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Real-world systems exhibit complex behavior, therefore novel mathematical approaches or modifications of classical ones have to be employed to precisely predict, monitor, and control complicated chaotic and stochastic processes. One of the most basic concepts that has to be taken into account while conducting research in all natural sciences is symmetry, and it is usually used to refer to an object that is invariant under some transformations including translation, reflection, rotation or scaling.The following Special Issue is dedicated to investigations of the concept of dynamical symmetry in the modelling and analysis of dynamic features occurring in various branches of science like physics, chemistry, biology, and engineering, with special emphasis on research based on the mathematical models of nonlinear partial and ordinary differential equations. Addressed topics cover theories developed and employed under the concept of invariance of the global/local behavior of the points of spacetime, including temporal/spatiotemporal symmetries.

Research & information: general --- Mathematics & science --- time delay --- third order differential equations --- difference scheme --- stability --- ϕc-Laplacian --- boundary value problem --- critical point theory --- three solutions --- multiple solutions --- fixed point theory --- boundary value problems --- generalized attracting horseshoe --- strange attractors --- poincaré map --- electronic circuits --- non-canonical differential equations --- second-order --- neutral delay --- mixed type --- oscillation criteria --- cell transplantation --- cytokines --- ischemic stroke --- numerical simulation --- runge-kutta method --- stability analysis --- ambient assisted living --- AAL --- ambient intelligence --- assisted living --- user-interfaces --- fuzzy logic --- vibrations --- symmetrical structures --- eigenmodes --- building --- concrete --- partial difference equation --- infinitely many small solutions --- (p,q)-Laplacian

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This Special Edition contains new results on Differential and Integral Equations and Systems, covering higher-order Initial and Boundary Value Problems, fractional differential and integral equations and applications, non-local optimal control, inverse, and higher-order nonlinear boundary value problems, distributional solutions in the form of a finite series of the Dirac delta function and its derivatives, asymptotic properties’ oscillatory theory for neutral nonlinear differential equations, the existence of extremal solutions via monotone iterative techniques, predator–prey interaction via fractional-order models, among others. Our main goal is not only to show new trends in this field but also to showcase and provide new methods and techniques that can lead to future research.

fourth-order differential equations --- neutral delay --- oscillation --- ψ-Caputo fractional derivative --- Cauchy problem extremal solutions --- monotone iterative technique --- upper and lower solutions --- third-order differential equation --- boundary value problem --- existence --- sign conditions --- mixed type nonlinear equation --- hilfer operator --- mittag–leffler function --- spectral parameter --- solvability --- equations of the pseudo-elliptic type of third order --- energy estimate --- analog of the Saint-Venant principle --- even-order differential equations --- Dirac delta function --- distributional solution --- Laplace transform --- power series solution --- integro-differential equation --- mixed type equation --- small parameter --- spectral parameters --- Caputo operators of different fractional orders --- inverse problem --- one value solvability --- Rosenzweig–MacArthur model --- fractional derivatives --- threshold harvesting --- distributed-order fractional calculus --- basic optimal control problem --- Pontryagin extremals

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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.

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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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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The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations.

Information technology industries --- odd-order differential equations --- Kneser solutions --- oscillatory solutions --- deviating argument --- fourth order --- differential equation --- oscillation --- advanced differential equations --- p-Laplacian equations --- comparison theorem --- oscillation criteria --- thrid-order --- delay differential equations --- oscillations --- Riccati transformations --- fourth-order delay equations --- differential operator --- unit disk --- univalent function --- analytic function --- subordination --- q-calculus --- fractional calculus --- fractional differential equation --- q-differential equation --- second order --- neutral differential equation --- (1/G′)-expansion method --- the Zhiber-Shabat equation --- (G′/G,1/G)-expansion method --- traveling wave solutions --- exact solutions --- Adomian decomposition method --- Caputo operator --- Natural transform --- Fornberg–Whitham equations --- generalized proportional fractional operator --- nonoscillatory behavior --- damping and forcing terms --- Volterra integral equations --- operational matrix of integration --- multi-wavelets --- time scales --- functional dynamic equations --- highly oscillatory integral --- Chebyshev polynomial --- nearly singular --- Levin quadrature rule --- adaptive mesh refinement --- la Cierva’s autogiro --- la Cierva’s equation --- stability --- differential equation with periodic coefficients --- interpolating scaling functions --- hyperbolic equation --- Galerkin method --- higher-order --- neutral delay --- center of mass --- conformal metric --- geodesic --- hyperbolic lever law --- non-canonical differential equations --- second-order --- mixed type