The space fractional diffusion equation with Feller’s operator

This paper investigates the space fractional diffusion equation with fractional Feller’s operator. The Green’s function is obtained by using Fourier transform, and the analytical solutions of some space fractional diffusion equations with initial (or initial and boundary) condition are obtained in terms of Green’s function. In addition, numerical simulations are discussed. The results indicate that the effect range of skewness parameter θ has more effect on probability density than that of parameter α. The results also explain the property of the skewness and long tail in the asymmetry diffusion process.     

Theory of fractional order in generalized thermoelectric MHD

A new mathematical model of thermoelectric MHD theory has been constructed in the context of a new consideration of heat conduction with fractional orders. This model is applied to Stokes’ first problem for a conducting fluid with heat sources. Laplace transforms and state-space techniques [1] will be used to obtain the general solution for any set of boundary conditions. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effects of the fractional order parameters on all the studied fields.     

Homotopy perturbation method for a limit case Stefan problem governed by fractional diffusion equation

The work presents a mathematical model describing the time fractional anomalous-diffusion process of a generalized Stefan problem which is a limit case of a shoreline problem. In this model, the governing equations include a fractional time derivative of order 0 < α ? 1 and variable latent heat. The approximate solution of the problem is obtained by homotopy perturbation method. The results thus obtained are compared graphically with the exact solutions. A brief sensitivity study is also performed.     

Localization of Wiener functionals of fractional regularity and applications

In this paper we localize some of Watanabe’s results on Wiener functionals of fractional regularity, and use them to give a precise estimate of the difference between two Donsker’s delta functionals even with fractional differentiability. As an application, the convergence rate of the density of the Euler scheme for non-Markovian stochastic differential equations is obtained.     

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.     

关于Menger图和Menger数的若干结果

该文研究Menger图和Menger数. 主要结果如下 : (1)\ 对任意的n≥4, n立方体Q\-n不是Menger图. 解决了Sampathkumar提出的未解问题2. (2)\ 如果G是一个偶图, 则m(G)=β\-0(G),其中m(G)是G的Menger数, β\-0(G)是G的独立数. 部分解决了Sampathkumar提出的未解问题3. (3)\ “确定图的Menger数”问题是NP困难的.     

Analytical relationships between metric and centrality measures of a network and its dual

The centrality and efficiency measures of a network G are strongly related to the respective measures on the dual G^? and the bipartite B(G) associated networks. We show some relationships between the Bonacich centralities c(G), c(G^?) and c(B(G)) and between the efficiencies E(G) and E(G^?) and we compute the behavior of these parameters in some examples.     

Leitmann’s direct method for fractional optimization problems

Based on a method introduced by Leitmann [G. Leitmann, A note on absolute extrema of certain integrals, Int. J. Non-Linear Mech. 2 (1967) 55-59], we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control.     

Existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations

Sufficient conditions are established for the existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations. The investigations are carried out by means of piecewise continuous functions of Lyapunov type and by using Markoff’s sets. We provide an example to demonstrate the effectiveness of our results.     

Existence theorems for variational inclusion problems and the set-valued vector Ekeland variational principle in a complete metric space

In this paper, we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in Ekeland’s sense and the existence results for some variants of set-valued vector Ekeland variational principles in a complete metric space. Our results contain Ekeland’s variational principle as a special case and our approaches are different to those for any existence theorems for such problems.     

A Short Note on the Quasilinearization Method for Fractional Differential Equations

Recent literature shows that for certain classes of fractional differential equations the monotone iterative technique fails to guarantee the quadratic convergence of the quasilinearization method. The present work proves the quadratic convergence of the quasilinearization method and the existence and uniqueness of the solution of such a class of fractional differential equations. Our analysis depends upon the classical Kantorovich theorem on Newton's method. Various examples are discussed in order to illustrate our approach.     

Stability Analysis of nth Order Nonlinear Impulsive Differential Equations in Quasi–Banach Space

Abstract

This article is about Ulam’s type stability of n^th order nonlinear differential equations with fractional integrable impulses. It is a best procession to the stability of higher order fractional integrable impulsive differential equations in quasi–normed Banach space. Different Ulam’s type stability results are obtained by using the definitions of Riemann–Liouville fractional integral, Hölder’s inequality and the beta integral inequality.     

A combinatorial proof of the Dense Hindman’s Theorem

The Dense Hindman’s Theorem states that, in any finite coloring of the natural numbers, one may find a single color and a “dense” set B₁, for each b₁∈B₁ a “dense” set (depending on b₁), for each a “dense” set (depending on b₁,b₂), and so on, such that for any such sequence of b_i, all finite sums belong to the chosen color. (Here density is often taken to be “piecewise syndetic”, but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.     

Gronwall's inequality for a nabla fractional difference system with a retarded argument and an application

ABSTRACT

We study the nabla fractional difference system with retarded argument. There are two major ingredients. A Gronwall's inequality for the nabla case is given. This allows us to evaluate the solution of nabla fractional difference system. We shall illustrate the validity of our results by means of examples.     

On the converse Jensen inequality

We give a survey on the converse Jensen inequality and we show that several recently published inequalities are simple consequences of certain long time known results. We also give a new refinement of the converse Jensen inequality as well as improvements of some related results.     

Generalized Abel type integral equations with localized fractional integrals and derivatives

Generalized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.     

Comments on generalized Heron polynomials and Robbins’ conjectures

Heron’s formula for a triangle gives a polynomial for the square of its area in terms of the lengths of its three sides. There is a very similar formula, due to Brahmagupta, for the area of a cyclic quadrilateral in terms of the lengths of its four sides. (A polygon is cyclic if its vertices lie on a circle.) In both cases if A is the area of the polygon, (4A)² is a polynomial function of the square in the lengths of its edges. David Robbins in [D.P. Robbins, Areas of polygons inscribed in a circle, Discrete Comput. Geom. 12 (2) (1994) 223-236. MR 95g:51027; David P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly 102 (6) (1995) 523-530. MR 96k:51024] showed that for any cyclic polygon with n edges, (4A)² satisfies a polynomial whose coefficients are themselves polynomials in the edge lengths, and he calculated this polynomial for n=5 and n=6. He conjectured the degree of this polynomial for all n, and recently Igor Pak and Maksym Fedorchuk [Maksym Fedorchuk, Igor Pak, Rigidity and polynomial invariants of convex polytopes, Duke Math. J. 129 (2) (2005) 371-404. MR 2006f:52015] have shown that this conjecture of Robbins is true. Robbins also conjectured that his polynomial is monic, and that was shown in [V.V. Varfolomeev, Inscribed polygons and Heron polynomials (Russian. Russian summary), Mat. Sb. 194 (3) (2003) 3-24. MR 2004d:51014]. A short independent proof will be shown here.     

A numerical approximation for Volterra’s population growth model with fractional order

This paper presents a numerical scheme for approximate solutions of the fractional Volterra’s model for population growth of a species in a closed system. In fact, the Bessel collocation method is extended by using the time-fractional derivative in the Caputo sense to give solutions for the mentioned model problem. In this extended of the method, a generalization of the Bessel functions of the first kind is used and its matrix form is constructed. And then, the matrix form based on the collocation points is formed for the each term of this model problem. Hence, the method converts the model problem into a system of nonlinear algebraic equations. We give some numerical applications to show efficiency and accuracy of the method. In applications, the reliability of the technique is demonstrated by the error function based on accuracy of the approximate solution.     

Relating De Morgan triples with Atanassov’s intuitionistic De Morgan triples via automorphisms

In this paper the relation between De Morgan triples on the unit interval and Atanassov’s intuitionistic De Morgan triples is presented, showing how to obtain, in a canonical way, Atanassov’s intuitionistic De Morgan triples from De Morgan triples. Moreover, we also show that the automorphisms on the unit interval and on L∗ (the intuitionistic value lattice) are in one-to-one correspondence and how automorphisms on L∗ act on Atanassov’s intuitionistic De Morgan triples. It is also proved that the action of automorphisms and the canonical construction of De Morgan triples on L∗ commutes.