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On an Interpolation Based Spectral Homotopy Analysis Method for PDE Based Unsteady Boundary Layer Flows

DOI: 10.1155/2014/848170

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Abstract:

This work presents a new approach to the application of the spectral homotopy analysis method (SHAM) in solving non-linear partial differential equations (PDEs). The proposed approach is based on an innovative idea of seeking solutions that obey a rule of solution expression that is defined in terms of bivariate Lagrange interpolation polynomials. The applicability and effectiveness of the expanded SHAM approach are tested on a non-linear PDE that models the problem of unsteady boundary layer flow caused by an impulsively stretching plate. Numerical simulations are conducted to generate results for the important flow properties such as the local skin friction. The accuracy of the present results is validated against existing results from the literature and against results generated using the Keller-box method. The preliminary results from the proposed study indicate that the present method is more accurate and computationally efficient than more traditional methods used for solving PDEs that describe nonsimilar boundary layer flow. 1. Introduction In this work, a new approach for applying the spectral homotopy analysis method (SHAM) is used to solve a partial differential equation (PDE) that models the problem of unsteady boundary layer flow caused by an impulsively stretching plate. The SHAM is a discrete numerical version of the homotopy analysis method (HAM) that has been widely applied to solve a wide variety of nonlinear ordinary and partial differential equations with applications in applied mathematics, physics, nonlinear mechanics, finance, and engineering. A detailed systematic description of the HAM and its applications can be found in two books (and a huge list of references cited therein) [1, 2] by Liao who is credited with developing the method. The SHAM was introduced in [3, 4] and it uses the principles of the traditional HAM and combines them with the Chebyshev spectral collocation method which is used to solve the so-called higher order deformation equations. One of the advantages of the SHAM is that it can accommodate very complex linear operators in its solution algorithm. In a recent application of the SHAM on solving PDE based problems [5], it was observed that the linear operator that gives the best results is one that is selected as the entire linear part of the governing PDEs. This is in sharp contrast to the traditional HAM which can only admit simple ordinary differential equation based linear differential operators with constant coefficient in its solution of PDEs describing the unsteady boundary layer problems of the type

References

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