Integrable system

Pure mathematics

Mathematics

Biology

Fractional calculus

Mathematical analysis

Philosophy

Artificial intelligence

Singularity

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Class (philosophy)

Type (biology)

Computer science

Linguistics

Zero (linguistics)

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the $n$-fold general fractional integrals and derivatives and study their properties.

General Fractional Integrals and Derivatives with the Sonine Kernels