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Coupling of optimal variation of parameters method with Adomian’s polynomials for nonlinear equations representing fluid flow in different geometries

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Abstract

In this study, we describe a modified analytical algorithm for the resolution of nonlinear differential equations by the variation of parameters method (VPM). Our approach, including auxiliary parameter and auxiliary linear differential operator, provides a computational advantage for the convergence of approximate solutions for nonlinear boundary value problems. We consume all of the boundary conditions to establish an integral equation before constructing an iterative algorithm to compute the solution components for an approximate solution. Thus, we establish a modified iterative algorithm for computing successive solution components that does not contain undetermined coefficients, whereas most previous iterative algorithm does incorporate undetermined coefficients. The present algorithm also avoid to compute the multiple roots of nonlinear algebraic equations for undetermined coefficients, whereas VPM required to complete calculation of solution by computing roots of undetermined coefficients. Furthermore, a simple way is considered for obtaining an optimal value of an auxiliary parameter via minimizing the residual error over the domain of problem. Graphical and numerical results reconfirm the accuracy and efficiency of developed algorithm.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan

    Waseem Sikander, Naveed Ahmed & Syed Tauseef Mohyud-Din

  2. Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan

    Umar Khan

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  1. Waseem Sikander
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  2. Naveed Ahmed
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  3. Umar Khan
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  4. Syed Tauseef Mohyud-Din
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Correspondence to Umar Khan.

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Sikander, W., Ahmed, N., Khan, U. et al. Coupling of optimal variation of parameters method with Adomian’s polynomials for nonlinear equations representing fluid flow in different geometries. Neural Comput & Applic 30, 3431–3444 (2018). https://doi.org/10.1007/s00521-017-2931-x

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