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On the Method of Inverse Mapping for Solutions of Coupled Systems of Nonlinear Differential Equations Arising in Nanofluid Flow, Heat and Mass Transfer

Mangalagama Dewasurendra

Mangalagama Dewasurendra

Department of Mathematics, University of Central FloridaOrlando, USA
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Dewasurendra, Mangalagama
 and 
Kuppalapalle Vajravelu

Kuppalapalle Vajravelu

Department of Mathematics, University of Central FloridaOrlando, USA
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Vajravelu, Kuppalapalle
  
Oct 03, 2018
On the Method of Inverse Mapping for Solutions of Coupled Systems of Nonlinear Differential Equations Arising in Nanofluid Flow, Heat and Mass Transfer's Cover Image
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Published Online: Oct 03, 2018
Page range: 1 - 14
Received: Nov 12, 2017
Accepted: Feb 05, 2018
DOI: https://doi.org/10.21042/AMNS.2018.1.00001
Keywords
© 2018 Mangalagama Dewasurendra and Kuppalapalle Vajravelu, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Fig. 1

Flow configuration.
Flow configuration.

Fig. 2

Plot of E(c0,δ), the squared residual error over η ∈ [0,499] as a function of c0 and δ using parameter values Le = 2, Nb = 2,Pr = 1, Nt = 1, n = 0.5, A = 0.1314. The error function has minimum E(c0,δ,A) = 9.71 × 10–5 where c0 = –0.6195 and δ = 0.8462963.
Plot of E(c0,δ), the squared residual error over η ∈ [0,499] as a function of c0 and δ using parameter values Le = 2, Nb = 2,Pr = 1, Nt = 1, n = 0.5, A = 0.1314. The error function has minimum E(c0,δ,A) = 9.71 × 10–5 where c0 = –0.6195 and δ = 0.8462963.

Fig. 3

Plot of E(c0,δ), the squared residual error over η ∈ [0,499] as a function of c0 and δ using parameter values Le = 3, Nb = 1,Pr = 5, Nt = 0, n = 1, A = 7.8902. The error function has minimum E(c0,δ,A) = 9.41 × 10–5 where c0 = –9.30195 and δ = 1.03944.
Plot of E(c0,δ), the squared residual error over η ∈ [0,499] as a function of c0 and δ using parameter values Le = 3, Nb = 1,Pr = 5, Nt = 0, n = 1, A = 7.8902. The error function has minimum E(c0,δ,A) = 9.41 × 10–5 where c0 = –9.30195 and δ = 1.03944.

Fig. 4

Plot of E(c0,δ), the squared residual error over η ∈ [0,499] as a function of c0 and δ using parameter values Le = 2, Nb = 2,Pr = 7, Nt = 0.5, n = 0.8, A = 0.24764. The error function has minimum E(c0,δ,A) = 8.28 × 10–5 where c0 = –0.690605 and δ = 0.8462963.
Plot of E(c0,δ), the squared residual error over η ∈ [0,499] as a function of c0 and δ using parameter values Le = 2, Nb = 2,Pr = 7, Nt = 0.5, n = 0.8, A = 0.24764. The error function has minimum E(c0,δ,A) = 8.28 × 10–5 where c0 = –0.690605 and δ = 0.8462963.

Fig. 5

Plot of f̂(η), where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.
Plot of f̂(η), where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.

Fig. 6

Plot of f̂′(η), where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.
Plot of f̂′(η), where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.

Fig. 7

Plot of θ̂(η), where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.
Plot of θ̂(η), where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.

Fig. 8

Plot of ϕ̂(η), where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.
Plot of ϕ̂(η), where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.

Fig. 9

Comparison of f(η), θ(η) and ϕ(η) obtained by the MDDiM 3-term approximation and shooting method solutions with Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, where Curve 1 is shooting method results of f(η), Curve 2 is MDDiM results of f(η), Curve 3 is shooting method results of θ(η), Curve 4 is MDDiM results of θ(η), Curve 5 is shooting method results of ϕ(η), Curve 6 is MDDiM results of ϕ(η).
Comparison of f(η), θ(η) and ϕ(η) obtained by the MDDiM 3-term approximation and shooting method solutions with Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, where Curve 1 is shooting method results of f(η), Curve 2 is MDDiM results of f(η), Curve 3 is shooting method results of θ(η), Curve 4 is MDDiM results of θ(η), Curve 5 is shooting method results of ϕ(η), Curve 6 is MDDiM results of ϕ(η).

Fig. 10

Plot of Residual Error function verses Terms of approximation, where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.
Plot of Residual Error function verses Terms of approximation, where Curve 1 has Le = 2, Nb = 2, Pr = 1, Nt = 1, n = 0.5, Curve 2 has Le = 3, Nb = 1, Pr = 5, Nt = 0, n = 1, and Curve 3 has Le = 2, Nb = 2, Pr = 7, Nt = 0.5, n = 0.8 using their respective error-minimizing convergence control parameter.

Fig. 11

Plot of |–f̂″(0)| versus n, using Le = 3, Nb = 1, Pr = 5 and Nt = 0.
Plot of |–f̂″(0)| versus n, using Le = 3, Nb = 1, Pr = 5 and Nt = 0.

Fig. 12

Plot of |–θ̂′(0)|, where Curve 1 is |–θ̂′(0)| versus Nt using Le = 3, Nb = 1, Pr = 5, n = 1, Curve 2 is |–θ̂′(0)| versus Nb using Le = 3, Pr = 5, Nt = 0, n = 1.
Plot of |–θ̂′(0)|, where Curve 1 is |–θ̂′(0)| versus Nt using Le = 3, Nb = 1, Pr = 5, n = 1, Curve 2 is |–θ̂′(0)| versus Nb using Le = 3, Pr = 5, Nt = 0, n = 1.

Fig. 13

Plot of |–ϕ̂′(0)|, where Curve 1 is |–ϕ̂′(0)| versus Nt using Le = 2, Nb = 2, Pr = 1, n = 0.5, Curve 2 is |–ϕ̂′(0)| versus Nb using Le = 2, Pr = 1, Nt = 1, n = 0.5.
Plot of |–ϕ̂′(0)|, where Curve 1 is |–ϕ̂′(0)| versus Nt using Le = 2, Nb = 2, Pr = 1, n = 0.5, Curve 2 is |–ϕ̂′(0)| versus Nb using Le = 2, Pr = 1, Nt = 1, n = 0.5.

Minimum of the squared residual error E(A,c0,δ) for three different sets of parameters_

LeNbPrNtnAc0δE(c0,δ,A)
22110.50.1314–0.61950.6739.71 × 10–5
315017.8902–9.30201.03949.71 × 10–5
2270.50.80.2476–0.69060.84638.28 × 10–5
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