On Finding a Solution of One Problem for the Heat Equation of on a Semi-Axis

Abstract

The presented article is devoted to finding a solution to one problem for a heat equation with discontinuous coefficients on the semiaxis. Applying an integral transformation to the problem, a boundary value problem is constructed for an ordinary differential equation depending on a complex parameter. In this case, it is assumed that the solution to the problem found at infinity is bounded. Since the coefficient of the equation is discontinuous, the solution to the problem is found in both finite and infinite intervals. In the general solutions found in both intervals, there are constants that are independent of each other. Although in the problem the number of boundary and associated conditions is three, the number of constants involved in the solution is four. This can lead to violation of the uniqueness of the solution to the problem. To eliminate this discrepancy, it is necessary to choose one of the found constants equal to zero, using the condition that the solution is bounded at infinity. The other three constants are found from the boundary and associated conditions and taken into account in the solution of an ordinary differential equation with discontinuous coefficients. The article shows that the solution of an ordinary differential equation with discontinuous coefficients is a meramorphic function on the complex plane. The singular points found for solving the problem are pole-type singular points located in a strip containing an imaginary axis. In the paper, the expansion theorem is proved and the solution of the problem is constructed in the form of a contour integral. The absolute and uniform convergence of the found contour integral for the solution is shown, which means the substantiation of the formal solution of the original problem.