**Abstract: **This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform.

**Abstract: **In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus (FOC) approaches. The numerical simulations were conducted and the time histories and pseudo-phase portraits presented. Both systems, the one that already had a damping behavior (Spring-Mass-Damper) and the system that did not present any sort of damping behavior (Simple Pendulum), showed signs indicating a possible better capacity of attenuation of their respective oscillation amplitudes. This implication could mean that if the selection of the order of the derivative is conveniently made, systems that need greater intensities of damping or vibrating absorbers may benefit from using fractional order in dynamics and possibly in control of the aforementioned systems. Thereafter, we believe that the results described in this paper may offer greater insights into the complex behavior of these systems, and thus instigate more research efforts in this direction.

**Abstract: **We review new electrodynamics models of interacting charged point particles and related fundamental physical aspects, motivated by the classical A.M. Ampère magnetic and H. Lorentz force laws electromagnetic field expressions. Based on the Feynman proper time paradigm and a recently devised vacuum field theory approach to the Lagrangian and Hamiltonian, the formulations of alternative classical electrodynamics models are analyzed in detail and their Dirac type quantization is suggested. Problems closely related to the radiation reaction force and electron mass inertia are analyzed. The validity of the Abraham-Lorentz electromagnetic electron mass origin hypothesis is argued. The related electromagnetic Dirac–Fock–Podolsky problem and symplectic properties of the Maxwell and Yang–Mills type dynamical systems are analyzed. The crucial importance of the remaining reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized.

**Abstract: **We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD.

**Abstract: **In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find the conditions under which the even moments are non-negative.

**Abstract: **We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the time variable \(t\), \(\textit{B}\) is a second order elliptic operator with respect to the space variable \(x\in\mathbb{R}^d\) and \(\dot W^H\) a time homogeneous fractional Gaussian noise of Hurst parameter \(H=(H_1, \cdots, H_d)\). We obtain conditions satisfied by \(\alpha\) and \(H\), so that the square integrable solution \(u\) exists uniquely.