Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

Study of reactor constitutive model and analysis of nuclear reactor kinetics by fractional calculus approach

The diffusion theory model of neutron transport plays a crucial role in reactor theory since it is simple enough to allow scientific insight, and it is sufficiently realistic to study many important design problems. The neutrons are here characterized by a single energy or speed, and the model allows preliminary design estimates. The mathematical methods used to analyze such a model are the same as those applied in more sophisticated methods such as multi-group diffusion theory, and transport theory. The neutron diffusion and point kinetic equations are most vital models of nuclear engineering which are included to countless studies and applications under neutron dynamics. By the help of neutron diffusion concept, we understand the complex behavior of average neutron motion. The simplest group diffusion problems involve only, one group of neutrons, which for simplicity, are assumed to be all thermal neutrons. A more accurate procedure, particularly for thermal reactors, is to split the neutrons into two groups; in which case thermal neutrons are included in one group called the thermal or slow group and all the other are included in fast group. The neutrons within each group are lumped together and their diffusion, scattering, absorption and other interactions are described in terms of suitably average diffusion coefficients and cross-sections, which are collectively known as group constants. We have applied Variational Iteration Method and Modified Decomposition Method to obtain the analytical approximate solution of the Neutron Diffusion Equation with fixed source. The analytical methods like Homotopy Analysis Method and Adomian Decomposition Method have been used to obtain the analytical approximate solutions of neutron diffusion equation for both finite cylinders and bare hemisphere. In addition to these, the boundary conditions like zero flux as well as extrapolated boundary conditions are investigated. The explicit solution for critical radius and flux distributions are also calculated. The solution obtained in explicit form which is suitable for computer programming and other purposes such as analysis of flux distribution in a square critical reactor. The Homotopy Analysis Method is a very powerful and efficient technique which yields analytical solutions. With the help of this method we can solve many functional equations such as ordinary, partial differential equations, integral equations and so many other equations. It does not require enough memory space in computer, free from rounding off errors and discretization of space variables. By using the excellence of these methods, we obtained the solutions which have been shown graphically

Propagation via a Peridynamics Formulation: A Stochastic\Deterministic Perspective

Novel numerical methods for treating fractional differential and integrodifferential equations arising in non local mechanics formulations are proposed. For fractional differential equations arising in modeling oscillatory systems incorporating viscoelastic elements governed by fractional derivatives, the devised scheme is based on the Grunwald-Letnikov fractional derivative representation, dual time meshing technique and Taylor expansion. The proposed algorithm transforms the governing fractional differential equation into a second order differential equation with appropriate effective coefficients. The enhanced efficiency of the scheme hinges upon circumventing the calculation of the non local fractional derivative operator. Several examples of application are provided. Further, the concept of non locality, specifically viscoelasticity, governed by fractional derivatives is utilized to accurately model polyester materials. Specifically, the linear standard solid (Zener model) is extended to capture non linear viscoelastic behavior. Then, experimental data of polyester ropes are utilized using the Gauss Newton and Levenberg-Marquart minimization algorithm to determine the model parameters. Next, for integrodifferential equations arising in peridynamics theory of mechanics, an approach is formulated based on the inverse multi-quadric (IMQ) radial basis function (RBF) expansion and the Kansa collocation method. The devised scheme utilizes interpolation functions and basis function expansion for the spatial discretization of the peri dynamics equation. This significantly reduces the computational effort required to numerically treat the peri dynamics equations. Further, the proposed method is extended to account for mechanical systems with random material properties operating under random excitation. For this, the stochastic peridynamics governing equation of motion is solved using the benchmark Monte Carlo analysis and tools of stochastic analysis. The stochastic analysis is done by numerical evaluation of the requisite Neumann expansion using pertinent Monte Carlo simulations. Further, the usefulness of the radial basis function (RBF) collocation method in conjunction with a polynomial chaos expansion (PCE) is explored in stochastic mechanics problems. It is shown that the proposed approach renders further solution improvements in solving stochastic mechanics problems vis-a-vis the stochastic finite element method and the element free Galerkin method