Some Applications of Fractional Calculus to Polymer Science

Douglas, Jack F.

doi:10.1002/9780470141618.ch3

Cited by 33 publications

(55 citation statements)

References 191 publications

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“…As quoted in [15], the series in (5.2) converges in C 0 (R * + , L 2 (Ω) 9 ) and in L 2 loc (R + , L 2 (Ω) 9 ). Here ρ = E α * t −β Γ (1−β) , and 0 < ρ(t) kt −δ (see [15]), δ = β − α.…”

Section: The Continuity Of the Solution (U S) At T =mentioning

confidence: 92%

“…It follows that S ∈ C 0 b (R * + , L 2 (Ω) 9 ). The existence of at least one solution belonging to the functional space of (3.12) is thus proved.…”

Section: Theorem 31 (First Existence and Uniqueness Theorem) Letmentioning

confidence: 94%

“…Most of the following estimates are already proved in [15], save for those derived from Lemma 5.1. 9 . Then exists ∃M > 0, such that, ∀t ∈ R + and ∀k ∈ N * ,…”

Section: 4)mentioning

confidence: 95%

“…(2.1)-(2.2). For any test function ϕ ∈ D(Ω) 9 , as a consequence of Eq. (2.2) and of the fact that u ∈ L 1 9 .…”

Section: The Solution Existence and Uniquenessmentioning

confidence: 99%

“…For any test function ϕ ∈ D(Ω) 9 , as a consequence of Eq. (2.2) and of the fact that u ∈ L 1 9 . Since S L 2 (Ω) 9 ∈ L and ∇u L 2 (Ω) 9 ∈ L , we infer that S|ϕ L 2 (Ω) 9 ∈ L , 9 ∈ L , and Proposition 3.2 leads to…”

Section: The Solution Existence and Uniquenessmentioning

confidence: 99%

See 4 more Smart Citations

Well posedness of a linearized fractional derivative fluid model

Heibig¹,

Palade²

2011

Journal of Mathematical Analysis and Applications

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Objective fractional derivative constitutive equation Viscoelasticity Rest state stability analysis Hadamard stability analysis Solution existence Uniqueness Smoothness The one-dimensional fractional derivative Maxwell model (e.g. Palade, et al., Rheol. Acta 35 (1996) 265), of importance in modeling the linear viscoelastic response in the glass transition region, has been generalized in Palade, et al., Internat. J. Engrg. Sci. 37 (1999) 315, to objective three-dimensional constitutive equations (CEs) for both fluids and solids. Regarding the rest state stability of the fluid CE, in Heibig and Palade, J. Math. Phys. 49 (2008) 043101, we gave a proof for the existence of weak solutions to the corresponding boundary value problem. The aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results.

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Section: The Continuity Of the Solution (U S) At T =mentioning

confidence: 92%

“…It follows that S ∈ C 0 b (R * + , L 2 (Ω) 9 ). The existence of at least one solution belonging to the functional space of (3.12) is thus proved.…”

Section: Theorem 31 (First Existence and Uniqueness Theorem) Letmentioning

confidence: 94%

“…Most of the following estimates are already proved in [15], save for those derived from Lemma 5.1. 9 . Then exists ∃M > 0, such that, ∀t ∈ R + and ∀k ∈ N * ,…”

Section: 4)mentioning

confidence: 95%

“…(2.1)-(2.2). For any test function ϕ ∈ D(Ω) 9 , as a consequence of Eq. (2.2) and of the fact that u ∈ L 1 9 .…”

Section: The Solution Existence and Uniquenessmentioning

confidence: 99%

Section: The Solution Existence and Uniquenessmentioning

confidence: 99%

See 3 more Smart Citations

Well posedness of a linearized fractional derivative fluid model

Heibig¹,

Palade²

2011

Journal of Mathematical Analysis and Applications

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New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine function

Geng

Yang

Alsolami

2022

Math Methods in App Sciences

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Cartilage polymers: From viscoelastic solutions to weak gels*

Horkay

Douglas

2021

Polymer Engineering & Sci

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The dynamic properties of the polymeric components of cartilage are important for understanding the many biological functions of this biological tissue. Knowledge of the rheology of these materials is also crucial for the development of therapeutic treatments for diseases, such as arthritis, in which cartilage loses its functional physiological properties. In the present work complementary noninvasive techniques, rheology and dynamic light scattering (DLS) are used to probe the structural and dynamic properties of solutions of the principal proteoglycan components of cartilage extracellular matrix at different levels of structural hierarchy (chondroitin sulfate, aggrecan, aggrecan‐hyaluronic acid complex). It is shown that the bottlebrush shaped aggrecan molecules form large, loosely organized polyelectrolyte domains comprised of molecules organized into diffuse branched polymer network structures permeated by a diffuse cloud of counter‐ions. The rheological behaviors of these polymer solutions vary from viscoelastic solutions, consistent with the formation of dynamic polymer clusters in solution, to “weak gel” materials exhibiting approximate power law shear stress relaxation or creep over a wide range of timescales.

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Some Applications of Fractional Calculus to Polymer Science

Cited by 33 publications

References 191 publications

Well posedness of a linearized fractional derivative fluid model

Well posedness of a linearized fractional derivative fluid model

New fractional integral formulas and kinetic model associated with the hypergeometric superhyperbolic sine function

Cartilage polymers: From viscoelastic solutions to weak gels*

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