Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution

Luciano Pandolfi

doi:10.1002/mma.4180

A linear algorithm for the identification of a weakly singular relaxation kernel using two boundary measurements

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2016-0064 ◽

2018 ◽

Vol 26 (2) ◽

pp. 299-310

Author(s):

Sergei Avdonin ◽

Luciano Pandolfi

Keyword(s):

Distributed System ◽

Diffusion Processes ◽

Experimental Measurements ◽

Relaxation Kernel ◽

Linear Algorithm ◽

The Past ◽

Weakly Singular ◽

Deconvolution Problem ◽

Boundary Measurements ◽

With Memory

AbstractWe consider a distributed system of a type which is encountered in the study of diffusion processes with memory and in viscoelasticity. The key feature of such a system is the persistence in the future of the past actions due the memory described via a certain relaxation kernel; see below. The parameters of the kernel have to be inferred from experimental measurements. Our main result in this paper is that by using two boundary measurements, the identification of a relaxation kernel which is a linear combination of Abel kernels (as often assumed in applications) can be reduced to the solution of a (linear) deconvolution problem.

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Two-Dimensional Advection–Diffusion Process with Memory and Concentrated Source

Symmetry ◽

10.3390/sym11070879 ◽

2019 ◽

Vol 11 (7) ◽

pp. 879 ◽

Cited By ~ 2

Author(s):

Najma Ahmed ◽

Nehad Ali Shah ◽

Dumitru Vieru

Keyword(s):

Differential Equation ◽

Fourier Transforms ◽

Diffusion Processes ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Two Dimensional ◽

Symmetry Center ◽

Advection Diffusion ◽

Initial Boundary Value ◽

With Memory

Two-dimensional advection–diffusion processes with memory and a source concentrated in the symmetry center of the domain have been investigated. The differential equation of the studied model is a fractional differential equation with short-tail memory (a differential equation with Caputo–Fabrizio time-fractional derivatives). An analytical solution of the initial-boundary value problem has been determined by employing the Laplace transform and double sine-Fourier transforms. A numerical solution of the studied problem has been determined using finite difference approximations. Numerical simulations for both solutions have been carried out using the software Mathcad.

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Existence Solution and Controllability of Sobolev Type Delay Nonlinear Fractional Integro-Differential System

Mathematics ◽

10.3390/math7010079 ◽

2019 ◽

Vol 7 (1) ◽

pp. 79 ◽

Cited By ~ 1

Author(s):

Hamdy Ahmed ◽

Mahmoud El-Borai ◽

Hassan El-Owaidy ◽

Ahmed Ghanem

Keyword(s):

Fractional Order ◽

Differential System ◽

Diffusion Processes ◽

Sufficient Conditions ◽

Null Controllability ◽

Sobolev Type ◽

Materials With Memory ◽

Impulsive Delay ◽

Physical Phenomena ◽

With Memory

Fractional integro-differential equations arise in the mathematical modeling of various physical phenomena like heat conduction in materials with memory, diffusion processes, etc. In this manuscript, we prove the existence of mild solution for Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. We establish the sufficient conditions for the approximate controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. In addition, we prove the exact null controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. Finally, an example is given to illustrate the obtained results.

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Meridional Circulation, Turbulence, and Diffusion Processes in Stars

International Astronomical Union Colloquium ◽

10.1017/s0252921100080477 ◽

1976 ◽

Vol 32 ◽

pp. 109-116 ◽

Cited By ~ 1

Author(s):

S. Vauclair

Keyword(s):

Convection Zone ◽

Diffusion Processes ◽

Meridional Circulation ◽

Diffusion Time ◽

Work In Progress ◽

First Results ◽

Stellar Envelopes ◽

And Diffusion ◽

Turbulence And Diffusion ◽

Hr Diagram

This paper gives the first results of a work in progress, in collaboration with G. Michaud and G. Vauclair. It is a first attempt to compute the effects of meridional circulation and turbulence on diffusion processes in stellar envelopes. Computations have been made for a 2 Mʘstar, which lies in the Am - δ Scuti region of the HR diagram.Let us recall that in Am stars diffusion cannot occur between the two outer convection zones, contrary to what was assumed by Watson (1970, 1971) and Smith (1971), since they are linked by overshooting (Latour, 1972; Toomre et al., 1975). But diffusion may occur at the bottom of the second convection zone. According to Vauclair et al. (1974), the second convection zone, due to He II ionization, disappears after a time equal to the helium diffusion time, and then diffusion may happen at the bottom of the first convection zone, so that the arguments by Watson and Smith are preserved.

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