Моделирование аппроксимирующего 16-точечного отсека поверхности отклика применительно к решению неоднородного уравнения теплопроводности

Abstract

The paper proposes a computational method for solving differential equations of mathematical physics by approximating the desired solution using geometric objects of multidimensional space passing through predetermined points. The essence of the method is to simulate an approximating geometric object of a multidimensional affine space constructed on a regular multidimensional network of points. In this case, the response function values satisfying the solution of the original differential equation are calculated at the nodal points of the network. Modeling of approximating geometric object is carried out by means the arcs of algebraic curves passing through predetermined points. It should be noted that taking into account the boundary conditions does not require changes in the geometric algorithm or point equations. It is sufficient to use the necessary coordinates of the nodal boundary points corresponding to the boundary conditions of the solution of the differential equation. To achieve the required accuracy of the solution of differential equations, it is sufficient to compact the reference network of points. Under such conditions, it is possible to use as a single geometric object to approximate the solution of the differential equation, and composite, based on the simulation of multidimensional contours on a regular network of points of multidimensional space. A geometric classification of differential equations depending on the number of parameters determining the approximating geometric object in multidimensional space is proposed. An example of solving the inhomogeneous heat equation by means of an approximating response surface passing through 16 predetermined points is given. In this case, the required approximating compartment of the response surface passes through 3 straight lines that correspond to the boundary conditions and satisfies the solution of the original differential equation at the nodal points of the 16-point network. A comparison of the results of solving the inhomogeneous heat equation approximated by a 16-point compartment of the response surface with the reference compartment of the surface obtained by the method of separating variables is also presented.