Fractional Euler Limits and Their Applications

Nabla Euler -Lagrange equations in discrete fractional variational calculus within Riemann and Caputo

Different fractional difference types of Euler-Lagrange equations are obtained within Riemann and Caputo by making use of different versions of integration by part forumlas in fractional difference calculus. An example is presented to illustrate part of the results.

Solutions of fractional logistic equations by Euler's numbers

In this paper, we solve in the convergence set, the fractional logistic equation making use of Euler's numbers. To our knowledge, the answer is still an open question. The key point is that the coefficients can be connected with Euler's numbers, and then they can be explicitly given. The constrained of our approach is that the formula is not valid outside the convergence set, The idea of the proof consists to explore some analogies with logistic function and Euler's numbers, and then to generalize them in the fractional case.

Fractional Euler-Lagrange differential equations via Caputo derivatives

We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.

分数和与Euler-like恒等式

Fractional Sums and Euler-like Identities

我们引入了一个自然的定义，为形式\[ \sum_{ν=1}^x f(ν) \]的总和时，项目的数量x是一个相当任意的实数或甚至复数。由此产生的理论包括由Γ函数或欧拉的鲜为人知的公式\sum_{ν=1}^{-1/2} \frac 1ν= -2 \ln 2的阶乘的已知插值。许多经典的恒等式，如等比级数和二项式定理，都很好地扩展到了这种更一般的情形。分数项的和与特殊函数密切相关，特别是Riemann和Hurwitz zeta函数。许多关于分数和的结果可以被解释为经典的无穷和或乘积或极限，包括像\[ \lim_{n\to\infty}[ e^{\frac n 4(4n+1)}n^{-\frac 1 8 - n(n+1)}(2π)^{-\frac n 2} \prod_{k=1}^{2n} Γ(1+\frac k 2)^{k(-1)^k} ] = \sqrt[12]{2} \exp({5/24} - \frac 3 2 ζ'(-1) -\frac{7ζ(3)}{16π^2}) \]这样的恒等式，其中一些似乎是新的。

带弱奇异核的随机分数阶中立型积分微分方程的Euler-Maruyama逼近

Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel

本文研究带弱奇异核的非线性随机分数阶中立型积分微分方程，在局部Lipschitz等条件下建立真解性质，开发Euler-Maruyama数值方法并证明其强收敛性，还在全局条件下确定了精确收敛率。

A Fractional Variational Approach for Modelling Dissipative Mechanical Systems: Continuous and Discrete Settings

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linear change of variables. This principle relies on a particular restriction upon the admissible variation of the curves. In the case of the half-derivative and mechanical Lagrangians, i.e. kinetic minus potential energy, the restricted fractional Euler-…