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Fractional calculus provides the possibility of introducing integrals and derivatives of an arbitrary order in the mathematical modelling of physical processes, and it has become a relevant subject with applications to various fields, such as anomalous diffusion, propagation in different media, and propogation in relation to materials with different properties. However, many aspects from theoretical and practical points of view have still to be developed in relation to models based on fractional operators. This Special Issue is related to new developments on different aspects of fractional differential equations, both from a theoretical point of view and in terms of applications in different fields such as physics, chemistry, or control theory, for instance. The topics of the Issue include fractional calculus, the mathematical analysis of the properties of the solutions to fractional equations, the extension of classical approaches, or applications of fractional equations to several fields.
fractional q-difference equation --- existence and uniqueness --- positive solutions --- fixed point theorem on mixed monotone operators --- fractional p-Laplacian --- Kirchhoff-type equations --- fountain theorem --- modified functional methods --- Moser iteration method --- fractional-order neural networks --- delays --- distributed delays --- impulses --- Mittag–Leffler synchronization --- Lyapunov functions --- Razumikhin method --- generalized convexity --- b-vex functions --- sub-b-s-convex functions --- oscillation --- nonlinear differential system --- delay differential system --- ?-fractional derivative --- positive solution --- fractional thermostat model --- fixed point index --- dependence on a parameter --- Hermite–Hadamard’s Inequality --- Convex Functions --- Power-mean Inequality --- Jenson Integral Inequality --- Riemann—Liouville Fractional Integration --- Laplace Adomian Decomposition Method (LADM) --- Navier-Stokes equation --- Caputo Operator --- fractional-order system --- model order reduction --- controllability and observability Gramians --- energy inequality --- integral conditions --- fractional wave equation --- existence and uniqueness --- initial boundary value problem --- conformable fractional derivative --- conformable partial fractional derivative --- conformable double Laplace decomposition method --- conformable Laplace transform --- singular one dimensional coupled Burgers’ equation
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The many technical and computational problems that appear to be constantly emerging in various branches of physics and engineering beg for a more detailed understanding of the fundamental mathematics that serves as the cornerstone of our way of understanding natural phenomena. The purpose of this Special Issue was to establish a brief collection of carefully selected articles authored by promising young scientists and the world's leading experts in pure and applied mathematics, highlighting the state-of-the-art of the various research lines focusing on the study of analytical and numerical mathematical methods for pure and applied sciences.
ultraparabolic equation --- ultradiffusion process --- probabilistic representation --- mathematical finance --- linear elastostatics --- layer potentials --- fredholmian operators --- fractional differential equations --- fractional derivative --- Abel-type integral --- time delay --- distributed lag --- gamma distribution --- macroeconomics --- Keynesian model --- integral transforms --- Laplace integral transform --- transmutation operator --- generating operator --- integral equations --- differential equations --- operational calculus of Mikusinski type --- Mellin integral transform --- fractional derivative --- fractional integral --- Mittag–Leffler function --- Riemann–Liouville derivative --- Caputo derivative --- Grünwald–Letnikov derivative --- space-time fractional diffusion equation --- fractional Laplacian --- subordination principle --- Mittag-Leffler function --- Bessel function --- exterior calculus --- exterior algebra --- electromagnetism --- Maxwell equations --- differential forms --- tensor calculus --- Fourier Theory --- DFT in polar coordinates --- polar coordinates --- multidimensional DFT --- discrete Hankel Transform --- discrete Fourier Transform --- Orthogonality --- multispecies biofilm --- biosorption --- free boundary value problem --- heavy metals toxicity --- method of characteristics --- relativistic diffusion equation --- Caputo fractional derivatives of a function with respect to another function --- Bessel-Riesz motion --- Mittag–Leffler function --- matrix function --- Schur decomposition --- Laplace transform --- fractional calculus --- central limit theorem --- anomalous diffusion --- stable distribution --- fractional calculus --- power law --- n/a
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This book is focused on fractional order systems. Historically, fractional calculus has been recognized since the inception of regular calculus, with the first written reference dated in September 1695 in a letter from Leibniz to L’Hospital. Nowadays, fractional calculus has a wide area of applications in areas such as physics, chemistry, bioengineering, chaos theory, control systems engineering, and many others. In all those applications, we deal with fractional order systems in general. Moreover, fractional calculus plays an important role even in complex systems and therefore allows us to develop better descriptions of real-world phenomena. On that basis, fractional order systems are ubiquitous, as the whole real world around us is fractional. Due to this reason, it is urgent to consider almost all systems as fractional order systems.
anomalous diffusion --- complexity --- magnetic resonance imaging --- fractional calculus --- fractional complex networks --- adaptive control --- pinning synchronization --- time-varying delays --- impulses --- reaction–diffusion terms --- fractional calculus --- mass absorption --- diffusion-wave equation --- Caputo derivative --- harmonic impact --- Laplace transform --- Fourier transform --- Mittag-Leffler function --- fractional calculus --- fractional-order system --- long memory --- time series --- Hurst exponent --- fractional --- control --- PID --- parameter --- meaning --- audio signal processing --- linear prediction --- fractional derivative --- musical signal --- optimal randomness --- swarm-based search --- cuckoo search --- heavy-tailed distribution --- global optimization
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Due to the influence of pore-throat size distribution, pore connectivity, and microscale fractures, the transport, distribution, and residual saturation of fluids in porous media are difficult to characterize. Petrophysical methods in natural porous media have attracted great attention in a variety of fields, especially in the oil and gas industry. A wide range of research studies have been conducted on the characterization of porous media covers and multiphase flow therein. Reliable approaches for characterizing microstructure and multiphase flow in porous media are crucial in many fields, including the characterization of residual water or oil in hydrocarbon reservoirs and the long-term storage of supercritical CO2 in geological formations. This book gathers together 15 recent works to emphasize fundamental innovations in the field and novel applications of petrophysics in unconventional reservoirs, including experimental studies, numerical modeling (fractal approach), and multiphase flow modeling/simulations. The relevant stakeholders of this book are authorities and service companies working in the petroleum, subsurface water resources, air and water pollution, environmental, and biomaterial sectors.
Wilkins equation --- non-laminar flow --- turbulence modelling --- porous media --- oil tanker --- temperature drop --- oscillating motion --- numerical simulation --- soil-water characteristic curve --- initial void ratio --- air-entry value --- fractal dimension --- fractal model --- oil properties --- diffusion coefficient --- supercritical CO2 --- Peng-Robinson equation of state (PR EOS) --- CT --- digital rock --- microfractures --- Lattice Boltzmann method --- pore-scale simulations --- tight sandstone --- pore structure --- multifractal --- classification --- Ordos Basin --- loose media --- coal --- porosity --- true density --- bulk density --- overburden pressure --- particle size --- tight conglomerate --- fracture characterization and prediction --- fractal method --- salt rock --- creep --- damage --- fractional derivative --- acoustic emission --- marine gas hydrate --- submarine landslide --- greenhouse gas emission --- lifecycle management --- hazard prevention --- multilayer reservoir --- interlayer interference --- producing degree --- seepage resistance --- wellbore multiphase flow --- inclined angle --- liquid rate --- gas rate --- pressure drawdown model with new coefficients --- base-level cycle --- pore structure --- mouth bar sand body --- Huanghua Depression --- isotopic composition --- methane --- gas hydrate --- South China Sea --- Bakken Formation --- pore structure --- controlling factors --- low-temperature nitrogen adsorption --- petrophysics --- fractal porous media --- unconventional reservoirs --- multiphase flow
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In recent decades, the study of groundwater flow and solute transport has advanced into new territories that are beyond conventional theories, such as Darcy’s law and Fick’s law. The studied media have changed from permeable porous and fractured ones to much less permeable ones, such as clay and shale. The studied pore sizes have also changed from millimetres to micro-meters or even nano-meters. The objective of this Special Issue is to report recent advances in groundwater flow and solute transport that push the knowledge boundary into new territories which include, but are not limited to, flow and transport in sloping aquifer/hillslopes, coupled unsaturated and saturated flow, coupled aquifer-vertical/horizontal/slant well flow, interaction of aquifer with connected and disconnected rivers, non-Darcian flow, anomalous transport beyond the Fickian scheme, and flow and transport in extremely small pore spaces such as shale and tight sandstones. Contributions focusing on innovative experimental, numerical, and analytical methods for understanding unconventional problems, such as the above-listed ones, are encouraged, and contributions addressing flow and transport at interfaces of different media and crossing multiple temporal and spatial scales are of great value
soil formation --- percolation --- infiltration --- erosion --- Levy stable distribution --- permeameter test --- hydraulic conductivity --- silty clay --- solute transport --- nuclear waste disposal --- the Beishan area --- TOUGH2 --- groundwater flow --- assessment --- rough single fracture --- solute transport --- non-Darcian --- non-Fickian --- heterogeneity --- bimsoils --- water flow --- slenderness effect --- permeability coefficient --- non-Darcy flow --- hydrologic exchange --- SW–GW interaction --- field measurements --- Columbia River --- steady-state vertical flux --- evaporation calculation --- unsaturated flow --- semi-analytical solution --- solute longitudinal dispersion --- evolving-scale log-conductivity --- first-order analytical approach --- stochastic Lagrangian framework --- fractured aquifers --- seawater intrusion --- flow modeling --- salinity map --- groundwater ERT --- groundwater flow model --- numerical simulation --- uncertainty --- IUV --- IUM --- perturbation method --- Monte Carlo --- GFModel --- radioactive contaminant --- fractional derivative --- analytical solution --- Ulan Buh Desert --- DSR --- infiltration --- desert farmland --- irrigation --- sustainable development --- water resource utilization efficiency --- n/a
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This Special Issue aims to be a compilation of new results in the areas of differential and difference Equations, covering boundary value problems, systems of differential and difference equations, as well as analytical and numerical methods. The objective is to provide an overview of techniques used in these different areas and to emphasize their applicability to real-life phenomena, by the inclusion of examples. These examples not only clarify the theoretical results presented, but also provide insight on how to apply, for future works, the techniques used.
Legendre wavelets --- collocation method --- three-step Taylor method --- asymptotic stability --- time-dependent partial differential equations --- non-instantaneous impulses --- Caputo fractional derivative --- differential equations --- state dependent delays --- lipschitz stability --- limit-periodic solutions --- difference equations --- exponential dichotomy --- strong nonlinearities --- effective existence criteria --- population dynamics --- discrete Lyapunov equation --- difference equations --- Hilbert space --- dichotomy --- exponential stability --- ?-Laplacian operator --- mean curvature operator --- heteroclinic solutions --- problems in the real line --- lower and upper solutions --- Nagumo condition on the real line --- fixed point theory --- coupled nonlinear systems --- functional boundary conditions --- Schauder’s fixed point theory --- Arzèla Ascoli theorem --- lower and upper solutions --- first order periodic systems --- SIRS epidemic model --- mathematical modelling --- Navier–Stokes equations --- global solutions --- regular solutions --- a priori estimates --- weak solutions --- kinetic energy --- dissipation --- Bäcklund transformation --- Clairin’s method --- generalized Liouville equation --- Miura transformation --- Korteweg-de Vries equation --- second-order differential/difference/q-difference equation of hypergeometric type --- non-uniform lattices --- divided-difference equations --- polynomial solution --- integro-differentials --- Sumudu decomposition method --- dynamical system
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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.
Banach space --- weighted-Newton method --- local convergence --- Fréchet-derivative --- ball radius of convergence --- Nondifferentiable operator --- nonlinear equation --- divided difference --- Lipschitz condition --- convergence order --- local and semilocal convergence --- scalar equations --- computational convergence order --- Steffensen’s method --- basins of attraction --- nonlinear equations --- multiple-root solvers --- Traub–Steffensen method --- fast algorithms --- Multiple roots --- Optimal iterative methods --- Scalar equations --- Order of convergence --- simple roots --- Newton’s method --- computational convergence order --- nonlinear equations --- split variational inclusion problem --- generalized mixed equilibrium problem --- fixed point problem --- maximal monotone operator --- left Bregman asymptotically nonexpansive mapping --- uniformly convex and uniformly smooth Banach space --- nonlinear equations --- multiple roots --- derivative-free method --- optimal convergence --- multiple roots --- optimal iterative methods --- scalar equations --- order of convergence --- Newton–HSS method --- systems of nonlinear equations --- semi-local convergence --- local convergence --- convergence order --- Banach space --- iterative method --- nonlinear equations --- Chebyshev’s iterative method --- fractional derivative --- basin of attraction --- nonlinear equations --- iterative methods --- general means --- basin of attraction
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Researches and investigations involving the theory and applications of integral transforms and operational calculus are remarkably wide-spread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences.
highly oscillatory --- convolution quadrature rule --- volterra integral equation --- Bessel kernel --- convergence --- higher order Schwarzian derivatives --- Janowski star-like function --- Janowski convex function --- bound on derivatives --- tangent numbers --- tangent polynomials --- Carlitz-type q-tangent numbers --- Carlitz-type q-tangent polynomials --- (p,q)-analogue of tangent numbers and polynomials --- (p,q)-analogue of tangent zeta function --- symmetric identities --- zeros --- Lommel functions --- univalent functions --- starlike functions --- convex functions --- inclusion relationships --- analytic function --- Hankel determinant --- exponential function --- upper bound --- nonlinear boundary value problems --- fractional-order differential equations --- Riemann-Stieltjes functional integral --- Liouville-Caputo fractional derivative --- infinite-point boundary conditions --- advanced and deviated arguments --- existence of at least one solution --- Fredholm integral equation --- Schauder fixed point theorem --- Hölder condition --- generalized Kuramoto–Sivashinsky equation --- modified Kudryashov method --- exact solutions --- Maple graphs --- analytic function --- Hadamard product (convolution) --- partial sum --- Srivastava–Tomovski generalization of Mittag–Leffler function --- subordination --- differential equation --- differential inclusion --- Liouville–Caputo-type fractional derivative --- fractional integral --- existence --- fixed point --- Bernoulli spiral --- Grandi curves --- Chebyshev polynomials --- pseudo-Chebyshev polynomials --- orthogonality property --- symmetric --- encryption --- password --- hash --- cryptography --- PBKDF --- q–Bleimann–Butzer–Hahn operators --- (p,q)-integers --- (p,q)-Bernstein operators --- (p,q)-Bleimann–Butzer–Hahn operators --- modulus of continuity --- rate of approximation --- K-functional --- Hurwitz-Lerch zeta function --- generalized functions --- analytic number theory --- ?-generalized Hurwitz-Lerch zeta functions --- derivative properties --- series representation --- basic hypergeometric functions --- generating functions --- q-polynomials --- analytic functions --- Mittag–Leffler functions --- starlike functions --- convex functions --- Hardy space --- vibrating string equation --- initial conditions --- spectral decomposition --- regular solution --- the uniqueness of the solution --- the existence of a solution --- analytic --- ?-convex function --- starlike function --- strongly-starlike function --- subordination --- q -Sheffer–Appell polynomials --- generating relations --- determinant definition --- recurrence relation --- q -Hermite–Bernoulli polynomials --- q -Hermite–Euler polynomials --- q -Hermite–Genocchi polynomials --- Volterra integral equations --- highly oscillatory Bessel kernel --- Hermite interpolation --- direct Hermite collocation method --- piecewise Hermite collocation method --- differential operator --- q-hypergeometric functions --- meromorphic function --- Mittag–Leffler function --- Hadamard product --- differential subordination --- starlike functions --- Bell numbers --- radius estimate --- (p, q)-integers --- Dunkl analogue --- generating functions --- generalization of exponential function --- Szász operator --- modulus of continuity --- function spaces and their duals --- distributions --- tempered distributions --- Schwartz testing function space --- generalized functions --- distribution space --- wavelet transform of generalized functions --- Fourier transform --- analytic function --- subordination --- Dziok–Srivastava operator --- nonlinear boundary value problem --- nonlocal --- multi-point --- multi-strip --- existence --- Ulam stability --- functions of bounded boundary and bounded radius rotations --- subordination --- functions with positive real part --- uniformly starlike and convex functions --- analytic functions --- univalent functions --- starlike and q-starlike functions --- q-derivative (or q-difference) operator --- sufficient conditions --- distortion theorems --- Janowski functions --- analytic number theory --- ?-generalized Hurwitz–Lerch zeta functions --- derivative properties --- recurrence relations --- integral representations --- Mellin transform --- natural transform --- Adomian decomposition method --- Caputo fractional derivative --- generalized mittag-leffler function --- analytic functions --- Hadamard product --- starlike functions --- q-derivative (or q-difference) operator --- Hankel determinant --- q-starlike functions --- fuzzy volterra integro-differential equations --- fuzzy general linear method --- fuzzy differential equations --- generalized Hukuhara differentiability --- spectrum symmetry --- DCT --- MFCC --- audio features --- anuran calls --- analytic functions --- convex functions --- starlike functions --- strongly convex functions --- strongly starlike functions --- uniformly convex functions --- Struve functions --- truncated-exponential polynomials --- monomiality principle --- generating functions --- Apostol-type polynomials and Apostol-type numbers --- Bernoulli, Euler and Genocchi polynomials --- Bernoulli, Euler, and Genocchi numbers --- operational methods --- summation formulas --- symmetric identities --- Euler numbers and polynomials --- q-Euler numbers and polynomials --- Hurwitz-Euler eta function --- multiple Hurwitz-Euler eta function --- higher order q-Euler numbers and polynomials --- (p, q)-Euler numbers and polynomials of higher order --- symmetric identities --- symmetry of the zero
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The use of scientific computing tools is currently customary for solving problems at several complexity levels in Applied Sciences. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and better performing numerical methods that are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.
time fractional differential equations --- mixed-index problems --- analytical solution --- asymptotic stability --- conservative problems --- Hamiltonian problems --- energy-conserving methods --- Poisson problems --- Hamiltonian Boundary Value Methods --- HBVMs --- line integral methods --- constrained Hamiltonian problems --- Hamiltonian PDEs --- highly oscillatory problems --- boundary element method --- finite difference method --- floating strike Asian options --- continuous geometric average --- barrier options --- isogeometric analysis --- adaptive methods --- hierarchical splines --- THB-splines --- local refinement --- linear systems --- preconditioners --- Cholesky factorization --- limited memory --- Volterra integral equations --- Volterra integro–differential equations --- collocation methods --- multistep methods --- convergence --- B-spline --- optimal basis --- fractional derivative --- Galerkin method --- collocation method --- spectral (eigenvalue) and singular value distributions --- generalized locally Toeplitz sequences --- discretization of systems of differential equations --- higher-order finite element methods --- discontinuous Galerkin methods --- finite difference methods --- isogeometric analysis --- B-splines --- curl–curl operator --- time harmonic Maxwell’s equations and magnetostatic problems --- low rank completion --- matrix ODEs --- gradient system --- ordinary differential equations --- Runge–Kutta --- tree --- stump --- order --- elementary differential --- edge-histogram --- edge-preserving smoothing --- histogram specification --- initial value problems --- one-step methods --- Hermite–Obreshkov methods --- symplecticity --- B-splines --- BS methods --- hyperbolic partial differential equations --- high order discontinuous Galerkin finite element schemes --- shock waves and discontinuities --- vectorization and parallelization --- high performance computing --- generalized Schur algorithm --- null-space --- displacement rank --- structured matrices --- stochastic differential equations --- stochastic multistep methods --- stochastic Volterra integral equations --- mean-square stability --- asymptotic stability --- numerical analysis --- numerical methods --- scientific computing --- initial value problems --- one-step methods --- Hermite–Obreshkov methods --- symplecticity --- B-splines --- BS methods
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Inequalities appear in various fields of natural science and engineering. Classical inequalities are still being improved and/or generalized by many researchers. That is, inequalities have been actively studied by mathematicians. In this book, we selected the papers that were published as the Special Issue ‘’Inequalities’’ in the journal Mathematics (MDPI publisher). They were ordered by similar topics for readers’ convenience and to give new and interesting results in mathematical inequalities, such as the improvements in famous inequalities, the results of Frame theory, the coefficient inequalities of functions, and the kind of convex functions used for Hermite–Hadamard inequalities. The editor believes that the contents of this book will be useful to study the latest results for researchers of this field.
analytic functions --- starlike functions --- convex functions --- Fekete-Szegö inequality --- Hilbert C*-module --- g-frame --- g-Bessel sequence --- adjointable operator --- analytic functions --- starlike functions --- convex functions --- Fekete-Szegö inequality --- operator inequality --- positive linear map --- operator Kantorovich inequality --- geometrically convex function --- frame --- weaving frame --- weaving frame operator --- alternate dual frame --- Hilbert space --- quantum estimates --- Hermite-Hadamard type inequalities --- quasi-convex --- Hermite–Hadamard type inequality --- strongly ?-convex functions --- Hölder’s inequality --- Power mean inequality --- Katugampola fractional integrals --- Riemann–Liouville fractional integrals --- Hadamard fractional integrals --- Steffensen’s inequality --- higher order convexity --- Green functions --- Montgomery identity --- Fink’s identity --- Hermite-Hadamard inequality --- interval-valued functions --- (h1, h2)-convex --- majorization inequality --- twice differentiable convex functions --- refined inequality --- Taylor theorem --- Gronwall–Bellman inequality --- proportional fractional derivative --- Riemann–Liouville and Caputo proportional fractional initial value problem --- convex functions --- Fejér’s inequality --- special means --- weaving frame --- weaving K-frame --- K-dual --- pseudo-inverse --- ?-variation --- one-sided singular integral --- commutator --- one-sided weighted Morrey space --- one-sided weighted Campanato space --- power inequalities --- exponential inequalities --- trigonometric inequalities --- weight function --- half-discrete Hardy-Hilbert’s inequality --- parameter --- Euler-Maclaurin summation formula --- reverse inequality
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