2 - Fractional derivatives with singular kernels

Abstract

In this chapter, we introduce the functional spaces and the fractional derivative and integral operators with singular kernels, for example, the power function and the function related to the power-law. We investigate the Riemann–Liouville, Osler, Liouville–Weyl, and Samko–Kilbas–Marichev fractional calculi, Liouville–Sonine–Caputo and Liouville fractional derivatives, Almeida and Liouville-type fractional derivatives with respect to another function, Liouville–Grünwald–Letnikov fractional derivatives, Kilbas–Srivastava–Trujillo fractional difference derivatives, Riesz fractional calculus, Liouville–Sonine–Caputo–Riesz-type fractional derivatives, Feller fractional calculus, Liouville–Sonine–Caputo–Feller-type fractional derivatives, Herrmann fractional calculus, Liouville–Sonine–Caputo–Herrmann-type fractional derivatives, Samko–Kilbas–Marichev symmetric fractional difference derivative, Grünwald–Letnikov–Herrmann-type symmetric fractional difference derivative, Grünwald–Letnikov–Feller-type symmetric fractional difference derivative, Erdelyi–Kober-type calculus, Hadamard fractional calculus, Hadamard type fractional calculus involving the exponential function, Marchaud fractional derivatives, Marchaud-type fractional derivatives with respect to another function, Riemann–Liouville-type tempered fractional calculus, Liouville–Weyl-type tempered fractional calculus, Liouville–Sonine–Caputo-type tempered fractional derivatives, Liouville–Weyl–Riesz-type tempered fractional calculus, Riemann–Liouville–Riesz-type fractional calculus, Liouville–Sonine–Caputo–Riesz-type tempered fractional derivatives, Liouville–Weyl–Feller tempered fractional calculus, Riemann–Liouville–Feller-type tempered type fractional calculus, Liouville–Sonine–Caputo–Feller-type tempered fractional derivatives, Liouville–Weyl–Herrmann tempered fractional calculus, Riemann–Liouville–Herrmann-type tempered type fractional calculus, Liouville–Sonine–Caputo–Herrmann-type tempered fractional derivatives, Riemann–Liouville-type tempered fractional calculus with respect to another function, Hilfer derivative, Liouville–Weyl–Hilfer-type derivative, Riesz–Hilfer-type fractional derivative, Feller–Hilfer-type fractional derivative, Herrmann–Hilfer-type fractional derivative, Sousa–de Oliveira fractional derivatives with respect to another function, Liouville–Weyl–Sousa–de Oliveira-type fractional derivatives, Hilfer–Riesz-type fractional derivatives with respect to another function, Hilfer–Feller-type fractional derivatives with respect to another function, and Hilfer–Herrmann-type fractional derivatives with respect to another function.