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This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention.

Research & information: general --- Mathematics & science --- bessel function --- harmonically convex function --- non-singular function involving kernel fractional operator --- Hadamard inequality --- Fejér–Hadamard inequality --- Elzaki transform --- Caputo fractional derivative --- AB-fractional operator --- new iterative transform method --- Fisher’s equation --- Hukuhara difference --- Atangana–Baleanu fractional derivative operator --- Mittag–Leffler kernel --- Fornberg–Whitham equation --- fractional div-curl systems --- Helmholtz decomposition theorem --- Riemann–Liouville derivative --- Caputo derivative --- fractional vector operators --- weighted (k,s) fractional integral operator --- weighted (k,s) fractional derivative --- weighted generalized Laplace transform --- fractional kinetic equation --- typhoid fever disease --- vaccination --- model calibration --- asymptotic stability --- fixed point theory --- nonlinear models --- efficiency index --- computational cost --- Halley’s method --- basin of attraction --- computational order of convergence --- Caputo–Hadamard fractional derivative --- thermostat modeling --- Caputo–Hadamard fractional integral --- hybrid Caputo–Hadamard fractional differential equation and inclusion --- prey-predator model --- boundedness --- period-doubling bifurcation --- Neimark-Sacker bifurcation --- hybrid control --- fractal dimensions --- cubic B-splines --- trigonometric cubic B-splines --- extended cubic B-splines --- Caputo–Fabrizio derivative --- Cattaneo equation --- Hermite-Hadamard-type inequalities --- Hilfer fractional derivative --- Hölder’s inequality --- fractional-order differential equations --- operational matrices --- shifted Vieta–Lucas polynomials --- Adomian decomposition method --- system of Whitham-Broer-Kaup equations --- Caputo-Fabrizio derivative --- Yang transform --- ϑ-Caputo derivative --- extremal solutions --- monotone iterative method --- sequences --- convex --- exponential convex --- fractional --- quantum --- inequalities --- Gould-Hopper-Laguerre-Sheffer matrix polynomials --- quasi-monomiality --- umbral calculus --- fractional calculus --- Euler’s integral of gamma functions --- beta function --- generalized hypergeometric series --- operational methods --- delta function --- Riemann zeta-function --- fractional transforms --- Fox–Wright-function --- generalized fractional kinetic equation --- n/a --- Fejér-Hadamard inequality --- Fisher's equation --- Atangana-Baleanu fractional derivative operator --- Mittag-Leffler kernel --- Fornberg-Whitham equation --- Riemann-Liouville derivative --- Halley's method --- Caputo-Hadamard fractional derivative --- Caputo-Hadamard fractional integral --- hybrid Caputo-Hadamard fractional differential equation and inclusion --- Hölder's inequality --- shifted Vieta-Lucas polynomials --- Euler's integral of gamma functions --- Fox-Wright-function

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In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia.

Research & information: general --- Mathematics & science --- condensing function --- approximate endpoint criterion --- quantum integro-difference BVP --- existence --- fractional Kadomtsev-Petviashvili system --- lie group analysis --- power series solutions --- convergence analysis --- conservation laws --- symmetry --- weighted fractional operators --- convex functions --- HHF type inequality --- fractional calculus --- Euler–Lagrange equation --- natural boundary conditions --- time delay --- MHD equations --- weak solution --- regularity criteria --- anisotropic Lorentz space --- Sonine kernel --- general fractional derivative of arbitrary order --- general fractional integral of arbitrary order --- first fundamental theorem of fractional calculus --- second fundamental theorem of fractional calculus --- ρ-Laplace variational iteration method --- ρ-Laplace decomposition method --- partial differential equation --- caputo operator --- fractional Fornberg–Whitham equation (FWE) --- Riemann–Liouville fractional difference operator --- boundary value problem --- discrete fractional calculus --- existence and uniqueness --- Ulam stability --- elastic beam problem --- tempered fractional derivative --- one-sided tempered fractional derivative --- bilateral tempered fractional derivative --- tempered riesz potential --- collocation method --- hermite cubic spline --- fractional burgers equation --- fractional differential equation --- fractional Dzhrbashyan–Nersesyan derivative --- degenerate evolution equation --- initial value problem --- initial boundary value problem --- partial Riemann–Liouville fractional integral --- Babenko’s approach --- Banach fixed point theorem --- Mittag–Leffler function --- gamma function --- nabla fractional difference --- separated boundary conditions --- Green’s function --- existence of solutions --- Caputo q-derivative --- singular sum fractional q-differential --- fixed point --- equations --- Riemann–Liouville q-integral --- Shehu transform --- Caputo fractional derivative --- Shehu decomposition method --- new iterative transform method --- fractional KdV equation --- approximate solutions --- Riemann–Liouville derivative --- concave operator --- fixed point theorem --- Gelfand problem --- order cone --- integral transform --- Atangana–Baleanu fractional derivative --- Aboodh transform iterative method --- φ-Hilfer fractional system with impulses --- semigroup theory --- nonlocal conditions --- optimal controls --- fractional derivatives --- fractional Prabhakar derivatives --- fractional differential equations --- fractional Sturm–Liouville problems --- eigenfunctions and eigenvalues --- Fredholm–Volterra integral Equations --- fractional derivative --- Bessel polynomials --- Caputo derivative --- collocation points --- Caputo–Fabrizio and Atangana-Baleanu operators --- time-fractional Kaup–Kupershmidt equation --- natural transform --- Adomian decomposition method

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In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia.

condensing function --- approximate endpoint criterion --- quantum integro-difference BVP --- existence --- fractional Kadomtsev-Petviashvili system --- lie group analysis --- power series solutions --- convergence analysis --- conservation laws --- symmetry --- weighted fractional operators --- convex functions --- HHF type inequality --- fractional calculus --- Euler–Lagrange equation --- natural boundary conditions --- time delay --- MHD equations --- weak solution --- regularity criteria --- anisotropic Lorentz space --- Sonine kernel --- general fractional derivative of arbitrary order --- general fractional integral of arbitrary order --- first fundamental theorem of fractional calculus --- second fundamental theorem of fractional calculus --- ρ-Laplace variational iteration method --- ρ-Laplace decomposition method --- partial differential equation --- caputo operator --- fractional Fornberg–Whitham equation (FWE) --- Riemann–Liouville fractional difference operator --- boundary value problem --- discrete fractional calculus --- existence and uniqueness --- Ulam stability --- elastic beam problem --- tempered fractional derivative --- one-sided tempered fractional derivative --- bilateral tempered fractional derivative --- tempered riesz potential --- collocation method --- hermite cubic spline --- fractional burgers equation --- fractional differential equation --- fractional Dzhrbashyan–Nersesyan derivative --- degenerate evolution equation --- initial value problem --- initial boundary value problem --- partial Riemann–Liouville fractional integral --- Babenko’s approach --- Banach fixed point theorem --- Mittag–Leffler function --- gamma function --- nabla fractional difference --- separated boundary conditions --- Green’s function --- existence of solutions --- Caputo q-derivative --- singular sum fractional q-differential --- fixed point --- equations --- Riemann–Liouville q-integral --- Shehu transform --- Caputo fractional derivative --- Shehu decomposition method --- new iterative transform method --- fractional KdV equation --- approximate solutions --- Riemann–Liouville derivative --- concave operator --- fixed point theorem --- Gelfand problem --- order cone --- integral transform --- Atangana–Baleanu fractional derivative --- Aboodh transform iterative method --- φ-Hilfer fractional system with impulses --- semigroup theory --- nonlocal conditions --- optimal controls --- fractional derivatives --- fractional Prabhakar derivatives --- fractional differential equations --- fractional Sturm–Liouville problems --- eigenfunctions and eigenvalues --- Fredholm–Volterra integral Equations --- fractional derivative --- Bessel polynomials --- Caputo derivative --- collocation points --- Caputo–Fabrizio and Atangana-Baleanu operators --- time-fractional Kaup–Kupershmidt equation --- natural transform --- Adomian decomposition method