BVP for the heat equation with a fractional integro-differentiation operator

Abstract

A boundary value problem for a loaded heat conduction equation is considered, when the loaded term has the form of a fractional Riemann-Liouville derivative with respect to a spatial variable, and the loading point moves with a variable velocity. The problem is reduced to a Volterra integral equation of the second kind, the kernel of which contains a special function, namely, a Wright-type function. The kernel of the resulting integral equation is estimated, and it is shown, under certain restrictions on the line along which the load moves, that the kernel of the equation has a weak singularity, which is the basis for the assertion that the loaded term in the equation of the problem is a weak perturbation of its differential part. The study is based on the asymptotic behavior of the Wright function at infinity and at zero.