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This monograph looks at several trends in the investigation of singular solutions of nonlinear elliptic and parabolic equations. It discusses results on the existence and properties of weak and entropy solutions for elliptic second-order equations and some classes of fourth-order equations with L1-data and questions on the removability of singularities of solutions to elliptic and parabolic second-order equations in divergence form. It looks at localized and nonlocalized singularly peaking boundary regimes for different classes of quasilinear parabolic second- and high-order equations in divergence form. The book will be useful for researchers and post-graduate students that specialize in the field of the theory of partial differential equations and nonlinear analysis. Contents: Foreword Part I: Nonlinear elliptic equations with L^1-dataNonlinear elliptic equations of the second order with L^1-dataNonlinear equations of the fourth order with strengthened coercivity and L^1-dataPart II: Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order Removability of singularities of the solutions of quasilinear elliptic equations Removability of singularities of the solutions of quasilinear parabolic equations Quasilinear elliptic equations with coefficients from the Kato class Part III: Boundary regimes with peaking for quasilinear parabolic equations Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations Method of functional inequalities in peaking regimes for parabolic equations of higher orders Nonlocalized regimes with singular peaking Appendix: Formulations and proofs of the auxiliary results Bibliography

Differential equations, Elliptic. --- Differential equations, Nonlinear. --- Differential equations, Parabolic. --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Nonlinear differential equations --- Nonlinear theories --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Nonlinear elliptic equations of the second and fourth order, -data, Dirichlet problem, existence, uniqueness and summability of solutions, Quasilinear parabolic equations of the second and high order, Isolated singularities, removability of singularities, Localized boundary blow-up regimes, nonlocalized boundary regimes with peaking.

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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.

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

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During the past decades, the subject of calculus of integrals and derivatives of any arbitrary real or complex order has gained considerable popularity and impact. This is mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. In connection with this, great importance is attached to the publication of results that focus on recent and novel developments in the theory of any types of differential and fractional differential equation and inclusions, especially covering analytical and numerical research for such kinds of equations. This book is a compilation of articles from a Special Issue of Mathematics devoted to the topic of “Recent Investigations of Differential and Fractional Equations and Inclusions”. It contains some theoretical works and approximate methods in fractional differential equations and inclusions as well as fuzzy integrodifferential equations. Many of the papers were supported by the Bulgarian National Science Fund under Project KP-06-N32/7. Overall, the volume is an excellent witness of the relevance of the theory of fractional differential equations.

weakly upper semicontinuous --- essential maps --- homotopy --- Riemann-Liouville fractional differential equation --- delay --- lower and upper solutions --- monotone-iterative technique --- homoclinic solutions --- fourth-order p-Laplacian differential equations --- minimization theorem --- Clark’s theorem --- exponential dichotomy --- roughness --- asymptotically constant matrices --- double fuzzy Sumudu transform --- partial Volterra fuzzy integro-differential equations --- n-th order fuzzy partial H-derivative --- m-dissipative operators --- limit solutions --- integral solutions --- one-sided Perron condition --- Banach spaces --- fixed point --- complete metric space --- fractional differential equations --- optimal feedback control --- Voigt model --- alpha-model --- fractional derivative --- Riemann–Liouville fractional differential equations --- nonlocal boundary conditions --- positive solutions --- existence --- multiplicity --- Caputo derivative --- Riemann–Liouville integral --- multipoint and sub-strip boundary conditions --- fixed point theorem --- fractional Navier–Stokes equations --- variable delay --- modified fractional Halanay inequality --- generalized comparison principle --- dissipativity --- Fourier-Laplace transforms --- porous material --- eigenvalues method --- fractional time derivative

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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