Parameter and differentiation order estimation for a two dimensional fractional partial differential equation

Aldoghaither, Abeer; Laleg‐Kirati, Taous‐Meriem

doi:10.1016/j.cam.2019.112570

Cited by 3 publications

(1 citation statement)

References 14 publications

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“…Inverse problems for two-dimensional time-fractional diffusion equation were also considered in [29,[31][32][33]. The study [34] considered the inverse problem for two-dimensional fractional partial differential equations. The method assumes the knowledge of the state function for a certain time interval at each point of the domain under consideration.…”

Section: Introductionmentioning

confidence: 99%

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

2021

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The article presents a method for solving the inverse problem of a two-dimensional anomalous diffusion equation with a Riemann–Liouville fractional-order derivative. In the first part of the present study, the authors present a numerical solution of the direct problem. For this purpose, a differential scheme was developed based on the alternating direction implicit method. The presented method was accompanied by examples illustrating its accuracy. The second part of the study concerned the inverse problem of recreating the model parameters, including the orders of the fractional derivative, in the anomalous diffusion equation. Equations of this type can be used to describe, inter alia, the heat conductivity in porous materials. The ant colony optimization algorithm was used to solve this problem. The authors investigated the impact of the distribution of measurement points, the use of different mesh sizes, and the input data errors on the obtained results.

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Section: Introductionmentioning

confidence: 99%

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

2021

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Estimation of parameters and valuation of options written on multiple assets described by uncertain fractional differential equations

Xin,

Zhang,

Noorani

et al. 2025

Applied Mathematics and Computation

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Bayesian calibration of order and diffusivity parameters in a fractional diffusion equation

et al. 2021

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This work focuses on parameter calibration of a variable-diffusivity fractional diffusion model. A random, spatially-varying diffusivity field with log-normal distribution is considered. The variance and correlation length of the diffusivity field are considered uncertain parameters, and the order of the fractional sub-diffusion operator is also taken uncertain and uniformly distributed in the range ( 0 , 1 ) . A Karhunen-Loève (KL) decomposition of the random diffusivity field is used, leading to a stochastic problem defined in terms of a finite number of canonical random variables. Polynomial chaos (PC) techniques are used to express the dependence of the stochastic solution on these random variables. A non-intrusive methodology is used, and a deterministic finite-difference solver of the fractional diffusion model is utilized for this purpose. The PC surrogates are first used to assess the sensitivity of quantities of interest (QoIs) to uncertain inputs and to examine their statistics. In particular, the analysis indicates that the fractional order has a dominant effect on the variance of the QoIs considered, followed by the leading KL modes. The PC surrogates are further exploited to calibrate the uncertain parameters using a Bayesian methodology. Different setups are considered, including distributed and localized forcing functions and data consisting of either noisy observations of the solution or its first moments. In the broad range of parameters addressed, the analysis shows that the uncertain parameters having a significant impact on the variance of the solution can be reliably inferred, even from limited observations.

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Parameter and differentiation order estimation for a two dimensional fractional partial differential equation

Cited by 3 publications

References 14 publications

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

Estimation of parameters and valuation of options written on multiple assets described by uncertain fractional differential equations

Bayesian calibration of order and diffusivity parameters in a fractional diffusion equation

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