Linear Viscoelastic Responses: The Prony Decomposition Naturally Leads Into the Caputo-Fabrizio Fractional Operator

Hristov, Jordan

doi:10.3389/fphy.2018.00135

Cited by 24 publications

(17 citation statements)

References 119 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the analysis in [38,39,62] and the final results related to the response functions, i.e. the stress relaxation and the compliance moduli, directly relate them to the discussed here Prony (Dirichlet) approximations and consequently to the Caputo-Fabrizio fractional operator (see the analysis in [75]).…”

Section: Some Final Commentsmentioning

confidence: 99%

Response functions in linear viscoelastic constitutive equations and related fractional operators

Hristov

2019

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

show abstract

Section: Some Final Commentsmentioning

confidence: 99%

Response functions in linear viscoelastic constitutive equations and related fractional operators

Hristov

2019

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

show abstract

“…Caputo and Fabrizio [32] responded that the definition presented in [25] satisfied all the general properties of fractional derivatives including nonlocality. Recently, Hristov [33] demonstrated the viscoelastic response functions and their relationship with the Caputo-Fabrizio fractional derivative, and addressed the criticism of [25]. The response of Caputo and Fabrizio and Hristov was appreciated by many researchers, and has been widely accepted.…”

Section: Introductionmentioning

confidence: 99%

Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles

et al. 2020

View full text Add to dashboard Cite

A nonlocal fractional model of Brinkman type fluid (BTF) containing a hybrid nanostructure was examined. The magnetohydrodynamic (MHD) flow of the hybrid nanofluid was studied using the fractional calculus approach. Hybridized silver (Ag) and Titanium dioxide (TiO2) nanoparticles were dissolved in base fluid water (H2O) to form a hybrid nanofluid. The MHD free convection flow of the nanofluid (Ag-TiO2-H2O) was considered in a microchannel (flow with a bounded domain). The BTF model was generalized using a nonlocal Caputo-Fabrizio fractional operator (CFFO) without a singular kernel of order α with effective thermophysical properties. The governing equations of the model were subjected to physical initial and boundary conditions. The exact solutions for the nonlocal fractional model without a singular kernel were developed via the fractional Laplace transform technique. The fractional solutions were reduced to local solutions by limiting α → 1 . To understand the rheological behavior of the fluid, the obtained solutions were numerically computed and plotted on various graphs. Finally, the influence of pertinent parameters was physically studied. It was found that the solutions were general, reliable, realistic and fixable. For the fractional parameter, the velocity and temperature profiles showed a decreasing trend for a constant time. By setting the values of the fractional parameter, excellent agreement between the theoretical and experimental results could be attained.

show abstract

“…From the definition (122) it follows that if f (t) = C = const., then CF D α t f (t) = 0, an expected results as in the classical Caputo derivative [117]. Analyzes of applications of the Caputo-Fabrizio operator are available in Hristov [96] and Hristov [119][120][121] and will skip reference quotations here.…”

Section: Caputo-fabrizio Fractional Operatormentioning

confidence: 99%

Bio-Heat Models Revisited: Concepts, Derivations, Nondimensalization and Fractionalization Approaches

Hristov

2019

Front. Phys.

View full text Add to dashboard Cite

The heat transfer in living tissues is an evergreen problem in mathematical modeling with great practical importance starting from the Pennes equation postulation. This study focuses on concept in model building, the correct scaling of the bio-heat equation (one-dimensional) by appropriate choice of time and length scales, and consequently order of magnitude analysis of effects, as well as fractionalization approaches. Fractionalization by different constitutive approaches, leading to application of different fractional differential operators, modeling the finite speed of the heat wave, is one of the principle problems discussed in the study. The correct choice of the damping (relaxation) function of the heat flux is of primarily importance in the formulation of the bio-heat equation with memory. Moreover, this affects the consequent scaling, order of magnitude analysis and solutions.

show abstract

Linear Viscoelastic Responses: The Prony Decomposition Naturally Leads Into the Caputo-Fabrizio Fractional Operator

Cited by 24 publications

References 119 publications

Response functions in linear viscoelastic constitutive equations and related fractional operators

Response functions in linear viscoelastic constitutive equations and related fractional operators

Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles

Bio-Heat Models Revisited: Concepts, Derivations, Nondimensalization and Fractionalization Approaches

Contact Info

Product

Resources

About