Explicit solutions to hyper-Bessel integral equations of second kind
type: Item
instance of: scholarly article
title: Explicit solutions to hyper-Bessel integral equations of second kind
zbMATH Open document ID: 0936.45003
DOI: 10.1016/S0898-1221(98)00243-0
publication date: 2000-01-20T00:00:00Z
review text: The equation of the title is put into three equivalent forms. One of these can be expressed in the format \[ y-\lambda G_1(y)=f, \] and the solution is then obtained in the form \[ y=f+\sum^\infty_{k=1}\lambda^kG_k(f), \] where the \(G_k\)-operator, which involves Meijer's \(G\)-function, is defined by \[ G_k(f)(x)=(x^\beta/\beta^m)^k\int^1_0 G^{m,0}_{m,m} \left[\sigma\left|\begin{matrix} (\gamma_i+k)^m_{i=1}\\ (\gamma_i)^m_{i=1}\end{matrix}\right.\right]f(x\sigma^{1/\beta})d\sigma. \] In order to relate this problem to a previously solved problem, an integral transformation is used. A generalized Erdélyi-Kober fractional integral, which can also be conveniently expressed in terms of the \(G\)-functions, provides the suitable transformation. Consequently, the solution can be expressed as the infinite series of generalized fractional integrals. Some interesting examples of special cases are given, including the solution of the non-homogeneous hyper-Bessel differential equation.
Mathematics Subject Classification ID: 45H05
Mathematics Subject Classification ID: 33C60
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34B30
zbMATH DE Number: 1391484
zbMATH Keywords: explicit solutions
zbMATH Keywords: hyper-Bessel integral equations of second kind
zbMATH Keywords: \(G\)-functions
zbMATH Keywords: Erdélyi-Kober fractional integral
zbMATH Keywords: hyper-Bessel differential equation
author: Virginia Kiryakova
author: B. N. Al-Saqabi
reviewed by: Robert G. Buschman
MaRDI profile type: Publication
cites work: Transmutation method for the solving Erdélyi-Kober fractional differintegral equations
cites work: Q4229986
cites work: Q4321403
cites work: Q3764451
cites work: Abel integral equations. Analysis and applications
cites work: Q4282622
cites work: The exact solution of certain differential equations of fractional order by using operational calculus
cites work: The special functions and their approximations. Vol. I, II
cites work: Q4233631
cites work: Generalized hypergeometric functions with applications in statistics and physical sciences
cites work: Fractional Powers of a Class of Ordinary Differentilal Operators
cites work: Q3997837
cites work: The Solution of Certain Integral Equations by Means of Operators of Arbitary Order
cites work: Q4726489
cites work: Q4866811
cites work: Q4023370
cites work: Solution of a fractional differintegral equation
cites work: Q4089045
cites work: Q5820721
cites work: Q4069254
cites work: Q5824770
cites work: Q4084185
cites work: The kernel representation of the fractional power of the strongly elliptic operator
cites work: Isomorphism of the quotient fields, generated by Bessel‐type differential operators
full work available at URL: s0898-1221(98)00243-0
OpenAlex ID: W1974470984
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1391484