Fractional Calculus and Special Functions with Applications

<p class=ql-align-justify><span style=color: rgba(0 0 0 1)>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.</span></p><p class=ql-align-justify><span style=color: rgba(0 0 0 1)> </span></p><p class=ql-align-justify><span style=color: rgba(0 0 0 1)>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.</span></p><p class=ql-align-justify><span style=color: rgba(0 0 0 1)> </span></p><p><span style=color: rgba(0 0 0 1)>The aim of this book is to explore and highlight the diverse connections between fractional calculus and special functions and their associated applications.</span></p>