Explicit solutions to hyper-Bessel integral equations of second kind
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.
hyper-Bessel integral equations of second kind
scientific article; zbMATH DE number 1391484
scientific article