On a Problem of 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 spatial 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. The problem is reduced to the Volterra integral equation of the second kind, the kernel of which contains the generalized hypergeometric series. 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 in the order of the fractional derivative.