Application of the method of lines for solving one-dimensional equation of parabolic type under the boundary conditions of the second and firs genera

In the article an algorithm for solving a one-dimensional inhomogeneous parabolic equation is described under boundary conditions of the second kind at the beginning and of the first kind at the end of the interval. By introduction of a grid with respect to the coordinate of the functions involved in the initial and boundary conditions, a matrix equation is built with respect to the grid function. The success of the work is the formation of fundamental and diagonal matrices, with the help of which a transition to individual ordinary equations with respect to the grid functions is carried out from the matrix equation. Formulas for the direct and inverse transition from the desired and newly formed functions are presented. The obtained ordinary differential equations admit an exact and approximate method of solution. The results are useful in solving one and many-dimensional equations of parabolic, elliptic and hyperbolic types under mixed boundary conditions of the second and first genera.