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This book focuses on applications of the theory of fractional calculus in numerical analysis and various fields of physics and engineering. Inequalities involving fractional calculus operators containing the Mittag–Leffler function in their kernels are of particular interest. Special attention is given to dynamical models, magnetization, hypergeometric series, initial and boundary value problems, and fractional differential equations, among others.

Research & information: general --- Mathematics & science --- fractional derivative --- generalized Mittag-Leffler kernel (GMLK) --- Legendre polynomials --- Legendre spectral collocation method --- dynamical systems --- random time change --- inverse subordinator --- asymptotic behavior --- Mittag–Leffler function --- data fitting --- magnetization --- magnetic fluids --- Gamma function --- Psi function --- Pochhammer symbol --- hypergeometric function 2F1 --- generalized hypergeometric functions tFu --- Gauss’s summation theorem for 2F1(1) --- Kummer’s summation theorem for 2F1(−1) --- generalized Kummer’s summation theorem for 2F1(−1) --- Stirling numbers of the first kind --- Hilfer–Hadamard fractional derivative --- Riemann–Liouville fractional derivative --- Caputo fractional derivative --- fractional differential equations --- inclusions --- nonlocal boundary conditions --- existence and uniqueness --- fixed point --- gamma function --- Beta function --- Mittag-Leffler function --- Generalized Mittag-Leffler functions --- generalized hypergeometric function --- Fox–Wright function --- recurrence relations --- Riemann–Liouville fractional calculus operators --- (α, h-m)-p-convex function --- Fejér–Hadamard inequality --- extended generalized fractional integrals --- Mittag–Leffler functions --- initial value problems --- Laplace transform --- exact solution --- Chebyshev inequality --- Pólya-Szegö inequality --- fractional integral operators --- Wright function --- Srivastava’s polynomials --- fractional calculus operators --- Lavoie–Trottier integral formula --- Oberhettinger integral formula --- fractional partial differential equation --- boundary value problem --- separation of variables --- Mittag-Leffler --- Abel-Gontscharoff Green’s function --- Hermite-Hadamard inequalities --- convex function --- κ-Riemann-Liouville fractional integral --- Dirichlet averages --- B-splines --- dirichlet splines --- Riemann–Liouville fractional integrals --- hypergeometric functions of one and several variables --- generalized Mittag-Leffler type function --- Srivastava–Daoust generalized Lauricella hypergeometric function --- fractional calculus --- Hermite–Hadamard inequality --- Fox H function --- subordinator and inverse stable subordinator --- Lamperti law --- order statistic --- n/a --- Gauss's summation theorem for 2F1(1) --- Kummer's summation theorem for 2F1(−1) --- generalized Kummer's summation theorem for 2F1(−1) --- Hilfer-Hadamard fractional derivative --- Riemann-Liouville fractional derivative --- Fox-Wright function --- Riemann-Liouville fractional calculus operators --- Fejér-Hadamard inequality --- Mittag-Leffler functions --- Pólya-Szegö inequality --- Srivastava's polynomials --- Lavoie-Trottier integral formula --- Abel-Gontscharoff Green's function --- Riemann-Liouville fractional integrals --- Srivastava-Daoust generalized Lauricella hypergeometric function --- Hermite-Hadamard inequality

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During the last decade, there has been an increased interest in fractional differential equations, inclusions, and inequalities, as they play a fundamental role in the modeling of numerous phenomena, in particular, in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electron-analytical chemistry, control theory, etc. This book presents collective works published in the recent Special Issue (SI) entitled "Fractional Differential Equation, Inclusions and Inequalities with Applications" of the journal Mathematics. This Special Issue presents recent developments in the theory of fractional differential equations and inequalities. Topics include but are not limited to the existence and uniqueness results for boundary value problems for different types of fractional differential equations, a variety of fractional inequalities, impulsive fractional differential equations, and applications in sciences and engineering.

Research & information: general --- Mathematics & science --- fractional evolution inclusions --- mild solutions --- condensing multivalued map --- arbitrary order differential equations --- multiple positive solution --- Perov-type fixed point theorem --- HU stability --- Caputo fractional derivative --- nonlocal --- integro-multipoint boundary conditions --- existence --- uniqueness --- Ulam-Hyers stability --- coupled system of fractional difference equations --- fractional sum --- discrete half-line --- non-instantaneous impulsive equations --- random impulsive and junction points --- continuous dependence --- Caputo–Fabrizio fractional differential equations --- Hyers–Ulam stability --- fractional derivative --- fixed point theorem --- fractional differential equation --- fractional sum-difference equations --- boundary value problem --- positive solution --- green function --- the method of lower and upper solutions --- three-point boundary-value problem --- Caputo’s fractional derivative --- Riemann-Liouville fractional integral --- fixed-point theorems --- Langevin equation --- generalized fractional integral --- generalized Liouville–Caputo derivative --- nonlocal boundary conditions --- fixed point --- fractional differential inclusions --- ψ-Riesz-Caputo derivative --- existence of solutions --- anti-periodic boundary value problems --- q-integro-difference equation --- fractional calculus --- fractional integrals --- Ostrowski type inequality --- convex function --- exponentially convex function --- generalized Riemann-liouville fractional integrals --- convex functions --- Hermite–Hadamard-type inequalities --- exponential kernel --- caputo fractional derivative --- coupled system --- impulses --- existence theory --- stability theory --- conformable derivative --- conformable partial derivative --- conformable double Laplace decomposition method --- conformable Laplace transform --- singular one dimensional coupled Burgers’ equation --- Green’s function --- existence and uniqueness of solution --- positivity of solution --- iterative method --- Riemann–Liouville type fractional problem --- positive solutions --- the index of fixed point --- matrix theory --- differential inclusions --- Caputo-type fractional derivative --- fractional integral --- time-fractional diffusion equation --- inverse problem --- ill-posed problem --- convergence estimates --- s-convex function --- Hermite–Hadamard inequalities --- Riemann–Liouville fractional integrals --- fractal space --- functional fractional differential inclusions --- Hadamard fractional derivative --- Katugampola fractional integrals --- Hermite–Hadamard inequality --- fractional q-difference inclusion --- measure of noncompactness --- solution --- proportional fractional integrals --- inequalities --- Qi inequality --- caputo-type fractional derivative --- fractional derivatives --- neutral fractional systems --- distributed delay --- integral representation --- fractional hardy’s inequality --- fractional bennett’s inequality --- fractional copson’s inequality --- fractional leindler’s inequality --- timescales --- conformable fractional calculus --- fractional hölder inequality --- sequential fractional delta-nabla sum-difference equations --- nonlocal fractional delta-nabla sum boundary value problem --- hadamard proportional fractional integrals --- fractional integral inequalities --- Hermite–Hadamard type inequalities --- interval-valued functions