Explicit solutions to hyper-Bessel integral equations of second kind

✍ Scribed by V. Kiryakova; B. Al-Saqabi

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 692 KB
• Volume: 37
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/s0898-1221(98)00243-0

✦ Synopsis

In earlier papers, the authors have used the transmutation method to find solutions to Volterra integral or differ-integral equations of second kind, involving Erd~lyi-Kober fractional integration operators (see ), as well as to dual integral equations and some Bessel-type differential equations (see ). Here we consider the so-called hyper-Bessel integral equations whose kernel-function is a rather general special function (a Meijer's G-function). Such an equation can be written also in a form involving a product of arbitrary number of Erd~lyi-Kober integrals. By means of a Poisson-type transmutation, we reduce its solution to the well-known solution of a simpler Volterra equation involving Riemann-Liouville integration only. In the general case, the solution is found as a series of integrals of G-functions, easily reducible to series of G-functions. For particular nonhomogeneous (right-hand side) parts, this solution reduces to some known special functions. The main techniques are based on the generalized fractional calculus.