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Complex systems with symmetry arise in many fields, at various length scales, including financial markets, social, transportation, telecommunication and power grid networks, world and country economies, ecosystems, molecular dynamics, immunology, living organisms, computational systems, and celestial and continuum mechanics. The emergence of new orders and structures in complex systems means symmetry breaking and transitions from unstable to stable states. Modeling complexity has attracted many researchers from different areas, dealing both with theoretical concepts and practical applications. This Special Issue fills the gap between the theory of symmetry-based dynamics and its application to model and analyze complex systems.

History of engineering & technology --- multi-agent system (MAS) --- reinforcement learning (RL) --- mobile robots --- function approximation --- Opportunistic complex social network --- cooperative --- neighbor node --- probability model --- social relationship --- adapted PageRank algorithm --- PageRank vector --- networks centrality --- multiplex networks --- biplex networks --- divided difference --- radius of convergence --- Kung–Traub method --- local convergence --- Lipschitz constant --- Banach space --- fractional calculus --- Caputo derivative --- generalized Fourier law --- Laplace transform --- Fourier transform --- Mittag–Leffler function --- non-Fourier heat conduction --- Mei symmetry --- conserved quantity --- adiabatic invariant --- quasi-fractional dynamical system --- non-standard Lagrangians --- complex systems --- symmetry-breaking --- bifurcation theory --- complex networks --- nonlinear dynamical systems

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A plethora of problems from diverse disciplines such as Mathematics, Mathematical: Biology, Chemistry, Economics, Physics, Scientific Computing and also Engineering can be formulated as an equation defined in abstract spaces using Mathematical Modelling. The solutions of these equations can be found in closed form only in special case. That is why researchers and practitioners utilize iterative procedures from which a sequence is being generated approximating the solution under some conditions on the initial data. This type of research is considered most interesting and challenging. This is our motivation for the introduction of this special issue on Iterative Procedures.

Lipschitz condition --- order of convergence --- Scalar equations --- local and semilocal convergence --- multiple roots --- Nondifferentiable operator --- optimal iterative methods --- Order of convergence --- convergence order --- fast algorithms --- iterative method --- computational convergence order --- generalized mixed equilibrium problem --- nonlinear equations --- systems of nonlinear equations --- Chebyshev’s iterative method --- local convergence --- iterative methods --- divided difference --- Multiple roots --- semi-local convergence --- scalar equations --- left Bregman asymptotically nonexpansive mapping --- basin of attraction --- maximal monotone operator --- Newton–HSS method --- general means --- Steffensen’s method --- derivative-free method --- simple roots --- fixed point problem --- split variational inclusion problem --- weighted-Newton method --- ball radius of convergence --- Traub–Steffensen method --- Newton’s method --- fractional derivative --- Banach space --- multiple-root solvers --- uniformly convex and uniformly smooth Banach space --- Fréchet-derivative --- optimal convergence --- Optimal iterative methods --- basins of attraction --- nonlinear equation

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This book presents collective works published in the recent Special Issue (SI) entitled "Multivariate Approximation for Solving ODE and PDE". These papers describe the different approaches and related objectives in the field of multivariate approximation. The articles in fact present specific contents of numerical methods for the analysis of the approximation, as well as the study of ordinary differential equations (for example oscillating with delay) or that of partial differential equations of the fractional order, but all linked by the objective to present analytical or numerical techniques for the simplification of the study of problems involving relationships that are not immediately computable, thus allowing to establish a connection between different fields of mathematical analysis and numerical analysis through different points of view and investigation. The present contents, therefore, describe the multivariate approximation theory, which is today an increasingly active research area that deals with a multitude of problems in a wide field of research. This book brings together a collection of inter-/multi-disciplinary works applied to many areas of applied mathematics in a coherent manner.

Research & information: general --- Mathematics & science --- nonlinear equations --- iteration methods --- one-point methods --- order of convergence --- oscillatory solutions --- nonoscillatory solutions --- second-order --- neutral differential equations --- multiple roots --- optimal convergence --- bivariate function --- divided difference --- inverse difference --- blending difference --- continued fraction --- Thiele–Newton’s expansion --- Viscovatov-like algorithm --- symmetric duality --- non-differentiable --- (G,αf)-invexity/(G,αf)-pseudoinvexity --- (G,αf)-bonvexity/(G,αf)-pseudobonvexity --- duality --- support function --- nondifferentiable --- strictly pseudo (V,α,ρ,d)-type-I --- unified dual --- efficient solutions --- Iyengar inequality --- right and left generalized fractional derivatives --- iterated generalized fractional derivatives --- generalized fractional Taylor’s formulae --- poisson equation --- domain decomposition --- asymmetric iterative schemes --- group explicit --- parallel computation --- even-order differential equations --- neutral delay --- oscillation --- Hilbert transform --- Hadamard transform --- hypersingular integral --- Bernstein polynomials --- Boolean sum --- simultaneous approximation --- equidistant nodes --- fourth-order --- delay differential equations --- riccati transformation --- parameter estimation --- physical modelling --- oblique decomposition --- least-squares

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Solving nonlinear equations in Banach spaces (real or complex nonlinear equations, nonlinear systems, and nonlinear matrix equations, among others), is a non-trivial task that involves many areas of science and technology. Usually the solution is not directly affordable and require an approach using iterative algorithms. This Special Issue focuses mainly on the design, analysis of convergence, and stability of new schemes for solving nonlinear problems and their application to practical problems. Included papers study the following topics: Methods for finding simple or multiple roots either with or without derivatives, iterative methods for approximating different generalized inverses, real or complex dynamics associated to the rational functions resulting from the application of an iterative method on a polynomial. Additionally, the analysis of the convergence has been carried out by means of different sufficient conditions assuring the local, semilocal, or global convergence. This Special issue has allowed us to present the latest research results in the area of iterative processes for solving nonlinear equations as well as systems and matrix equations. In addition to the theoretical papers, several manuscripts on signal processing, nonlinear integral equations, or partial differential equations, reveal the connection between iterative methods and other branches of science and engineering.

Lipschitz condition --- heston model --- rectangular matrices --- computational efficiency --- Hull–White --- order of convergence --- signal and image processing --- dynamics --- divided difference operator --- engineering applications --- smooth and nonsmooth operators --- Newton-HSS method --- higher order method --- Moore–Penrose --- asymptotic error constant --- multiple roots --- higher order --- efficiency index --- multiple-root finder --- computational efficiency index --- Potra–Pták method --- nonlinear equations --- system of nonlinear equations --- purely imaginary extraneous fixed point --- attractor basin --- point projection --- fixed point theorem --- convex constraints --- weight function --- radius of convergence --- Frédholm integral equation --- semi-local convergence --- nonlinear HSS-like method --- convexity --- accretive operators --- Newton-type methods --- multipoint iterations --- banach space --- Kantorovich hypothesis --- variational inequality problem --- Newton method --- semilocal convergence --- least square problem --- Fréchet derivative --- Newton’s method --- iterative process --- Newton-like method --- Banach space --- sixteenth-order optimal convergence --- nonlinear systems --- Chebyshev–Halley-type --- Jarratt method --- iteration scheme --- Newton’s iterative method --- basins of attraction --- drazin inverse --- option pricing --- higher order of convergence --- non-linear equation --- numerical experiment --- signal processing --- optimal methods --- rate of convergence --- n-dimensional Euclidean space --- non-differentiable operator --- projection method --- Newton’s second order method --- intersection --- planar algebraic curve --- Hilbert space --- conjugate gradient method --- sixteenth order convergence method --- Padé approximation --- optimal iterative methods --- error bound --- high order --- Fredholm integral equation --- global convergence --- iterative method --- integral equation --- ?-continuity condition --- systems of nonlinear equations --- generalized inverse --- local convergence --- iterative methods --- multi-valued quasi-nonexpasive mappings --- R-order --- finite difference (FD) --- nonlinear operator equation --- basin of attraction --- PDE --- King’s family --- Steffensen’s method --- nonlinear monotone equations --- Picard-HSS method --- nonlinear models --- the improved curvature circle algorithm --- split variational inclusion problem --- computational order of convergence --- with memory --- multipoint iterative methods --- Kung–Traub conjecture --- multiple zeros --- fourth order iterative methods --- parametric curve --- optimal order --- nonlinear equation

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This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods.

Research & information: general --- Mathematics & science --- mathematical modeling --- infiltration well --- differential equations --- porous medium --- fractal conductivity model --- incomplete rankings --- Kendall’s tau --- permutation graph --- competitive balance --- Spotify --- collocation --- volterra integral equation --- highly oscillatory --- convergence --- areal porosity --- volumetric porosity --- fractal area-volume relationship --- tortuosity factor --- joint probability --- corrugated box printing machine --- modified Delphi method --- analytic network process (ANP) --- supplier --- nonlinear system --- iterative method --- divided difference operator --- stability --- parameter plane --- dynamical plane --- random hyperbolic model --- random laplace transform --- numerical integration --- monte carlo method --- numerical simulation --- talbot algorithm --- stochastic perturbations --- random nonlinear oscillator --- maximum entropy principle --- probability density function --- stationary Gaussian noise --- random mean square parabolic model --- finite degree of randomness --- random finite difference scheme --- relativistic harmonic oscillator --- kinematics of a particle --- special relativity --- nonlinear problems in mechanics --- equations of motion in gravitational theory --- virus propagation --- stochastic modeling --- Gillespie algorithm --- conservative formulation --- multidimensional fragmentation equation --- weight functions --- finite volume scheme --- contamination plume --- advection-diffusion --- universal curves --- Dirichlet-to-Neumann map --- Schrödinger operator --- contagion effect --- difference equation --- elections --- labor condition --- mathematical compartmental discrete model --- political corruption --- revolving doors --- sensitivity analysis --- simulation --- numerical methods --- integro-interpolation method --- splitting method --- convergence of models --- standard deviation of the error --- diabetic retinopathy --- ocular fundus --- laser coagulation --- optical coherence tomography --- image processing --- segmentation --- safe treatment --- Hermite interpolation --- nodal systems --- unit circle --- circular membrane --- fluid-structure interaction --- differential-integral equations --- power series method --- closed-form solution --- time series model --- wavelet transform --- ARIMA model --- neural network NARX --- ionospheric parameters --- courtyard --- climate change --- microclimate --- Support Vector Regression (SVR) --- machine learning --- matrix functions --- matrix hyperbolic tangent --- matrix exponential --- Taylor series --- matrix polynomial evaluation --- n/a --- Kendall's tau --- Schrödinger operator