Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Integral transforms. --- Fractional integrals. --- Integrals, Fractional --- Integrals --- Transform calculus --- Integral equations --- Transformations (Mathematics)

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Sobolev spaces --- Sobolev, Espaces de. --- Espaces fonctionnels. --- Function spaces --- Intégrales fractionnaires. --- Fractional integrals

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

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

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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 --- 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

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications.

Research & information: general --- Mathematics & science --- Caputo-Hadamard fractional derivative --- coupled system --- Hadamard fractional integral --- boundary conditions --- existence --- fixed point theorem --- fractional Langevin equations --- existence and uniqueness solution --- fractional derivatives and integrals --- stochastic processes --- calculus of variations --- Mittag-Leffler functions --- Prabhakar fractional calculus --- Atangana-Baleanu fractional calculus --- complex integrals --- analytic continuation --- k-gamma function --- k-beta function --- Pochhammer symbol --- hypergeometric function --- Appell functions --- integral representation --- reduction and transformation formula --- fractional derivative --- generating function --- physical problems --- fractional derivatives --- fractional modeling --- real-world problems --- electrical circuits --- fractional differential equations --- fixed point theory --- Atangana-Baleanu derivative --- mobile phone worms --- fractional integrals --- Abel equations --- Laplace transforms --- mixed partial derivatives --- second Chebyshev wavelet --- system of Volterra-Fredholm integro-differential equations --- fractional-order Caputo derivative operator --- fractional-order Riemann-Liouville integral operator --- error bound --- Caputo-Hadamard fractional derivative --- coupled system --- Hadamard fractional integral --- boundary conditions --- existence --- fixed point theorem --- fractional Langevin equations --- existence and uniqueness solution --- fractional derivatives and integrals --- stochastic processes --- calculus of variations --- Mittag-Leffler functions --- Prabhakar fractional calculus --- Atangana-Baleanu fractional calculus --- complex integrals --- analytic continuation --- k-gamma function --- k-beta function --- Pochhammer symbol --- hypergeometric function --- Appell functions --- integral representation --- reduction and transformation formula --- fractional derivative --- generating function --- physical problems --- fractional derivatives --- fractional modeling --- real-world problems --- electrical circuits --- fractional differential equations --- fixed point theory --- Atangana-Baleanu derivative --- mobile phone worms --- fractional integrals --- Abel equations --- Laplace transforms --- mixed partial derivatives --- second Chebyshev wavelet --- system of Volterra-Fredholm integro-differential equations --- fractional-order Caputo derivative operator --- fractional-order Riemann-Liouville integral operator --- error bound