On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential equations

Magagula, Vusi Mpendulo; Motsa, S. S.; Sibanda, Precious

doi:10.1016/b978-0-12-817949-9.00016-5

Cited by 12 publications

(7 citation statements)

References 20 publications

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“…Since U in equation ( 70) is known at the boundaries of the domain, rewriting equation (70) for the interior points Journal of Applied Mathematics of the domain, yields the new expression…”

Section: Stabilitymentioning

confidence: 99%

“…Jamil et al [66] studied the unsteady MHD micropolar-nanofluids with homogeneousheterogeneous chemical reactions over a stretching surface using the BI-SLLM. Since both BI-SQLM and BI-SLLM face challenges with problems of large time scales, Motsa et al [67] and Magagula et al [68] proposed the nonoverlapping multidomain spectral collocation methods, i.e., multidomain BI-SLLM [69] and multidomain BI-SQLM [65,70]. Later, Mkhatshwa et al [71] proposed an overlapping multidomain spectral method to study conjugate problems of conduction and MHD free convection flow of nanofluids over flat plates.…”

Section: Introductionmentioning

confidence: 99%

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A Novel Multivariate Spectral Local Quasilinearization Method (MV-SLQLM) for Modelling Flow, Moisture, Heat, and Solute Transport in Soil

Mwakilama

Magagula

Gathungu

2023

Journal of Applied Mathematics

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Conventionally, the problem of studying the transport of water, heat, and solute in soil or groundwater systems has been numerically solved using finite difference (FD) or finite element (FE) methods. FE methods are attractive over FD methods because they are geometrically flexible. However, recent studies demonstrate that spectral collocation (SC) methods converge exponentially faster than FD or FE methods using a few grid points or on coarse grids. This work proposes and applies a multivariate spectral local quasilinearization method (MV-SLQLM) to model the transportation and interaction of soil moisture, heat, and solute concentration in a nonbare soil ridge. The MV-SLQLM uses a quasilinearization method (QLM) to linearize any nonlinear equations and then employs a local linearization method (LLM) to decouple the linearized system of PDEs to form a sequence of equations that are solved in a computationally efficient manner. The MV-SLQLM is thus an extension of the bivariate spectral local linearization method (BI-SLLM) that fails to deal with a 2D problem and is a modification of the MV-SQLM whose efficiency is compromised when operating on high dense solution matrices. We use the residual error norms of the difference between successive iterations to affirm convergence to the expected solution. To illustrate the application and check the solution accuracy, we conduct systematic analyses of the effect of model parameters on distribution profiles. Findings are in good agreement with theory and literature, thereby revealing suitability of the MV-SLQLM to solve coupled nonlinear PDEs with environmental fluid dynamics applications.

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“…Since U in equation ( 70) is known at the boundaries of the domain, rewriting equation (70) for the interior points Journal of Applied Mathematics of the domain, yields the new expression…”

Section: Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Novel Multivariate Spectral Local Quasilinearization Method (MV-SLQLM) for Modelling Flow, Moisture, Heat, and Solute Transport in Soil

Mwakilama

Magagula

Gathungu

2023

Journal of Applied Mathematics

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“…The bivariate Lagrange interpolation polynomial interpolates

ω_{k} (η, ξ)

at selected grid points

(η_{i}, ξ_{j})

in both the

η

and

ξ

directions, for

i = 0, 1, 2, …, N_{η}

and

j = 0, 1, 2, …, N_{ξ}

. These selected grid points are called Chebyshev‐Gauss‐Lobatto points 39–41 and are given by

{η_{i}} = {\{\cos (\frac{π i}{N_{η}})\}}_{i = 0}^{N_{η}}, {ξ_{j}} = {\{\cos (\frac{π j}{N_{ξ}})\}}_{j = 0}^{N_{ξ}},

and

ℒ_{i} (η) = \prod_{\begin{matrix} i = 0 \\ i \neq k \end{matrix}}^{N_{η}} \frac{η - η_{k}}{η_{i} - η_{k}},

where

ℒ_{i} (η_{k}) = δ_{i k} = ({, \begin{matrix} 0 & if i \neq k \\ 1 & if i = k . \end{matrix})

The nonlinear operators

Ω_{k}

, for

k = 1, 2,

…”

Section: Numerical Solutionmentioning

confidence: 99%

“…The values of the

ξ

derivatives are computed at the Chebyshev‐Gauss‐Lobatto grid points

(η_{i}, ξ_{j})

, as (for

j = 0, 1, 2, …, N_{ξ}

)

\begin{matrix} center center left0.33em \\ {(∣, \frac{\partial ω_{n}}{\partial ξ})}_{(η_{i}, ξ_{j})} & = & \sum_{ω = 0}^{N_{η}} \sum_{μ = 0}^{N_{ξ}} ω_{n} (η_{ω}, ξ_{μ}) {MJX-tex-caligraphicscriptL}_{ω} (η_{i}) \frac{d {MJX-tex-caligraphicscriptL}_{μ} (ξ_{j})}{d ξ} \\ = & \sum_{μ = 0}^{N_{ξ}} d_{j μ} ω_{n} (η_{i}, ξ_{μ}), \end{matrix}

where

d_{j μ} = \frac{d ℒ_{μ} (ξ_{j})}{d ξ}

is the

j

th and

μ

th entry of the standard first derivative Chebyshev differentiation matrix of size

(N_{ξ} + 1) \times (N_{ξ} + 1)

, given by References [39–41,43]. The values of the space derivatives at the Chebyshev‐Gauss‐Lobatto points

(η_{i}, ξ_{j})

(for

i = 0, 1, 2, …, N_{η}

) are similarly computed as

{\frac{\partial ω_{n}}{\partial η}∣}_{(η_{i}, ξ_{j})} = \sum ω = 0...

…”

Section: Numerical Solutionmentioning

confidence: 99%

Influence of Soret and Dufour on mixed convection flow across a vertical cone

2021

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This contribution examines the influence of Soret and Dufour on an incompressible viscous fluid flow across a vertical cone. The flow model is framed in the form of mathematical governing equations and a nondimensionalization is performed on them for ease of the numerical computations' examination; the obtained nonsimilarity equations are solved numerically through the bivariate Chebyshev spectral collocation quasi‐linearization method. Outcomes of the flow characteristics, velocity, temperature, concentration, skin friction rate, heat, and mass transfer rates are analyzed with the variations of governing parameters, Prandtl number, buoyancy parameter, Schmidt number, buoyancy ratio, Soret and Dufour parameters at various stream‐wise spots of the flow. To certify the exactness of the listed computations, we performed a comparison with prior published computations, which were found with great agreement, and the residual analysis study was also portrayed to reflect the convergence and stability of the adopted numerical technique.

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“…The non-Newtonian fluid flows past a vertical plate with the flow being influenced by viscous dissipation and a heat source or sink. We intend to implement the BSQLM in this current study as it is reported by Magagula et al [30] that the method is computationally efficient and gives solutions that are more uniformly accurate than traditional methods like the finite difference methods. Although the BSQLM does not handle periodic boundary conditions properly, the method is considered because it is easy to implement and it takes little time to converge to the exact solution with few grid points, Rai and Mondal [31].…”

Section: Introductionmentioning

confidence: 99%

MHD Laminar Boundary Layer Flow of a Jeffrey Fluid Past a Vertical Plate Influenced by Viscous Dissipation and a Heat Source/Sink

Muzara

Shateyi

2021

Mathematics

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This study investigates the effects of viscous dissipation and a heat source or sink on the magneto-hydrodynamic laminar boundary layer flow of a Jeffrey fluid past a vertical plate. The governing boundary layer non-linear partial differential equations are reduced to non-linear ordinary differential equations using suitable similarity transformations. The resulting system of dimensionless differential equations is then solved numerically using the bivariate spectral quasi-linearisation method. The effects of some physical parameters that include the Schmidt number, Eckert number, radiation parameter, magnetic field parameter, heat generation parameter, and the ratio of relaxation to retardation times on the velocity, temperature, and concentration profiles are presented graphically. Additionally, the influence of some physical parameters on the skin friction coefficient, local Nusselt number, and the local Sherwood number are displayed in tabular form.

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On the bivariate spectral quasilinearization method for nonlinear boundary layer partial differential equations

Cited by 12 publications

References 20 publications

A Novel Multivariate Spectral Local Quasilinearization Method (MV-SLQLM) for Modelling Flow, Moisture, Heat, and Solute Transport in Soil

A Novel Multivariate Spectral Local Quasilinearization Method (MV-SLQLM) for Modelling Flow, Moisture, Heat, and Solute Transport in Soil

Influence of Soret and Dufour on mixed convection flow across a vertical cone

MHD Laminar Boundary Layer Flow of a Jeffrey Fluid Past a Vertical Plate Influenced by Viscous Dissipation and a Heat Source/Sink

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