A nonlinear viscoelastic fractional derivative model of infant hydrocephalus

Infant communicating hydrocephalus is a clinical condition where the cerebral ventricles become enlarged causing the developing brain parenchyma of the newborn to be displaced outwards into the soft, unfused skull. In this paper, a hyperelastic, fractional derivative viscoelastic model is derived to describe infant brain tissue under conditions consistent with the development of hydrocephalus. An incremental numerical technique is developed to determine the relationship between tissue deformation and applied pressure gradients. Using parameter values appropriate for infant parenchyma, it is shown that pressure gradients of the order of 1 mm Hg are sufficient to cause hydrocephalus. Predicting brain tissue deformations resulting from pressure gradients is of interest and relevance to the treatment and management of hydrocephalus, and to the best of our knowledge, this is the first time that results of this nature have been established.     

Differential versus integral formulation of fractional hyperviscoelastic constitutive laws for brain tissue modelling

For years, interest has been constantly growing in biological tissue modelling. Particularly, the mechanical study of the brain has become a major topic in the field of biomechanics. A global model of this organ, including a realistic mesh and suitable constitutive laws for the different tissues, would find applications in various domains such as neurosurgery, haptic device design or car manufacturing to evaluate the possible trauma due to an impact.Several constitutive models have already been designed; regarding the strong strain-rate dependence of the stress-strain curves available in the literature, we decided to describe brain tissue as a viscoelastic medium through the use of the fractional derivation operator. Thanks to this approach, we can derive a convolution-based model with the Mittag-Leffler function as the regularized kernel.     

A sufficient condition of viability for fractional differential equations with the Caputo derivative

In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give the sufficient condition that guarantees fractional viability of a locally closed set with respect to nonlinear function. As an example we discuss positivity of solutions, particularly in linear case.     

Comments on the concept of existence of solution for impulsive fractional differential equations

In some recent works dealing with the existence of solutions for impulsive fractional differential equations, it is pointed out that the concept of solutions for such equations in some preceding papers is incorrect. In support of this claim, the authors of these papers begin with a counterexample. The objective of this note to indicate the mistake in these counterexamples and show the plausibility of the previous results.     

A general study on random integro-differential equations of arbitrary order

Here the broad study is depending on random integro-differential equations (RIDE) of arbitrary order. The fractional order is in terms of $\psi$-Hilfer fractional operator. This work reveals the dynamical behaviour such as existence, uniqueness and stability solutions for RIDE involving fractional order. Thus initial value problem (IVP), boundary value problem (BVP), impulsive effect and nonlocal conditions are taken in account to prove the results.     

A note on the stability criterion for a class of nonlinear fractional differential systems

This paper is devoted to dealing with a flaw that existed in a recent paper (Zhou et al. 2014). We give a new proof of Th. 3.1 in Zhou et al. (2014), which is a correction of the original proof.     

A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation

This paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations. An algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed. Several illustrative examples, with linear and nonlinear fractional differential equations, are given to demonstrate the effectiveness of the method.     

关于多元函数可微性的一个注记

研究了多元函数的可微性,给出了多元函数连续、可微的一些条件.     

A study on the convergence conditions of generalized differential transform method

This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non‐linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.     

两个函数之比单调性判别法注记

利用左、右导数,研究弱化条件下两个函数之比的单调性判别法。给出两个函数之比的单调性判别法的一种推广形式。     

A Note on the Local Time of Fractional Brownian Motion

Asymptotic behavior of the local time at the origin of q-dimensional fractional Brownian motion is considered when the index approaches the critical value 1/q. It is proved that, under a suitable (temporally inhomogeneous) normalization, it converges in law to the inverse of an extremal process which appears in the extreme value theory.     

A note on the identity operators of fractional calculus

The application of an identity operator for Saigo’s fractional calculus operators is shown by evaluating the limit of an indeterminate form. Its special case yields the result which has been used as an infinitesimal generator in the semigroup theory. Also, an identity operator for the recently introduced multi-dimensional fractional operators (due to Srivastava and Raina [8]) is discussed.     

A note on property of the Mittag-Leffler function

Recently the authors have found in some publications that the following property (0.1) of Mittag-Leffler function is taken for granted and used to derive other properties.

(0.1)     

The effect of noise on the Chafee-Infante equation: A nonlinear case study

We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, , by noise. While a single multiplicative Itô noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of multiplicative Stratonovich terms can make the origin exponentially stable, while an additive noise of sufficient richness reduces the random attractor to a single point.

     

A fractional tumor-growth model and the determination of the power law for different cancers based on data fitting

We study a fractional model for tumor growth and derive its general solution, the blow-up time and the radius of convergence. The model is simplified then to fit human data. The results show that there is a noticeable variation in the value of the scaling exponent depending on whether the model is fractional or integer. This supports the idea that the inclusion of memory effects may become relevant in the study of tumor growth via the scaling exponent.     

Second derivatives of convex functions in the sense of A. D. Aleksandrov on infinite-dimensional spaces with measure

We consider convex functions on infinite-dimensional spaces equipped with measures. Our main results give some estimates of the first and second derivatives of a convex function, where second derivatives are considered from two different points of view: as point functions and as measures.     

A numerical study on the effect of cavitation erosion in a diesel injector

The consequences of geometry alterations in a diesel injector caused by cavitation erosion are investigated with numerical simulations. The differences in the results between the nominal design geometry and the eroded one are analyzed for the internal injector flow and spray formation. The flow in the injector is modeled with a three-phase Eulerian approach using a compressible pressure-based multiphase flow solver. Cavitation is simulated with a nonequilibrium mass transfer rate model based on the simplified form of the Rayleigh–Plesset equation. Slip velocity between the liquid-vapor mixture and air is included in the model by solving two separate momentum conservation equations. The eroded injector is found to result in a loss in the rate of injection but also lower cavitation volume fraction inside the nozzle. The injected sprays are then simulated with a Lagrangian method considering as initial conditions the predicted flow characteristics at the exit of the nozzle. The results obtained show wider spray dispersion for the eroded injector and shorter spray tip penetration.     

A study on the effect of yield uncertainty in price-setting newsvendor models with additive-multiplicative demand

Random yield and uncertain demand usually both exist in many industries, such as the semiconductor industry. In this paper, the price-setting newsvendor model is studied which involves a single manufacturer and a single retailer with random yield and uncertain demand respectively. Under the condition of additive-multiplicative demand, we investigate the varying effects of random yield on the optimal price, order quantity, and expected profit in two situations with different cost structures. The first case is an in-house production case where the firm pays for the raw material quantity it has ordered, and the second one is a procurement case where the firm pays for the real product quantity it receives only. By using the theory of stochastic comparisons, we find that a less variable and a stochastically larger yield rate both lead to a lower optimal price and a higher expected profit for the in-house production case. Moreover, a less variable yield rate also results in a lower optimal price and a higher profit for the procurement case, but this is not true for a stochastically larger yield rate. Numerical examples illustrate that the effect of yield randomness on the optimal order quantity is not general.