Operational method for solving generalized Abel integral equation of second kind. (English) Zbl 0887.44003

Gorenflo, Rudolf; Luchko, Yuri

Integral Transforms Spec. Funct. 5, No. 1-2, 47-58 (1997).

The authors solve explicitly linear integral equations of second kind of Abel type by means of a modified Mikusiński operational calculus. In particular, the solution of \[ y(x)- \sum^m_{i=1} \lambda_i (I^{i\mu} y)(x)= f(x), \quad \mu>0,\;x>0 \] with \[ (I^\mu y)(x)= {1 \over \Gamma (\mu)} \int^x_0 (x-t)^{\mu-1} f(t)dt \] is expressed in terms of the Mittag-Leffler function \[ E^m_{\alpha, \beta}= \sum^\infty_{i=0} {m+i- 1\choose i} z^i/ \Gamma (\alpha i+ \beta). \] Reviewer’s remark: It is not necessary to modify the Mikusiński operational calculus, if one starts with the ring of Lebesgue-integrable functions instead of continuous functions.

Reviewer: Lothar Berg (Rostock)

MSC:

44A40	Calculus of Mikusiński and other operational calculi
45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)

Keywords:

generalized Abel integral equation of second kind; Mikusiński operational calculus; Mittag-Leffler function

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