Recovering Parameters of Generalized Fractional Diffusion Models

Scientists of Tallinn University of Technology recover parameters of generalized fractional diffusion models

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Fractional differential equations is a popular tool to model slow diffusion with nonlocal properties. Recent development of these models goes to the direction of more generality in density kernels characterizing the nonlocality. For instance, models with multiple fractional derivatives (distributed derivatives) are utilized to describe more complex diffusion processes.

Mathematical tools for differential equations with usual fractional derivatives are relatively well-developed (e.g. application of Mittag-Leffler functions in time fractional models). Handling generalized models requires substantial enchancement of mathematical techniques.

In Tallinn University of Technology, a group of researchers studies inverse problems to reconstruct parameters of generalized fractional diffusion models. In particular, problems to recover source terms, coefficients and characteristics of generalized derivatives are considered. Well-posedness issues of the inverse problems as well as numerical methods are in the focus.

An immediate generalization of a distributed time fractional time derivatives is the following operator:

Dα + m* Dα  

where  Dα  is a (Caputo or Riemann-Liouville) time fractional derivative of the order α,  m is a perturbation kernel and * denotes the time convolution.

The figure illustrates a result of reconstruction of a space-dependent factor  f(x) of a source from final measurements in case  α = 0.4 and  m(t) = t-0.9 .  The problem is regularized by means of the method of least error using the discrepancy rule of parameter choice. The noise level of data is  0.01.

Related project in ResearchGate.

Contact: Prof. Jaan Janno, Department of Cybernetics, Tallinn University of Technology. E-mail:  jaan.janno (at) ttu.ee

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