A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

Fuzzy Geometry of Phase Space and Quantization of Massive Fields

The quantum space-time and the phase space with fuzzy structure is investigated as the possible quantization formalism. In this theory the state of nonrelativistic particle corresponds to the element of fuzzy ordered set (Foset) - fuzzy point. Due to Foset partial (weak) ordering, particle's space coordinate x acquires principal uncertainty dx. It's shown that Shroedinger formalism of Quantum Mechanics can be completely derived from consideration of particle evolution in fuzzy phase space with minimal number of axioms.Comment: 13 pages, Talk given at QFEXT07 Workshop, Leipzig, Sept. 200

Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

The management of intelligence-assisted finite element analysis technology

Artificial Intelligence (AI) approaches to Finite Element Analysis (FEA), have had tentative degrees of success over the last few years and some authors have argued that effective FEA can help in the manufacture reliability and safety aspects of engineered artefacts. The author of this paper reviews how such AI techniques have been applied and in this light, the author then uses a Fuzzy Cognitive Mapping (FCM), to develop a framework for the management of intelligence-assisted FEA

Rigorous treatment of electrostatics for spatially varying dielectrics based on energy minimization

A novel energy minimization formulation of electrostatics that allows computation of the electrostatic energy and forces to any desired accuracy in a system with arbitrary dielectric properties is presented. An integral equation for the scalar charge density is derived from an energy functional of the polarization vector field. This energy functional represents the true energy of the system even in non-equilibrium states. Arbitrary accuracy is achieved by solving the integral equation for the charge density via a series expansion in terms of the equation's kernel, which depends only on the geometry of the dielectrics. The streamlined formalism operates with volume charge distributions only, not resorting to introducing surface charges by hand. Therefore, it can be applied to any spatial variation of the dielectric susceptibility, which is of particular importance in applications to biomolecular systems. The simplicity of application of the formalism to real problems is shown with analytical and numerical examples.Comment: 27 pages, 5 figure