Construction of an Approximate-Analytical Solution for Boundary Value Problems of a Parabolic Equation

Abstract

The article considers an approximate analytical solution (AAS) of a linear parabolic equation with initial and boundary conditions with one spatial variable. Many problems in engineering applications are reduced to solving an initial boundary value problem of a parabolic type. There are various analytical, approximate-analytical and numerical methods for solving such problems. Here we consider ways to obtain an AAS based on the movable node method. Three approaches to obtaining an AAS are considered. In all approaches, an arbitrary point (a movable node) is considered inside the area where the solution is being sought. In the first approach, the parabolic equation is approximated by a difference scheme with a movable node in both variables. As a result, the differential problem is reduced to one algebraic equation (in the case of a boundary condition of the first kind), solving which we obtain an AAS. If one of the boundary conditions is of the second or third kind, assuming the fulfillment of the difference equation up to the boundary, we determine the unknown value of the solution on the boundary. The second and third approaches are based on the idea of the method of lines for differential equations in combination with a movable node. In the second approach, in a parabolic equation, the time derivative is approximated by a difference relation with the node being moved, and as a result we obtain an ordinary second-order differential equation with boundary conditions. Solving the obtained ordinary differential equation, we have an AAS. In the third approach, the approximation of the parabolic equation is performed only by the spatial variable. As a result, we obtain an ordinary differential equation of the first order with an initial condition and the solution of which gives an AAS to the original problem. The simplicity of the described approach allows the use of its engineering calculations. Comparisons have been made.