Using the Bayesian Framework for Inference in Fractional Advection-Diffusion Transport System

Boone, Edward L.; Ghanam, Ryad; Malik, Nadeem A.; Whitlinger, J.

doi:10.12732/ijam.v33i5.4

Cited by 4 publications

(6 citation statements)

References 25 publications

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“…One of the main barriers to application is determining the fractional parameter α. In [16], Ghanam et al show how to estimate this parameter using a Bayesian framework. In addition this work considers whether or not a fractional calculus based model is supported by data.…”

Section: Introductionmentioning

confidence: 99%

Statistical Inference on Fractional Advection-Diffusion Transport Systems

Boone¹,

Ghanam²

2023

Int. J. of Appl. Math.

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The field fractional partial differential equations has been greatly expanding with in the last 20 years as evidenced by the growing body of literature. The work has primarily been theoretical in nature as there has been limited availability of experimental data. This work demonstrates using simulated data, how one can utilize these models and apply them to real world data to make inferences about the parameter values, predictions at future values and even what forcing mechanism was used to generate the data. This is done using a Bayesian approach that employs Markov chain Monte Carlo techniques. The proposed approach is evaluated using simulation studies concerning credible interval coverage probabilities and highest posterior model probabilities. The simulation study shows that the proposed method is effective for many parameters and that more work is needed to better estimate some physical coefficients.

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Section: Introductionmentioning

confidence: 99%

Statistical Inference on Fractional Advection-Diffusion Transport Systems

Boone¹,

Ghanam²

2023

Int. J. of Appl. Math.

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“…Fractional Partial Differential Equations (FPDE) are growing in interest among researchers confronted with modeling flow through porous materials [3,12,16,17,18,23,24,28,14,6]. The reason behind that is there are many phenomena in nature which are not described adequately by regualar derivatives and so there is a need for fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, estimatation of various parameters associated with the model such as the fractional derivative parameter α as well as measurement error. Once empirical data is collected the parameters can be estimated using techniques discussed in [6]. However, generating this empirical data may require experimentation on samples of the porous materials under study.…”

Section: Introductionmentioning

confidence: 99%

“…In this article we consider the mathematical model presented in [6] for which a parameter estimation for the fractional derivative order α is obtained and apply that to solve an optimization problem related to design of experiment. In this work, we consider how to place sensors for observation along the distance from the pressure source axis x using a follow-up study approach.…”

Section: Introductionmentioning

confidence: 99%

“…The main barrier to employing these models is the choice of the fractional order α. Recently, [6] show how to both estimate and make inferences about α from a Bayesian perspective. However, for experimental settings one needs to be able to design experiments that will provide optimal information about α.…”

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confidence: 99%

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Optimal Follow Up Designs for Fractional Partial Differential Equations with Application to a Convection-Advection Model

Boone,

Ryad A. Ghanam,

Lee

2023

GLM

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As the mathematical properties of Fractional Partial Differential Equations are rapidly being developed, there is an increasing desire by researchers to employ these models in real world data oriented contexts. The main barrier to employing these models is the choice of the fractional order alpha. Recently, show how to both estimate and make inferences about alpha from a Bayesian perspective. However, for experimental settings one needs to be able to design experiments that will provide optimal information about alpha. This work demonstrates how to use information based criteria, namely A, D and E optimality, to choose sampling locations in follow-up designs. Specifically, the simultaneous addition of one, two and three measurement locations is considered for a simple example. The simultaneous addition of four and five measurement locations is also considered across a variety of values of alpha. The results show that each of the criteria provide different optimal measurement locations as the number of additional measurement locations is increased. Indicating that the choice of optimality criteria should not be decidedarbitrarily.

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Optimal Sampling Locations for Fractional Partial Differential Equations Using D-Optimality

Boone,

Ghanam

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Using the Bayesian Framework for Inference in Fractional Advection-Diffusion Transport System

Cited by 4 publications

References 25 publications

Statistical Inference on Fractional Advection-Diffusion Transport Systems

Statistical Inference on Fractional Advection-Diffusion Transport Systems

Optimal Follow Up Designs for Fractional Partial Differential Equations with Application to a Convection-Advection Model

Optimal Sampling Locations for Fractional Partial Differential Equations Using D-Optimality

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