In this article we examine the existence of a unique smooth solution to a singular fractional differential equation. We reformulate the singular equation with the help of cordial Volterra integral operators and then extend a result from cordial Volterra integral operator theory.

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The concepts of the finite part (f.p.) and analytic finite part (a.f.p.) of divergent integrals are defined in the situation where the singular function in the integral has a logarithmic factor. The change of variables in f.p.- and a.f.p-integrals is examined.

The aim of this note is to examine the relationship between ϕ-topologies and β-topologies in FK-spaces. Every ϕ-topology on an FK-space is a β-topology, the converse statement is not always true. Still, in AB-BK-spaces the statement holds, i.e., every β-topology is aϕ-topology. We establish that an analogous statement is true for ABFK-spaces.

A boundary value problem for fractional integro-differential equations with weakly singular kernels is considered. The problem is reformulated as an integral equation of the second kind with respect to, the Caputo fractional derivative of y of order α, with 1 < α < 2, where y is the solution of the original problem. Using this reformulation, the regularity properties of both y and its Caputo derivative z are studied. Based on this information a piecewise polynomial collocation method is developed for finding an approximate solution zN of the reformulated problem. Using zN, an approximation yN for y is constructed and a detailed convergence analysis of the proposed method is given. In particular, the attainable order of convergence of the proposed method for appropriate values of grid and collocation parameters is established. To illustrate the performance of our approach, results of some numerical experiments are presented.