Application of the direct method for solving a one-dimensional parabolic equation with third-order boundary conditions

The article describes an algorithm for solving a one-dimensional non-homogeneous parabolic equation with third-order boundary conditions at the beginning and end of the interval. By introducing a grid in the coordinate direction and functions involved in the initial and boundary conditions, a matrix equation is constructed with respect to the grid functions. The success of the work lies in the formation of fundamental and diagonal matrices, using which the matrix equation is transformed into separate ordinary equations with respect to the grid functions. Formulas for direct and inverse transformations of the desired and newly formed functions are presented. The obtained ordinary differential equations allow for exact and approximate methods of solution. The results are useful for solving one- and multi-dimensional equations of parabolic, elliptic, and hyperbolic types with mixed third-order boundary conditions.