Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model

Berdyshev, Abdumauvlen; Baigereyev, Dossan; Boranbek, Kulzhamila

doi:10.3390/math11173763

Cited by 5 publications

(2 citation statements)

References 48 publications

(94 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many other applications and tests of finite difference derivative approximations are provided in the literature [21,[32][33][34][35][36][37][38][39][40][41].…”

Section: Comparison Of the Real Solution With The Complex One By Fini...mentioning

confidence: 99%

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

Almeida Magalhães,

Brito,

Lamon

et al. 2024

Mathematics

View full text Add to dashboard Cite

The paper expands the finite difference method to the complex plane, and thus obtains an improvement in the resolution of differential equations with an increase in numerical precision and a generalization in the mathematical modeling of problems. The article begins with a selection of the best techniques for obtaining finite difference coefficients for approximating derivatives in the real domain. Then, the calculation is expanded to the complex domain. The research expands forward, backward, and central difference approximations of the real case by a quadrant approximation in the complex plane, which facilitates the use in boundary conditions of differential equations. The article shows many real and complex finite difference equations with their respective order of error, intended to serve as a basis and reference, which have been tested in practical examples of solving differential equations used in engineering. Finally, a comparison is made between the real and complex techniques of finite difference methods applied in the Theory of Elasticity. As a surprising result, the article shows that the finite difference method has great advantages in numerical precision, diversity of formulas, and modeling generalities in the complex domain when compared to the real domain.

show abstract

“…Many other applications and tests of finite difference derivative approximations are provided in the literature [21,[32][33][34][35][36][37][38][39][40][41].…”

Section: Comparison Of the Real Solution With The Complex One By Fini...mentioning

confidence: 99%

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

Almeida Magalhães,

Brito,

Lamon

et al. 2024

Mathematics

View full text Add to dashboard Cite

show abstract

“…First, a reliable determination of the atmospheric turbulence coefficient K is a non-trivial problem. This is due to the fact that this parameter depends on local features of the area under study and is characterized by a rapid change from point to point [50]. This problem is especially complicated in urban areas where high-rise buildings are built up, which can lead to the formation of turbulent movements.…”

Section: The Proposed Approachmentioning

confidence: 99%

A Combined Approach for Predicting the Distribution of Harmful Substances in the Atmosphere Based on Parameter Estimation and Machine Learning Algorithms

Madiyarov,

Temirbekov,

Alimbekova

et al. 2023

Computation

View full text Add to dashboard Cite

This paper proposes a new approach to predicting the distribution of harmful substances in the atmosphere based on the combined use of the parameter estimation technique and machine learning algorithms. The essence of the proposed approach is based on the assumption that the concentration values predicted by machine learning algorithms at observation points can be used to refine the pollutant concentration field when solving a differential equation of the convection-diffusion-reaction type. This approach reduces to minimizing an objective functional on some admissible set by choosing the atmospheric turbulence coefficient. We consider two atmospheric turbulence models and restore its unknown parameters by using the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm. Three ensemble machine learning algorithms are analyzed for the prediction of concentration values at observation points, and comparison of the predicted values with the measurement results is presented. The proposed approach has been tested on an example of two cities in the Republic of Kazakhstan. In addition, due to the lack of data on pollution sources and their intensities, an approach for identifying this information is presented.

show abstract

Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model

Baishemirov,

Berdyshev,

Baigereyev

et al. 2024

Fractal Fract

View full text Add to dashboard Cite

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions.

show abstract

Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model

Cited by 5 publications

References 48 publications

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

A Combined Approach for Predicting the Distribution of Harmful Substances in the Atmosphere Based on Parameter Estimation and Machine Learning Algorithms

Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model

Contact Info

Product

Resources

About