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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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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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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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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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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