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Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue

Freed, Alan, Diethelm, Kai (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 74B20, 74D10, 74L15The popular elastic law of Fung that describes the non-linear stress- strain behavior of soft biological tissues is extended into a viscoelastic material model that incorporates fractional derivatives in the sense of Caputo. This one-dimensional material model is then transformed into a three-dimensional constitutive model that is suitable for general analysis. The model is derived in a configuration that differs from the current, or spatial,...

Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative

Luchko, Yury, Trujillo, Juan (2007)

Fractional Calculus and Applied Analysis

2000 Math. Subject Classification: 26A33; 33E12, 33E30, 44A15, 45J05The Caputo fractional derivative is one of the most used definitions of a fractional derivative along with the Riemann-Liouville and the Grünwald- Letnikov ones. Whereas the Riemann-Liouville definition of a fractional derivative is usually employed in mathematical texts and not so frequently in applications, and the Grünwald-Letnikov definition – for numerical approximation of both Caputo and Riemann-Liouville fractional derivatives,...

Category theorems concerning Z-density continuous functions

K. Ciesielski, L. Larson (1991)

Fundamenta Mathematicae

The ℑ-density topology T ℑ on ℝ is a refinement of the natural topology. It is a category analogue of the density topology [9, 10]. This paper is concerned with ℑ-density continuous functions, i.e., the real functions that are continuous when the ℑ-densitytopology is used on the domain and the range. It is shown that the family C ℑ of ordinary continuous functions f: [0,1]→ℝ which have at least one point of ℑ-density continuity is a first category subset of C([0,1])= f: [0,1]→ℝ: f is continuous equipped...

Cauchy Problem for Differential Equation with Caputo Derivative

Kilbas, Anatoly, Marzan, Sergei (2004)

Fractional Calculus and Applied Analysis

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution...

Cauchy's residue theorem for a class of real valued functions

Branko Sarić (2010)

Czechoslovak Mathematical Journal

Let [ a , b ] be an interval in ℝ and let F be a real valued function defined at the endpoints of [ a , b ] and with a certain number of discontinuities within [ a , b ] . Assuming F to be differentiable on a set [ a , b ] ∖ E to the derivative f , where E is a subset of [ a , b ] at whose points F can take values ± ∞ or not be defined at all, we adopt the convention that F and f are equal to 0 at all points of E and show that 𝒦ℋ -vt ∫ a b f = F ( b ) - F ( a ) , where 𝒦ℋ -vt denotes the total value of the Kurzweil-Henstock integral. The paper ends with a few examples that illustrate...

Chain rules and p-variation

R. Norvaiša (2002)

Studia Mathematica

The main result is a Young-Stieltjes integral representation of the composition ϕ ∘ f of two functions f and ϕ such that for some α ∈ (0,1], ϕ has a derivative satisfying a Lipschitz condition of order α, and f has bounded p-variation for some p < 1 + α. If given α ∈ (0,1], the p-variation of f is bounded for some p < 2 + α, and ϕ has a second derivative satisfying a Lipschitz condition of order α, then a similar result holds with the Young-Stieltjes integral replaced by its extension.

Characterization of compact subsets of curves with ω-continuous derivatives

Marcin Pilipczuk (2011)

Fundamenta Mathematicae

We give a characterization of compact subsets of finite unions of disjoint finite-length curves in ℝⁿ with ω-continuous derivative and without self-intersections. Intuitively, our condition can be formulated as follows: there exists a finite set of regular curves covering a compact set K iff every triple of points of K behaves like a triple of points of a regular curve. This work was inspired by theorems by Jones, Okikiolu, Schul and others that characterize compact subsets of...

Characterization of ω -limit sets of continuous maps of the circle

David Pokluda (2002)

Commentationes Mathematicae Universitatis Carolinae

In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of ω -limit sets from the class 𝒞 ( I , I ) of continuous maps of the interval to the class 𝒞 ( 𝕊 , 𝕊 ) of continuous maps of the circle. Among others we give geometric characterization of ω -limit sets and then we prove that the family of ω -limit sets is closed with respect to the Hausdorff metric.

Currently displaying 361 – 380 of 2145