06291

0.07416

25

0.3

0.5

0.3

-

-

-

0.07938

0.08906

0.10534

0.08225

0.08367

0.09719

25

0.7

0.5

0.3

-

-

-

0.02842

0.03561

0.04606

0.06093

0.06291

0.07416

25

0.3

0.5

0.3

-

-

-

-

-

-

0.06093

0.06291

0.07416

25

0.3

4

0.3

-

-

-

-

-

-

0.05969

0.06200

0.07335

25

0.3

0.5

0.3

-

-

-

-

-

-

0.06093

0.06291

0.07416

25

0.3

0.5

0.7

-

-

-

-

-

-

0.06161

0.06339

0.07456

\(\hat{G}_{T}\)

\(\hat{G}_{C}\)

\(\dddot M\)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{K}\)

Stehfest’s \(\tilde{U}^{ * }\)(C)

Stehfest’s \(\tilde{U}^{ * }\)(CF)

Stehfest’s \(\tilde{U}^{ * }\)(AB)

1

5

10

5

0.02242

0.02415

0.02973

5

5

10

5

0.06093

0.06291

0.07416

5

1

10

5

0.05071

0.05134

0.05927

5

5

10

5

0.06093

0.06291

0.07416

5

5

3

5

0.15145

0.12570

0.14778

5

5

10

5

0.06093

0.06291

0.07416

5

5

10

3

0.06043

0.06243

0.07357

5

5

10

5

0.06093

0.06291

0.07416

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