On Spectral Homotopy Analysis Method for Solving Linear Volterra and Fredholm Integrodifferential Equations

Atabakan, Z. Pashazadeh; Kılıçman, Adem; Nasab, A. Kazemi

doi:10.1155/2012/960289

Cited by 6 publications

(3 citation statements)

References 31 publications

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“…Te execution of SHAM invites schemes of the standard protocol, specifcally convergence-controlling indices, in the attempt to lighten the constraints associated with the implementation of the traditional Homotopy analysis method [66][67][68][69]. Te report of Abdelmalek et al employed SHAM to solve higher-order Fredholm-type integrodiferential problems [70,71]. Makukula applied SHAM to study MHD fow restrictions over a stretching surface, while the study in [72] explored SHAM to investigate bioconvection in plasma-mediated nanoparticle fow under magnetic feld exposure.…”

Section: Spectral Relaxation Methodmentioning

confidence: 99%

Magneto-Rotational Augmentation of Bioconvective Transport in Plasma-Nanofluid Flowing through a Penetrable Spinning Disc

Ige

Odesola

Ayedun

et al. 2022

Journal of Engineering

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The phenomenon of bioconvective transport through the manipulation of motile microorganisms is considered a promising process control technique in several biological processes and microdevices. Inducing convective transport in self-propelling microbes could be tailored to improve mixing, reaction propensity, and concentration transport within the media. This paper examined the combined effect of magnetic and rotational fields on the argumentation of bioconvective transport in the nanofluid-mediated plasma flow. A detailed analysis of the transport and dynamics of reactive forces during bioconvection in a rotary disc-like microchannel is presented. The physics of the problem was described by coupled nonlinear ordinary differential equations, which were numerically computed using the spectral relaxation scheme of the spectral homotopy analysis method. It was observed that the imposition of a magnetic field constituted viscous drag in the plasma-nanofluid media, which consequently increases the thermophoretic parameter in the bioconvective flow. It was ascertained that coupled magnetic and rotational effects significantly augmented the motility of microorganisms and translated to growth in momentum and concentration fields which is noticeable in the generation of stretching effect on the bacterium-containing plasma-nanofluid flow. The findings of this study could provide an essential basis for the design of bioreactors, centrifugal microfluidics technologies, and microdevices for use in a broad spectrum of biotechnology.

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Section: Spectral Relaxation Methodmentioning

confidence: 99%

Magneto-Rotational Augmentation of Bioconvective Transport in Plasma-Nanofluid Flowing through a Penetrable Spinning Disc

Ige

Odesola

Ayedun

et al. 2022

Journal of Engineering

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“…Spectral methods are now becoming the preferred tools for solving ordinary and partial differential equations because of their elegance and high accuracy in resolving problems with smooth functions [6,33]. The use of the SHAM has largely been restricted to the solution of nonlinear boundary value problems [1,8,13,18,[21][22][23][24][25]31] However, Atabakan et al [3] recently used the method to solve Volterra and Fredholm integro-differential equations. A slightly different version of the SHAM that uses Chebyshev-Tau method to convert a BVP to algebraic equations is proposed in Kazem and Shaban [12].…”

Section: Introductionmentioning

confidence: 99%

On the practical use of the spectral homotopy analysis method and local linearisation method for unsteady boundary-layer flows caused by an impulsively stretching plate

Motsa

2013

Numer Algor

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This paper expands the ideas of the spectral homotopy analysis method to apply them, for the first time, on non-linear partial differential equations. The spectral homotopy analysis method (SHAM) is a numerical version of the homotopy analysis method (HAM) which has only been previously used to solve non-linear ordinary differential equations. In this work, the modified version of the 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 robustness of the SHAM approach is demonstrated by its flexibility to allow linear operators that are partial derivatives with variable coefficients. This is seen to significantly improve the convergence and accuracy of the method. To validate accuracy of the the present SHAM results, the governing PDEs are also solved using a novel local linearisation technique coupled with an implicit finite difference approach. The two approaches are compared in terms of accuracy, speed of convergence and computational efficiency.

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“…In SHAM the initial approximation is taken to be the solution of the nonhomogeneous linear part of the given equation. In 2012, Pashazadeh Atabakan et al solved linear Volterra and Fredholm integro-differential equations using spectral homotopy analysis method; see [22].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

Atabakan

Nasab

Kılıçman

et al. 2013

Mathematical Problems in Engineering

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Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.

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On Spectral Homotopy Analysis Method for Solving Linear Volterra and Fredholm Integrodifferential Equations

Cited by 6 publications

References 31 publications

Magneto-Rotational Augmentation of Bioconvective Transport in Plasma-Nanofluid Flowing through a Penetrable Spinning Disc

Magneto-Rotational Augmentation of Bioconvective Transport in Plasma-Nanofluid Flowing through a Penetrable Spinning Disc

On the practical use of the spectral homotopy analysis method and local linearisation method for unsteady boundary-layer flows caused by an impulsively stretching plate

Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

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