Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given.The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given.The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail,and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.

Noether's theorems of a fractional Birkhoffian system within Riemann-Liouville derivatives

The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff＇s equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether＇s theorems of a standard Birhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether＇s theorems are established. Finally, an example is given to illustrate the application of the results.

Fractional Optical Solitons and Fractional Noether’s Theorem with Ortigueira’s Centered Derivatives

This paper shows that the centered fractional derivatives introduced by Manuel Duarte Ortigueira in 2006 are useful in the description of optical solitons. It is shown that we can construct a fractional extension of the nonlinear Schrödinger (NLS) equation which incorporates Ortigueira’s derivatives and has soliton solutions. It is also shown that this fractional NLS equation has a Lagrangian density and can be derived from a variational principle. Finally, a fractional extension of Noether’s theorem is formulated to determine the conserved quantities associated to the invariances of the action integral under infinitesimal transformations.

Fractional Noether's Theorems for El-Nabulsi's Fractional Birkhoffian Systems in Terms of Riemann-Liouville Derivatives

The fractional Pfaffian variational problem and Noether’s theorems were investigated in terms of Riemann-Liouville derivatives on the basis of El-Nabulsi fractional model.The problem of the calculus of variations with fractional derivatives is a hot topic recently.Firstly,within Riemann-Liouville derivatives,the ElNabulsi Pfaffian variational problem was presented,the fractional Pfaff-Birkhoff-d’Alembert principle was established,and the fractional Birkhoff equations and the corresponding transversality conditions were obtained.Then,the Noether’s theorems in terms of Riemann-Liouville derivatives for the Birkhoffian system on the basis of El-Nabulsi fractional model are investigated under the special and the general transformations respectively.Finally,an example is given to illustrate the methods and results appeared in this paper.

EXISTENCE OF SOLUTION FOR BOUNDARY VALUE PROBLEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.

Variational Calculus With Conformable Fractional Derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.

EXACT MAXIMUM LIKELIHOOD ESTIMATOR FOR DRIFT FRACTIONAL BROWNIAN MOTION AT DISCRETE OBSERVATION

This paper deals with the problems of consistency and strong consistency of the maximum likelihood estimators of the mean and variance of the drift fractional Brownian motions observed at discrete time instants. Both the central limit theorem and the Berry-Ess′een bounds for these estimators are obtained by using the Stein’s method via Malliavin calculus.

Existence of Positive Solutions of Three-point Boundary Value Problem for Higher Order Nonlinear Fractional Differential Equations

In this paper, we consider the positive solutions of fractional three-point boundary value problem of the form Dο^α＋u（t）＋f（t,u（t）,u＇（t）,…,u^（n-3）（5）,u^（n-2）（t））=0,u^（i）（0）=0,0≤i≤n-2,u^（n-2）（1）-βu^（n-2）（ξ）=0,where 0〈t〈1,n-1〈α≤n,n≥2,ξ Е（0,1）,βξ^a-n〈1. We first transform it into another equivalent boundary value problem. Then, we derive the Green＇s function for the equivalent boundary value problem and show that it satisfies certain properties. At last, by using some fixed-point theorems, we obtain the existence of positive solution for this problem. Example is given to illustrate the effectiveness of our result.

Existence and Continuous Dependence of Mild Solutions for Some Fractional Neutral Differential Equations with Nonlocal Initial Conditions

The existence,uniqueness,and continuous dependence to the mild solutions of the nonlocal Cauchy problem were proved for a class of semilinear fractional neutral differential equations.The results are obtained by using the Krasnoselskii's fixed point theorem and the theory of resolvent operators for integral equations.

A Cauchy Problem for Some Fractional q-Difference Equations with Nonlocal Conditions

In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principle are used. At the end of the manuscript, we have an example that illustrates the key findings.

分数阶时滞广义Logistic方程解的研究

基于Banach不动点定理与分数阶微积分的相关性质,首先研究了分数阶时滞广义Logistic方程解的存在唯一性,同时得到解的一致稳定性的充分条件。最后,利用改进的Adams-Bashforth-Moulton预估-校正算法得到其数值解。

Calculations for Density of Quark Core Consisting of Mono Flavored Closely Packed Quarks inside Neutron Star

The attempt has been taken to calculate the density of stars possessing quark matter core using sphere packing concept of crystallography. The quark matter has been taken as solid in nature as predicted in references 36 and 37, and due to immense gravitational pressure at the core of the star the densest packing of quarks as spheres has been assumed to calculate the packing fraction Φ, thus the density ρ of the matter. Three possible types of pickings—mono-sized sphere packing, binary sphere packing and ternary sphere packing, have been worked out using three possible types of quark matter. It has been concluded that no value about the ρ of quark matter can be calculated using binary and ternary packing conditions and for mono-sized packing condition different flavor quark matters of different values in the density have been calculated using results from the experiments done by HI, ZEUS, L3 and CDF Collaborations about the radius limit of quark. For example, for u quark matter ρ ranges from 4.0587 × 1048 - 7.40038 × 1048 MeV/c2 cm3 using results of L3 Collaboration, for s quark matter 15.91794 × 1048 - 17.6866 × 1048 MeV/c2 cm3, etc.

整数阶电路定理到分数阶电路推广及分析

近几十年来,对分数阶电路的研究逐渐深入,但对其中电路定理的分析较少,因此针对分数阶电路需要进一步探究其规律,将一些经典的电路定理推广到分数阶电路中,使得在以后的分析过程中能直接使用。在整数阶电路定理的基础上,运用基尔霍夫定律在分数阶电路中证明了叠加定理、替代定理、等效电源定理和互易定理,并进行了应用分析。

Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6where 3 ≤ a 〈 4, D^ is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f（x, t） = -p（x）t^σ, with cr ~ （-1, 1） and p being a nonnegative continuous function that may be singular at x - 0 or x - 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch^uder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem （P0,0）. Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a ＋ b 〉 0 and f（x, t） -= tφ（x, t）, with φ（x, t） being a nonnegative continuous function in （0, 1） × [0, ∞） that is required to satisfy some suitable integrability condition. Using estimates on the Green＇s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem （Pa,b）, which behaves like the unique solution of the homogeneous problem corresponding the existence results. to （Pa,b）. Some examples are given to illustrate the existence results.,

Noether's Symmetries and Its Inverse for Fractional Logarithmic Lagrangian Systems

In this paper, Noether's theorem and its inverse theorem are proved for the fractional variational problems based on logarithmic Lagrangian systems. The Hamilton principle of the systems is derived. And the definitions and the criterions of Noether's symmetry and Noether's quasi-symmetry of the systems based on logarithmic Lagrangians are given. The intrinsic relation between Noether's symmetry and the conserved quantity is established. At last an example is given to illustrate the application of the results.

Quasi Sure Large Deviation for Increments of Fractional Brownian Motion in H¨older Norm

In this paper, we first prove Schilder＇s theorem in H？lder norm （0 ≤ α 〈1） with respect to Cr,p-capacity. Then, based on this result, we further prove a sharpening of large deviation principle for increments of fractional Brownian motion for Cr,p-capacity in the stronger topology.

Ulam-Hyers stability and analytical approach for m-dimensional Caputo space-time variable fractional order advection-dispersion equation

This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration.Moreover,with the help of fixed point theory,we study the existence and uniqueness conditions for the positive solution and prove some new results.Also,obtain the Ulam–Hyers stabilities for the proposed problem.Two gen-eralized examples are considered to show the method’s applicability and compared with other existing numerical methods.The present method performs exceptionally well in terms of efficiency and simplicity.Further,we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.

一类Riemann-Liouville型分数阶微分方程正解的存在性

应用格林函数的性质和Guo-Krasnoselskiis不动点定理,研究了一类非线性Riemann-Liouville型分数阶微分方程边值问题{D^αu(t)+f(t,u(t))=0(0<t<1),u(0)=D……βu(0)=D^βu(1)=0正解的存在性,其中2<α≤3,1<β≤2,1+β≤α,fC([0,1]×[0,∞),[0,∞)),Dα和Dβ分别表示α阶和β阶Riemann-Liouville分数阶导数,得到了该边值问题正解存在性的2个充分条件,并举例说明主要结论的适用性.

具有适型分数阶导数的非线性特征值问题的正解

本文研究具有适型分数阶导数的非线性特征值问题正解的存在性。首先给出Green函数G(t,s)并且证明其非负标和有界性;其次,利用Krasnosel’skii不动点定理对该问题的特征值区间给以刻划,得到正解的存在性和多解性。