A new class of A-stable numerical techniques for ordinary differential equations: Application to boundary-layer flow

Abodayeh

Advances in Mechanical Engineering

et al. 2021

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A two-stage third-order numerical scheme is proposed for solving ordinary differential equations. The scheme is explicit and implicit type in two stages. First, the stability region of the scheme is found when it is applied to the linear equation. Further, the stability conditions of the scheme are found using a linearized homogenous set of differential equations. This set of equations is obtained by applying transformations on the governing equations of heat and mass transfer of incompressible, laminar, steady, two-dimensional, and non-Newtonian power-law fluid flows over a stretching sheet with effects of thermal radiations and chemical reaction. The proposed scheme with an iterative method is employed in two different forms called linearized and non-linearized. But it is found that the non-linearized approach performs better than the linearized one when residuals are compared through plots. Additionally, the proposed scheme is compared to the second-order central finite difference method for second-order non-linear differential equations and the Keller-Box/trapezoidal method for a linear differential equation. It is determined that the proposed scheme is more effective and computationally less expensive than the standard/classical finite difference methods. Moreover, the impact of magnetic parameter, radiation parameter, modified Prandtl and Schmidt numbers for power-law fluid, and chemical reaction rate parameter on velocity, temperature, and concentration profiles are displayed through graphs and discussed. The power-law fluid’s heat and mass transfer simulations are also carried out with varying flow behavior index, sheet velocity, and mass diffusivity. We hoped that this effort would serve as a guide for investigators tasked with resolving unresolved issues in the field of enclosures used in industry and engineering.

Section: Discussionmentioning

confidence: 99%

A third-order two-step numerical scheme for heat and mass transfer of chemically reactive radiative MHD power-law fluid

Abodayeh

Advances in Mechanical Engineering

et al. 2021

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“…Due to this reason, the comparison plots over spatial variable were shown to be in straight lines. Following the completion of this work, it is possible to propose additional applications for the current methodology (Nawaz and Arif 2021 ; Nawaz et al 2020 ; 2021a ; 2021b ).…”

Section: Discussionmentioning

confidence: 99%

“…The scheme was the predictor-corrector type, but it did not satisfy some features of the model. Some more work on the non-standard finite difference method and compact numerical schemes can be seen in Yanga (2016); Raza et al 2021;Martin-Vaquero et al 2018;Arenas and Gilberto Gonz'alez-Parra, Benito M. Chen-Charpentier, 2017 ;Muhammad Rafiq et al 2021;Arif et al 2019;Nawaz andArif 2021 Nawaz et al 2021a). A Caputo fractional derivative-based COVID-19 pandemic model has been formulated in Mohammad et al (2021).…”

Section: Introductionmentioning

confidence: 99%

Development of Explicit Schemes for Diffusive SEAIR COVID-19 Epidemic Spreading Model: An Application to Computational Biology

Iran J Sci Technol Trans Sci

Ashraf

2021

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In this contribution, a first-order time scheme is proposed for finding solutions to partial differential equations (PDEs). A mathematical model of the COVID-19 epidemic is modified where the recovery rate of exposed individuals is also considered. The linear stability of the equilibrium states for the modified COVID-19 model is given by finding its Jacobian and applying Routh–Hurwitz criteria on characteristic polynomial. The proposed scheme provides the first-order accuracy in time and second-order accuracy in space. The stability of the proposed scheme is given using the von Neumann stability criterion for standard parabolic PDEs. The consistency for the proposed scheme is also given by expanding the involved terms in it using the Taylor series. The scheme can be used to obtain the condition of getting a positive solution. The stability region of the scheme can be enlarged by choosing suitable values of the contained parameter. Finally, a comparison of the proposed scheme is made with the existing non-standard finite difference method. The results indicate that the non-standard classical technique is incapable of preserving the unique characteristics of the model’s epidemiologically significant solutions, whereas the proposed approaches are capable of doing so. A computational code for the proposed discrete model scheme may be made available to readers upon request for convenience.

A Class of Unconditionally Stable Shooting Methods with Application to Radiative Darcy–Forchheimer Flow

Int. J. Comput. Methods

Ashraf

et al. 2022

Since there are shooting methods in the literature to solve ordinary differential equations (ODEs), those are mainly based on Runge–Kutta explicit methods. It is a proven fact that explicit Runge–Kutta methods are conditionally stable. The present contribution consists of constructing a class of shooting methods based on implicit methods that are unconditionally stable with their application to solving the problem of Darcy–Forchheimer’s flow over the stretching sheet under the influence of thermal radiations. Governing equations of the flow phenomena are presented in the form of partial differential equations (PDEs). Further, these PDEs are reduced into ODEs which are tackled by the present constructing shooting methods based on the class of Adam Moulton’s techniques with the Gauss–Siedel iterative method. The stability and convergence for the system of differential equations are also given. Thus, the main advantage of the presented methods is to turn the implicit methods into explicit methods using the shooting approach. Also, the impact of inertia coefficient, porosity parameter, radiation parameter, and Prandtl number on velocity and temperature profiles is given graphically. It is seen that the increment of the porosity parameter escalates the temperature profile. A computational code for the proposed model scheme may be made available to readers upon request for convenience.