Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation

In this paper we present the operational matrices of the left Caputo fractional derivative, right Caputo fractional derivative and Riemann–Liouville fractional integral for shifted Legendre polynomials. We develop an accurate numerical algorithm to solve the two-sided space–time fractional advection–dispersion equation (FADE) based on a spectral shifted Legendre tau (SLT) method in combination with the derived shifted Legendre operational matrices. The fractional derivatives are described in the Caputo sense. We propose a spectral SLT method, both in temporal and spatial discretizations for the two-sided space–time FADE. This technique reduces the two-sided space–time FADE to a system of algebraic equations that simplifies the problem. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithm. By selecting relatively few Legendre polynomial degrees, we are able to get very accurate approximations, demonstrating the utility of the new approach over other numerical methods.

Pagine di buona fede a proposito di musica : precedute da alcune altre pagine di ugale buona fede ed intieramente inedite : state dirette all'autore da F. dr. Filippi, F. d'Arcais, M.C. Caputo, St. Tempia, F. Hiller, Alb. de Lasalle, Chev. van Elewyck, Ch. Bosselet, Carl Kipke /

Mode of access: Internet.

Della vita del P. Sertorio Caputo della Compagnia di Giesu /

Mode of access: Internet.

Cauchy Problem for Differential Equation with Caputo Derivative

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 of this problem. Two numerical examples are given.

Impulsive Fractional Differential Inclusions Involving the Caputo Fractional Derivative

Mathematics Subject Classification: 26A33, 34A37.

On the Equivalence of the Riemann-Liouville and the Caputo Fractional Order Derivatives in Modeling of Linear Viscoelastic Materials

Mathematics Subject Classification: 26A33

Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations

Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10

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

Mathematics Subject Classification: 26A33, 74B20, 74D10, 74L15

Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative

2000 Math. Subject Classification: 26A33; 33E12, 33E30, 44A15, 45J05

On q–Analogues of Caputo Derivative and Mittag–Leffler Function

Mathematics Subject Classification: 33D60, 33E12, 26A33

Eigenfunctions and fundamental solutions of the Caputo fractional Laplace and Dirac operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

A Caputo fractional derivative of a function with respect to another function

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor’s Theorem, Fermat’s Theorem, etc., are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.

Fractional differential equations with dependence on the Caputo-Katugampola derivative

In this paper we present a new type of fractional operator, the Caputo–Katugampola derivative. The Caputo and the Caputo–Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy type problem, with dependence on the Caputo–Katugampola derivative, is proven. A decomposition formula for the Caputo–Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation.

Assessment of matrix metalloproteinase (MMP)-2, MMP-8, MMP-9, and their inhibitors, the tissue inhibitors of metalloproteinase (TIMP)-1 and TIMP-2 in obese children and adolescents

Objectives: To compare the circulating levels of matrix metalloproteinase (MMP)-8, pro-MMP-2, pro-MMP-9, and total MMP-9, their endogenous inhibitors, the tissue inhibitors of metalloproteinases (TIMP)-1 and TIMP-2, and the MMP-8/TIMP-1, MMP-9/TIMP-1, and MMP-2/TIMP-2 ratios in normotensive obese children and adolescents with those found in non obese children and adolescents. Design and methods: We studied 40 obese and 40 non obese (controls) children and adolescents in this cross-sectional study. MMP and TIMP concentrations were measured in plasma samples by gelatin zymography and ELISA. Results: Obese children and adolescents had higher circulating MMP-8 concentrations, lower plasma TIMP-1 concentrations, and higher MMP-8/TIMP-1 ratios than non obese controls (P < 0.05). We found no differences in pro-MMP-9 or total MMP-9 levels, or in MMP-9/TIMP-1 ratios between groups (P > 0.05). While we found no significant differences in pro-MMP-2 levels (P > 0.05) obese Subjects had higher TIMP-2 concentrations and lower pro-MMP-2/TIMP-2 ratios (P < 0.05) than non obese controls. Conclusions: In conclusion, we found evidence indicating higher net MMP-8 (but not MMP-9 and MMP-2) activity in childhood obesity. The increased MMP-8 levels found in obese children suggest a possibly relevant pathophysiological mechanism that may be involved in the increase of cardiovascular risk associated with childhood obesity. (c) 2009 The Canadian Society of Clinical Chemists. Published by Elsevier Inc. All rights reserved.