Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

Bao, Wendi; Song, Yisheng

doi:10.1002/num.21797

Cited by 9 publications

(4 citation statements)

References 30 publications

(39 reference statements)

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“…We also analyzed other solving strategies for PDEAs; however, most of them focus on linear PDEs [41], [43], [44], which are not suitable for our model. An appealing alternative solution for nonlinear PDEAs is described in [42] for a chromatography column (one PDE, one ODE, and one implicit algebraic equation), taking 10 times less CPU time than the method of lines (the strategy we used in this study); however, our model is larger (three ODE, four PDE, and two implicit equations); moreover, nonlinearities are much more complex.…”

Section: ) Numerical Solution and Simulationmentioning

confidence: 99%

Modeling and Simulation of a Packed Column Batch Still for Fruit Wine Distillations

2022

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Batch distillations are extensively used worldwide to produce fruit wine spirits with distinctive aromas. These processes are typically operated manually, based on the experience of the distiller. However, dynamic optimization and automatic control strategies could be valuable for ensuring a more consistent product and quickly adapting to new market tendencies. Consequently, reliable process models are required for the application of these methods. The novelty of this study is developing a highly efficient method to reliably simulate and calibrate a rigorous first-principles model of a packed batch column still. We applied the method of lines (MOL) to transform the partial differential equations describing the packed column dynamics into an ordinary differential equation (ODE) system. The nonlinear phase equilibrium equations were pre-solved, and several polynomials were fitted to get explicit algebraic equations. The final differentialalgebraic system (DAE) comprises 31 ODEs, 113 explicit algebraic equations, and two implicit algebraic equations. Variables scaling, smoothing of discontinuities, and applying an algebraic loop within Simulink to solve the implicit algebraic equations resulted in 600 times faster simulations than a naïve approach. The model was regressed using pilot-scale experimental data and subjected to extensive validation using independent data. Sensitivity and residual analysis established that a better heat loss model could improve ethanol concentration prediction. This new heat losses model reduced the average ethanol concentration error by more than five times. The developed model can be applied to design reliable control systems and optimal operating strategies for packed column batch distillations.INDEX TERMS Differential algebraic systems, model calibration, algebraic loops, first-principles models.

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Section: ) Numerical Solution and Simulationmentioning

confidence: 99%

Modeling and Simulation of a Packed Column Batch Still for Fruit Wine Distillations

2022

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“…By utilizing the definition of MQ B-splines, Beatson and Dyn (1996) investigated MQ quasi-interpolation theoretically. The quasi-interpolation operator defined via MQ functions has many applications such as the solution of PDEs (Chen and Wu 2007;Bao and Song 2014), the approximation of derivatives of a function (Ma and Wu 2009), and the solution of shock wave equations (Hon and Wu 2000).…”

Section: Introductionmentioning

confidence: 99%

Shape preserving fractal multiquadric quasi-interpolation

Kumar,

Chand,

Massopust

2024

Comp. Appl. Math.

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In this article, we construct a novel self-referential fractal multiquadric function which is symmetric about the origin. The scaling factors are suitably restricted to preserve the differentiability and the convexity of the underlying classical multiquadric function. Based on the translates of a fractal multiquadric function defined on a grid, we propose two fractal quasi-interpolants $$L_C^{\alpha }f$$ L C α f and $$L_D^{\alpha }f$$ L D α f to approximate smooth and irregular functions. We study the convergence of $$L_C^{\alpha }f$$ L C α f and $$L_D^{\alpha }f$$ L D α f to f using uniform error estimates. We investigate the linear polynomial reproducing property, convexity/concavity and monotonicity features of these quasi-interpolation operators. The advantages of fractal quasi-interpolants over the classical quasi-interpolants are demonstrated by various examples.

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“…e authors have given that the approximating capacity of the operator L Λ n is comparable with that of the operator L H 2m− 1 . Furthermore, many researchers applied multiquadric quasiinterpolants to solve differential equations [15][16][17][18][19][20][21][22][23][24][25][26]. Meanwhile, Ali et al [27] constructed the SDI using Timmer triangular patches, which are used to visualize the energy data, i.e., spatial interpolation in visualizing rainfall data.…”

Section: Introductionmentioning

confidence: 99%

Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

2021

Journal of Mathematics

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A kind of Abel–Goncharov type operators is surveyed. The presented method is studied by combining the known multiquadric quasi-interpolant with univariate Abel–Goncharov interpolation polynomials. The construction of new quasi-interpolants ℒ m AG f has the property of m m ∈ ℤ , m > 0 degree polynomial reproducing and converges up to a rate of m + 1 . In this study, some error bounds and convergence rates of the combined operators are studied. Error estimates indicate that our operators could provide the desired precision by choosing the suitable shape-preserving parameter c and a nonnegative integer m. Several numerical comparisons are carried out to verify a higher degree of accuracy based on the obtained scheme. Furthermore, the advantage of our method is that the associated algorithm is very simple and easy to implement.

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Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

Cited by 9 publications

References 30 publications

Modeling and Simulation of a Packed Column Batch Still for Fruit Wine Distillations

Modeling and Simulation of a Packed Column Batch Still for Fruit Wine Distillations

Shape preserving fractal multiquadric quasi-interpolation

Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

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