Method for solving mixed boundary value problems for parabolic type equations by using modifications Lagrange-Sturm-Liouville operators

A new method for solving a mixed boundary value problem with a parabolic type equation is obtained. Boundary conditions of the third kind are considered. The inhomogeneity of the equation and the initial conditions of the problem are arbitrary continuous functions. They are not required to satisfy boundary conditions. The solution is constructed using interpolation operators of Lagrange-Sturm-Liouville functions. A sequential approach is used to construct a generalized solution. The solution is presented as a series. The series converges uniformly on any compact set contained within the domain of definition of the solution. The coefficients of the series are linear combinations of the values of the functions from the equation and the initial conditions of the boundary value problem. A simple method for finding the coefficients of these linear combinations is proposed.