The method of lines for solving equations of mathematical physics with boundary conditions of the first and third types

The article demonstrates the main aspects of the method of lines on the example of solving a one-dimensional parabolic equation. The transition to the grid functions is performed along the x coordinate when a boundary condition of the first type is imposed at the beginning of the segment, and a condition of the third type is imposed at the end of it at x = l, according to the statements of the method of lines. The ways of searching for eigenvalues and vectors of a tridiagonal matrix at various values of the complex αl are shown. With them, the transition to autonomous ordinary differential equations relative to new grid functions is performed. The ordinary differential equations derived are solved numerically. Formulas for the forward transition from the original grid function to the new sought-for function and the inverse transition are given. Comparison with the discontinuous solution of the test problem showed good agreement. The results can be used to solve multidimensional parabolic equations and one-and multidimensional equations of elliptic and hyperbolic types if mixed boundary conditions hold at least one of the coordinates.