PPubMed24 Nov 2014

Approximate Solutions for Flow with a Stretching Boundary due to Partial Slip

U. Filobello-Nino, H. Vazquez-Leal, A. Sarmiento-Reyes, B. Benhammouda, V. M. Jimenez-Fernandez, D. Pereyra-Diaz, A. Perez-Sesma, J. Cervantes-Perez, J. Huerta-Chua, J. Sanchez-Orea, A. D. Contreras-Hernandez

Abstract

'B. Benhammouda' 'V. M. Jimenez-Fernandez' 'D. Pereyra-Diaz' 'A. Perez-Sesma' 'J. Cervantes-Perez' 'J. Huerta-Chua' 'J. Sanchez-Orea' 'A. D. Contreras-Hernandez'] The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient.

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Coupling HPM with Laplace‑Padé and Padé yields accurate, efficient approximations for flow over a stretching sheet with partial slip.

What’s new

i.Coupled homotopy perturbation method with Laplace‑Padé (LPHPM) for the nonlinear flow equation
ii.Coupled homotopy perturbation method with Padé approximants (PHPM) to extend solution validity
iii.Demonstrated low relative errors (as low as –0.006) compared to Runge‑Kutta numerical solutions
iv.Recovered the exact solution y=1‑e^{‑x} for the case k=0 using the Laplace‑Padé scheme
v.Provided seventh‑order approximations valid for all k≥0

How they did it

The authors formulate the governing nonlinear differential equation for a stretching sheet with partial slip and construct a homotopy perturbation series. They truncate the series and apply Padé approximants, and for LPHPM also perform a Laplace transform followed by an inverse Laplace to obtain rational approximations. The resulting expressions are compared with fourth‑order Runge‑Kutta numerical solutions.

What they found

Largest relative error for LPHPM is –0.006 at k=5
Largest relative error for PHPM is –0.0250 at k=0
Seventh‑order approximation is valid for all k≥0
Exact solution y=1‑e^{‑x} recovered for k=0

Caveats

✕HPM series is accurate only for a restricted domain of the independent variable x and small k values
✕Higher‑order approximations improve accuracy but become cumbersome
✕Truncated series has a finite region of convergence

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