Identification of the trajectory of a moving point source when heating a one-dimensional rod

The process of heating an inhomogeneous one-dimensional rod by a moving point heat source described by a parabolic equation with a singular right-hand side is considered. The problem of identifying the trajectory of a moving source in the given temperature mode at a given point of the rod is posed. To solve the resulting difference analog of the problem posed, a special representation is proposed that allows dividing the problem into two mutually independent second-order difference problems. As a result, an explicit formula is obtained for determining the position of a moving point source for each value of the time variable.