الملخص الإنجليزي
We discuss standard approaches to the problem of fractional derivatives and fractional integrals (simply called differintegrals) namely, the Riemann-Liouville, the Caputo and the Grunwald -Letnikov approaches. We study the basic properties of differintegrals including the rules for their compositions and the conditions for the equivalence of various definitions with examples. Further, we give a brief survey of the basic methods for solving fractional differential equations. In addition, we state the existence and uniqueness theorems with methods of solving initial value problems with fractional derivative with consideration of dependence of solutions on initial conditions. Finally, we consider some physical applications, the Tautochrone curve and modelling heat transfer in heterogeneous media where fractional derivatives are involved.
