New trends on fractional integral and differential equations

A. A. Kilbas

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Труды геометрического семинара, Tome 147 (2005) no. 1, pp. 72-106

Résumé

One- and multi-dimensional integral equations and ordinary and partial differential equations with fractional integrals and derivatives by Riemann–Liouville, Liouville, Caputo, Hadamard and Riesz are considered. The method based on the reduction of the Cauchy-type and Cauchy problems for the one-dimensional nonlinear fractional differential equations to Volterra integral equations is discussed. A unified approach is presented to solve in close form of some classes of one- and multi-dimensional linear integral equations and linear ordinary and partial differential equations of fractional order. This approach is based on compositional relations, operational calculus and integral transforms by Laplace, Fourier and Mellin. Problems and new trends of research are discussed.

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