Fractional Differential Equations

https://www.chester.ac.uk/sites/files/chester/rep377.pdf

Analysis of Fractional Differential Equations

In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.

http://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf

A new Definition of Fractional Derivative without Singular Kernel

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.

https://ntrs.nasa.gov/api/citations/20020024453/downloads/20020024453.pdf

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

Finite difference/spectral approximations for the time-fractional diffusion equation

We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value theorem for integrals by allowing the corresponding fractional integral, viz.\ the Riemann-Liouville operator, instead of a classical (first-order) integral. As an application of the former result we then prove a uniqueness theorem for initial value problems involving Caputo-type fractional differential operators. This theorem generalizes the classical Nagumo theorem for first-order differential equations.

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo's fractional calculus (Corrected Version)

A viscoelastic model of the K-BKZ (Kaye 1962; Bernstein et al. 1963) type is developed for isotropic biological tissues, and applied to the fat pad of the human heel. To facilitate this pursuit, a class of elastic solids is introduced through a novel strain-energy function whose elements possess strong ellipticity, and therefore lead to stable material models. The standard fractional-order viscoelastic (FOV) solid is used to arrive at the overall elastic/viscoelastic structure of the model, while the elastic potential via the K-BKZ hypothesis is used to arrive at the tensorial structure of the model. Candidate sets of functions are proposed for the elastic and viscoelastic material functions present in the model, including a regularized fractional derivative that was determined to be the best. The Akaike information criterion (AIC) is advocated for performing multi-model inference, enabling an objective selection of the best material function from within a candidate set.

https://ntrs.nasa.gov/api/citations/20060005000/downloads/20060005000.pdf

A K-BKZ Formulation for Soft-Tissue Viscoelasticity

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials. In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual. The standard version of the Peano-Sard theory is...

A fractional version of the peano-sard theorem

Let $I=(c,d)$, $c < 0 < d$, $Q\in C^1: I\rightarrow[0,\infty)$ be a function with given regularity behavior on $I$. Write $w:=\exp(-Q)$ on $I$ and assume that $\int_I x^nw^2(x)dx<\infty$ for all $n=0,1,2,\ldots$. For $x\in I$, we consider the problem of the analytic and numerical approximation of the Cauchy principal value integral: \begin{equation*} I[f;x]:=\lim_{\varepsilon \to 0+} \left( \int_{c}^{x-\varepsilon} w^2(t)\frac{f(t)}{t-x}dt+ \int_{x+\varepsilon}^{d} w^2(t)\frac{f(t)}{t-x}dt. \right) \end{equation*} for a class of functions $f: I\rightarrow \mathbb{R^+}$ for which $I[f;x]$ is finite. In [1-4], the first two authors studied this problem and some of its applications for even exponential weights $w$ on $(-\infty,\infty)$ of smooth polynomial decay at $\pm \infty$ and given regularity.

Analytic and Numerical Analysis of Singular Cauchy integrals with exponential-type weights

We consider the representation of error functionals in numerical quadrature by the Peano kernel method. It is easily observed that the usual expressions for Peano kernels of order s still make sense if s is not a natural number. In this paper, we discuss how to interpret these Peano kernels, we state their main properties, and we compare them to the (classical) Peano kernels of integer order.

https://www.ems-ph.org/journals/show_pdf.php?issn=0232-2064&vol=16&iss=3&rank=13

Kai Diethelm

Papers

The mean value theorems and a Nagumo-type uniqueness theorem for Caputo's fractional calculus (Corrected Version)

A K-BKZ Formulation for Soft-Tissue Viscoelasticity

A fractional version of the peano-sard theorem

Analytic and Numerical Analysis of Singular Cauchy integrals with exponential-type weights

Peano Kernels of Non-Integer Order