Polynomial based differential quadrature method for numerical solution of nonlinear Burgers' equation

Korkmaz, Alper; Dagˇ, İdris

doi:10.1016/j.jfranklin.2011.09.008

Cited by 71 publications

(40 citation statements)

References 28 publications

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“…So far various numerical algorithms such as finite difference and cubic spline finite element methods [6], the group-explicit method [7], the generalized boundary element approach [8], quartic B-splines collocation method [9], quadratic B-splines finite element method [10], finite element method [11], spectral method [32], fourthorder finite difference method [12], a novel numerical scheme [13], explicit and exactexplicit finite difference methods [14], automatic differentiation method [15], Galerkin finite element method [16], cubic B-splines collocation method [17], spectral collocation method [18], Polynomial based differential quadrature method [19], quartic B-splines differential quadrature method [20], least-squares quadratic B-splines finite element method [21], implicit fourth-order compact finite difference scheme [22], some implicit methods [23], variational iteration method [24], homotopy analysis method [25], differential transform method and the homotopy analysis method [26], a numerical method based on Crank-Nicolson [27], modified cubic B-splines collocation method [28] ,differential quadrature method [29][30][31]46,47], some new semi-implicit finite difference schemes [33], Haar wavelet quasilinearization approach [34] etc. have been developed for the numerical solutions of Burgers' equation.…”

Section: Introductionmentioning

confidence: 99%

A hybrid numerical scheme for the numerical solution of the Burgers’ equation

Jiwari

2015

Computer Physics Communications

141

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Section: Introductionmentioning

confidence: 99%

A hybrid numerical scheme for the numerical solution of the Burgers’ equation

Jiwari

2015

Computer Physics Communications

141

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“…In Table , the numerical solution obtained for kinematic viscosities

ν = 1, 0.1, 0.01, 0.005, 0.004

are compared with exact and results obtained using modified cubic B‐spline DQM , automatic differentiation method , cubic Hermite collection method , Haar wavelet quasilinearization method , Crank‐Nicolson , polynomial based DQM and least‐squares quadratic B‐spline FEM . For small values of kinematic viscosities like

ν = 0.005

and

0.004

, the computation has carried up to

T = 15

using 15‐stage computation with 30 computational time steps per stage.…”

Section: Resultsmentioning

confidence: 99%

“…Table V is used for the comparison of errors with that of pseudospectral method [61]. In [61], Darvishi PDQ [40] PDQ-TS [58] PDQ [40] PDQ-TS [58] PDQ [40] PDQ-TS [42].…”

Section: Step 3: Substitutionmentioning

confidence: 99%

A differential quadrature based numerical method for highly accurate solutions of Burgers' equation

Aswin

Awasthi

Rashidi

2017

Numerical Methods Partial

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In this article, we introduce a new, simple, and accurate computational technique for one‐dimensional Burgers' equation. The idea behind this method is the use of polynomial based differential quadrature (PDQ) for the discretization of both time and space derivatives. The quasilinearization process is used for the elimination of nonlinearity. The resultant scheme has simulated for five classic examples of Burgers' equation. The simulation outcomes are validated through comparison with exact and secondary data in the literature for small and large values of kinematic viscosity. The article has deduced that the proposed scheme gives very accurate results even with less number of grid points. The scheme is found to be very simple to implement. Hence, it applies to any domain requires quick implementation and computation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2023–2042, 2017

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“…In this segment, we exhibit the computational technique employed to solve the nonlinear system of PDEs (14)- (15) with boundary conditions (16). Newton's linearization method (NLM) was utilized to linearize the non-linear system (14)- (16), which was subsequently solved using the differential quadrature method (DQM) [27][28][29][30][31][32][33][34][35][36][37][38][39] and two-point backward finite difference method. Applying NLM on (14)- (16) gives:…”

Section: Hybrid Linearization-differential Quadrature Methods (Hldqm)mentioning

confidence: 99%

The Impact of Sinusoidal Surface Temperature on the Natural Convective Flow of a Ferrofluid along a Vertical Plate

2019

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The spotlight of this investigation is primarily the effectiveness of the magnetic field on the natural convective for a Fe 3 O 4 ferrofluid flow over a vertical radiate plate using streamwise sinusoidal variation in surface temperature. The energy equation is reduplicated by interpolating the non-linear radiation effectiveness. The original equations describing the ferrofluid motion and energy are converted into non-dimensional equations and solved numerically using a new hybrid linearization-differential quadrature method (HLDQM). HLDQM is a high order semi-analytical numerical method that results in analytical solutions in η-direction, and so the solutions are valid overall in the η domain, not only at grid points. The dimensionless velocity and temperature curves are elaborated. Furthermore, the engineering curiosity of the drag coefficient and local Nusselt number are debated and sketched in view of various emerging parameters. The analyzed numerical results display that applying the magnetic field to the ferroliquid generates a dragging force that diminishes the ferrofluid velocity, whereas it is found to boost the temperature curves. Furthermore, the drag coefficient sufficiently minifies, while an evolution in the heat transfer rate occurs as nanoparticle volume fraction builds. Additionally, the augmentation in temperature ratio parameter signifies a considerable growth in the drag coefficient and Nusselt number. The current theoretical investigation may be beneficial in manufacturing processes, development of transport of energy, and heat resources. Mathematics 2019, 7, 1014 2 of 12 naturalistic kind of boundary condition which considers the effectiveness of thermophoresis Brownian movement. Khan and Pop [4] utilized the idea in [3] to contemplate the principal deal with nanofluid flow over an extending sheet. Bachok et al. [5] implemented an investigation of nanofluid flow past a semi-infinite plate. Rashad et al. [6] inspected the enhancement of heat transfer in a Darcy nanofluid convective flow past a porous cone. Rashad et al. [7] also probed the non-Darcy problem of a thermally stratified nanofluid flow over a vertical cylinder. They explored how heat transfer rate declined as the Brownian motion parameter boosted. However, abundant fundamental investigations are performed in this field, as can be seen in [8-13].Magnetic nanofluids (or ferrofluids) are the colloidal suspension of magneto-nanoparticles in a regular fluid (water, kerosene, mineral oil). Ferrofluids are magnetically controllable nanofluids, which contain magnetic nanoparticles, such as iron, cobalt, nickel, magnetite, ferrite, spinel-type. Since ferrofluids can be easily manipulated by means of an external magnetic force, the use of these fluids is promoted in numerous significant industrial applications, such as in vacuum seals, vibration-dampers, loudspeakers, shock absorbers, transformers coolant, and stepper motors [14][15][16][17][18][19]. Hayat et al. [20,21] probed the magneto-flow of Newtonian and viscoelastic nanofluids over ...

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Polynomial based differential quadrature method for numerical solution of nonlinear Burgers' equation

Cited by 71 publications

References 28 publications

A hybrid numerical scheme for the numerical solution of the Burgers’ equation

A hybrid numerical scheme for the numerical solution of the Burgers’ equation

A differential quadrature based numerical method for highly accurate solutions of Burgers' equation

The Impact of Sinusoidal Surface Temperature on the Natural Convective Flow of a Ferrofluid along a Vertical Plate

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