To solving the heat equation with fractional load

Abstract

In the paper, the solvability problems of an nonhomogeneous boundary value problem
in the first quadrant for a fractionally loaded heat equation are studied. Feature of this problem
is that, firstly, the loaded term is presented in the form of the Caputo fractional derivative with
respect to the time variable, secondly, the order of the derivative in the loaded term is less than
the order of the differential part and, thirdly, the point of load is moving (with constant or variable
velocity). By inverting the differential part, the problem is reduced to the Volterra integral equation
of the second kind, the kernel of which contains the function of a parabolic cylinder. The kernel of
the obtained integral equation is estimated and it is shown that the kernel of the equation has a
weak singularity (under certain restrictions on the load), this is the basis for the statement that the
loaded term in the equation is a weak perturbation of its differential part. In addition, the limiting
cases of the order of the fractional derivative are considered. It is proved that there is continuity
on the right in the order of the fractional derivative. Continuity on the left is broken. The results
of the paper may turn out to be useful in the study of fractionally loaded heat equations in the
case, when the loaded term is presented in the form of a Caputo fractional derivative with respect
to the spatial variable.