Vladimir Kulish

Application of Fractional Calculus to Fluid Mechanics

We present the application of fractional calculus, or the calculus of arbitrary (noninteger) differentiation, to the solution of time-dependent, viscous-diffusion fluid mechanics problems. Together with the Laplace transform method, the application of fractional calculus to the classical transient viscous-diffusion equation in a semi-infinite space is shown to yield explicit analytical (fractional) solutions for the shear-stress and fluid speed anywhere in the domain

Fractional-Diffusion Solutions for Transient Local Temperature and Heat Flux

Applying properties of the Laplace transform, the transient heat diffusion equation can be transformed into a fractional (extraordinary) differential equation. This equation can then be modified, using the Fourier Law, into a unique expression relating the local value of the time-varying temperature (or heat flux) and the corresponding transient heat flux (or temperature). We demonstrate that the transformation into a fractional equation requires the assumption of unidirectional heat transport through a semiinfinite domain. Even considering this limitation, the transformed equation leads to a very simple relation between local timevarying temperature and heat flux. When applied along the boundary of the domain, the analytical expression determines the local time-variation of surface temperature (or heat flux) without having to solve the diffusion equation within the entire domain. The simplicity of the solution procedure, together with some introductory concepts of fractional derivatives, is highlighted considering some transient heat transfer problems with known analytical solutions. S0022-14810001002-1

A Fractional-Diffusion Theory for Calculating Thermal Properties of Thin Films From Surface Transient Thermoreflectance Measurements

The transient thermoreflectance (TTR) method consists of measuring changes in the reflectivity of a material (thin film) under pulsed laser heating, and relating these changes to the corresponding surface temperature variations. Analytical solutions of the diffusion problem are then used to determine the thermal conductivity of the material following an iterative matching process between the solutions and the experimental results. Analytical solutions are attainable either when the material absorbs the laser energy volumetrically or when the material absorbs the laser energy at the surface. Either solution allows for the determination of only one thermal property (thermal conductivity or diffusivity), with the other one assumed to be known. A new, single, analytical solution to the transient diffusion equation with simultaneous surface and volumetric heating, found using fractional calculus, is presented in a semi-derivative form. This complete solution provides the means to determine the two thermal properties of the material (thermal conductivity and diffusivity) concomitantly

Diffusion within a porous medium with randomly distributed heat sinks

Abstract Heat sinks filled with phase-change material, a design concept suitable for the thermal management of electronic airborne systems, find limited applicability at present due to lack of structural rigidity and poor isothermal cooling characteristics. A composite heat sink device (CHSD) is a new design concept proposed in this work, which combines the thermal benefit of the phase-change material with the structural rigidity provided by a solid porous matrix. The device consists of a solid porous matrix with pores filled with capsules containing a phase-change material. Numerical simulations of a CHSD steady-model indicate the existence of a maximum capsule density beyond which a transition from disperse diffusion (when all capsules actively participate in the diffusion process) to concentrate diffusion (when the capsules close to the heated surface shield the capsules placed further away) occurs. This observation is important for optimizing the volume of the CHSD.

Mathematical Based Calculation of Drug Penetration Depth in Solid Tumors

Cancer is a class of diseases characterized by out-of-control cells growth which affect cells and make them damaged. Many treatment options for cancer exist. Chemotherapy as an important treatment option is the use of drugs to treat cancer. The anticancer drug travels to the tumor and then diffuses in it through capillaries. The diffusion of drugs in the solid tumor is limited by penetration depth which is different in case of different drugs and cancers. The computation of this depth is important as it helps physicians to investigate about treatment of infected tissue. Although many efforts have been made on studying and measuring drug penetration depth, less works have been done on computing this length from a mathematical point of view. In this paper, first we propose phase lagging model for diffusion of drug in the tumor. Then, using this model on one side and considering the classic diffusion on the other side, we compute the drug penetration depth in the solid tumor. This computed value of drug penetration depth is corroborated by comparison with the values measured by experiments.

Three-dimensional, Unsteady Simulation of Alveolar Respiration

A novel macroscopic gas transport model, derived from fundamental engineering principles, is used to simulate the three-dimensional, unsteady respiration process within the alveolar region of the lungs. The simulations, mimicking the single-breath technique for measuring the lung diffusing capacity for carbon-monoxide (CO), allow the prediction of the red blood cell (RBC) distribution effects on the lung diffusing capacity. Results, obtained through numerical simulations, unveil a strong relationship between the type of distribution and the lung diffusing capacity. Several RBC distributions are considered, namely: normal (random), uniform, center-cluster, and corner-cluster red cell distributions. A nondimensional correlation is obtained in terms of a geometric parameter characterizing the RBC distribution, and presented as a useful tool for predicting the RBC distribution effect on the lung diffusing capacity. The effect of red cell movement is not considered in the present study because CO does not equilibrate with capillary blood within the time spent by blood in the capillary. Hence, blood flow effect on CO diffusion is expected to be only marginal.

FORECASTING THE BEHAVIOR OF FRACTAL TIME SERIES: HURST EXPONENT AS A MEASURE OF PREDICTABILITY

The Hurst exponent (H) is a statistical measure used to classify time series. H = 0.5 indicates a random series while H > 0.5 indicates a trend reinforcing series. The larger the H value is the stronger trend. In this paper we investigate the use of the Hurst exponent to classify series of financial data representing different periods of time. In this paper we show that series with large values of the Hurst exponent can be predicted more accurately than those series with H value close to 0.5. Thus the Hurst exponent provides a measure for predictability.

Retraction: Fractal Based Analysis of the Influence of Odorants on Heart Activity

Scientific Reports 6: Article number: 38555; published online: 08 December 2016; updated: 01 June 2018 This Article has been retracted by Scientific Reports at the request of Nanyang Technological University. An investigation at Nanyang Technological University found that ethical approval for the reported experiments was not sought from their Internal Review Board.

Retraction: The Analysis of the Influence of Odorant’s Complexity on Fractal Dynamics of Human Respiration

Scientific Reports 6: Article number: 26948; published online: 31 May 2016; updated: 01 June 2018 This Article has been retracted by Scientific Reports at the request of Nanyang Technological University. An investigation at Nanyang Technological University found that ethical approval for the reported experiments was not sought from their Internal Review Board.

Retraction: Mathematical Modelling and Prediction of the Effect of Chemotherapy on Cancer Cells

Scientific Reports 5: Article number: 13583; published online: 28 August 2015; updated: 01 June 2018 This Article has been retracted by Scientific Reports at the request of Nanyang Technological University. An investigation at Nanyang Technological University found that ethical approval for the reported experiments was not sought from their Internal Review Board.