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Generalized Abel type integral equations with localized fractional integrals and derivatives. (English) Zbl 1392.45008

Summary: Generalized Abel type integral equations with Gauss, Kummer’s and Humbert’s confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.

MSC:

45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45H05 Integral equations with miscellaneous special kernels
33C20 Generalized hypergeometric series, \({}_pF_q\)

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