Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

Fractional State Space Analysis of Temperature Time Series

Atmospheric temperatures characterize Earth as a slow dynamics spatiotemporal system, revealing long-memory and complex behavior. Temperature time series of 54 worldwide geographic locations are considered as representative of the Earth weather dynamics. These data are then interpreted as the time evolution of a set of state space variables describing a complex system. The data are analyzed by means of multidimensional scaling (MDS), and the fractional state space portrait (fSSP). A centennial perspective covering the period from 1910 to 2012 allows MDS to identify similarities among different Earth’s locations. The multivariate mutual information is proposed to determine the “optimal” order of the time derivative for the fSSP representation. The fSSP emerges as a valuable alternative for visualizing system dynamics.

Fractional pennes' bioheat equation: theoretical and numerical studies

In this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperature-dependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bioheat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.

Fractional bioheat equation

In this work we develop a new mathematical model for the Pennes’ bioheat equation assuming a fractional time derivative of single order. A numerical method for the solu- tion of such equations is proposed, and, the suitability of the new model for modelling real physical problems is studied and discussed

Comparison of different numerical methods for the solution of the time-fractional reaction-diffusion equation with variable diffusion coefficient

In this work we perform a comparison of two different numerical schemes for the solution of the time-fractional diffusion equation with variable diffusion coefficient and a nonlinear source term. The two methods are the implicit numerical scheme presented in [M.L. Morgado, M. Rebelo, Numerical approximation of distributed order reaction- diffusion equations, Journal of Computational and Applied Mathematics 275 (2015) 216-227] that is adapted to our type of equation, and a colocation method where Chebyshev polynomials are used to reduce the fractional differential equation to a system of ordinary differential equations

A prelude to the fractional calculus applied to tumor dynamic

In order to refine the solution given by the classical logistic equation and extend its range of applications in the study of tumor dynamics, we propose and solve a generalization of this equation, using the so-called Fractional Calculus, i.e., we replace the ordinary derivative of order 1, in one version of the usual equation, by a non-integer derivative of order 0 < α < 1, and recover the classical solution as a particular case. Finally, we analyze the applicability of this model to describe the growth of cancer tumors.

Fractional Calculus of the Generalized Wright Function

Mathematics Subject Classification: 26A33, 33C20.

Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations

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

Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey

2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05,

Generalized Fractional Evolution Equation

2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20

Solution of Space-Time Fractional Schrödinger Equation Occurring in Quantum Mechanics

Dedicated to Professor A.M. Mathai on the occasion of his 75-th birthday. Mathematics Subject Classi¯cation 2010: 26A33, 44A10, 33C60, 35J10.

Clifford-Hermite polynomials in fractional Clifford analysis

In this paper we generalize radial and standard Clifford-Hermite polynomials to the new framework of fractional Clifford analysis with respect to the Riemann-Liouville derivative in a symbolic way. As main consequence of this approach, one does not require an a priori integration theory. Basic properties such as orthogonality relations, differential equations, and recursion formulas, are proven.

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.

The Applicability Of Ordered Mesoporous Sba-15 And Its Hydrophobic Glutaraldehyde-bridge Derivative To Improve Ibuprofen-loading In Releasing System.

The mesoporous SBA-15 silica with uniform hexagonal pore, narrow pore size distribution and tuneable pore diameter was organofunctionalized with glutaraldehyde-bridged silylating agent. The precursor and its derivative silicas were ibuprofen-loaded for controlled delivery in simulated biological fluids. The synthesized silicas were characterized by elemental analysis, infrared spectroscopy, (13)C and (29)Si solid state NMR spectroscopy, nitrogen adsorption, X-ray diffractometry, thermogravimetry and scanning electron microscopy. Surface functionalization with amine containing bridged hydrophobic structure resulted in significantly decreased surface area from 802.4 to 63.0 m(2) g(-1) and pore diameter 8.0-6.0 nm, which ultimately increased the drug-loading capacity from 18.0% up to 28.3% and a very slow release rate of ibuprofen over the period of 72.5h. The in vitro drug release demonstrated that SBA-15 presented the fastest release from 25% to 27% and SBA-15GA gave near 10% of drug release in all fluids during 72.5 h. The Korsmeyer-Peppas model better fits the release data with the Fickian diffusion mechanism and zero order kinetics for synthesized mesoporous silicas. Both pore sizes and hydrophobicity influenced the rate of the release process, indicating that the chemically modified silica can be suggested to design formulation of slow and constant release over a defined period, to avoid repeated administration.

A New Goniothalamin N-acylated Aza-derivative Strongly Downregulates Mediators Of Signaling Transduction Associated With Pancreatic Cancer Aggressiveness.

In this study, a novel concise series of molecules based on the structure of goniothalamin (1) was synthesized and evaluated against a highly metastatic human pancreatic cancer cell line (Panc-1). Among them, derivative 8 displayed a low IC50 value (2.7 μM) and its concentration for decreasing colony formation was 20-fold lower than goniothalamin (1). Both compounds reduced the levels of the receptor tyrosine kinase (AXL) and cyclin D1 which are known to be overexpressed in pancreatic cancer cells. Importantly, despite the fact that goniothalamin (1) and derivative 8 caused pancreatic cancer cell cycle arrest and cell death, only derivative 8 was able to downregulate pro-survival and proliferation pathways mediated by mitogen activated protein kinase ERK1/2. Another interesting finding was that Panc-1 cells treated with derivative 8 displayed a strong decrease in the transcription factor (c-Myc), hypoxia-inducible factor-1α (HIF-1α) and vascular endothelial growth factor (VEGF) protein levels. Notably, the molecular effects caused by derivative 8 might not be related to ROS generation, since no significant production of ROS was observed in low concentrations of this compound (from 1.5 up to 3 μM). Therefore, the downregulation of important mediators of pancreatic cancer aggressiveness by derivative 8 reveals its great potential for the development of new chemotherapeutic agents for pancreatic cancer treatment.