TY - BOOK ID - 138159206 TI - Fractional Calculus and Special Functions with Applications AU - Özarslan, Mehmet Ali AU - Fernandez, Arran AU - Area, Ivan PY - 2022 PB - Basel MDPI - Multidisciplinary Digital Publishing Institute DB - UniCat KW - Caputo-Hadamard fractional derivative KW - coupled system KW - Hadamard fractional integral KW - boundary conditions KW - existence KW - fixed point theorem KW - fractional Langevin equations KW - existence and uniqueness solution KW - fractional derivatives and integrals KW - stochastic processes KW - calculus of variations KW - Mittag-Leffler functions KW - Prabhakar fractional calculus KW - Atangana–Baleanu fractional calculus KW - complex integrals KW - analytic continuation KW - k-gamma function KW - k-beta function KW - Pochhammer symbol KW - hypergeometric function KW - Appell functions KW - integral representation KW - reduction and transformation formula KW - fractional derivative KW - generating function KW - physical problems KW - fractional derivatives KW - fractional modeling KW - real-world problems KW - electrical circuits KW - fractional differential equations KW - fixed point theory KW - Atangana–Baleanu derivative KW - mobile phone worms KW - fractional integrals KW - Abel equations KW - Laplace transforms KW - mixed partial derivatives KW - second Chebyshev wavelet KW - system of Volterra–Fredholm integro-differential equations KW - fractional-order Caputo derivative operator KW - fractional-order Riemann–Liouville integral operator KW - error bound KW - n/a KW - Atangana-Baleanu fractional calculus KW - Atangana-Baleanu derivative KW - system of Volterra-Fredholm integro-differential equations KW - fractional-order Riemann-Liouville integral operator UR - https://www.unicat.be/uniCat?func=search&query=sysid:138159206 AB - 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. ER -