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Table 3 Comparison solution by MDTM, FDM, and shooting method with GM, LSM, and CM [32]) for θ (η) when \(\varphi\) = 0.01, δ = 1, P = 0, Ha = 3, Rd = 2, Ec = 1, Pr = 6.2, and \(\alpha\) = 0

From: Radiative MHD flow of Rivlin–Ericksen nanofluid of grade three through porous medium with uniform heat source

η

Present θ (η)

θ (η) [32]

MDTM

FDM

Shooting method

G.M

L.S.M

C.M

 − 1

0.5

0.5

0.5

0.5

0.5

0.5

 − 0.9

0.45048334393744

0.4500557057

0.449970142

0.449982

0.449947

0.449999

 − 0.8

0.400767395390827

0.40020133769

0.399959597

0.399969

4.00E-01

0.399998

 − 0.7

0.350989746391596

0.35039626873

0.349957975

0.349962

0.349892

0.349997

 − 0.6

0.301201167815963

0.30061028898

0.299961713

0.29996

0.299885

0.299997

 − 0.5

0.251410810147527

0.25082103146

0.249968222

0.249961

0.24989

0.249997

 − 0.4

0.201610063585561

0.20101206823

0.199975753

0.199966

0.199905

0.199997

 − 0.3

0.151784736266043

0.15117151419

0.149983257

0.149973

0.149926

0.149998

 − 0.2

0.101920941655585

0.10129101917

0.099990246

0.099983

0.099954

0.099998

 − 0.1

0.052007749159865

0.05136506198

0.049996659

0.049994

0.049985

0.049999

0

0.002038317205475

0.00139048604

2.7177E-06

4.73 × 10−6

1.74 × 10−5

2.05 × 10−8

0.1

 − 0.04798969711509

 − 0.04863376327

 − 0.049991207

 − 0.04998

 − 0.04995

 − 0.05

0.2

 − 0.09807401948685

 − 0.09870672228

 − 0.099984735

 − 0.09997

 − 0.09992

 − 0.10

0.3

 − 0.14820784031401

 − 0.14882532079

 − 0.149977617

 − 0.14996

 − 0.14989

 − 0.15

0.4

 − 0.19838020932551

 − 0.19898411506

 − 0.199969867

 − 0.19996

 − 0.19988

 − 0.20

0.5

 − 0.24857715789604

 − 0.24917481927

 − 0.249961909

 − 0.24995

 − 0.24986

 − 0.25

0.6

 − 0.29878442328292

 − 0.29938559613

 − 0.299954708

 − 0.29995

 − 0.29986

 − 0.30

0.7

 − 0.3489934561497

 − 0.34960004659

 − 0.34994991

 − 0.34996

 − 0.34987

 − 0.35

0.8

 − 0.3992138754586

 − 0.39979581311

 − 0.399949984

 − 0.39997

 − 0.3999

 − 0.40

0.9

 − 0.44949797889672

 − 0.44994267864

 − 0.449958355

 − 0.44998

 − 0.44994

 − 0.45

1

 − 0.4999868653492

 − 0.5

 − 0.5

 − 0.5

 − 0.5

 − 0.5