On a difference method of potential fields' extension

Background. The problem of potential fields' extension rises in many branches of physics and technology: in geophysics in extension of fields measured on the Earth's surface, in the depth of the Earth, in meteorology when determining the limits of atmospheric fields, in defectoscopy for research of materials inner properties without destruction thereof and in a number of other branches. In spite of the fact that for all such problems researchers suggest different methods, as a rule, all of them are reduced to Fredholm integral equations of first kind, which are the ill-conditioned problems. As proved by numerical experiments, application of classical difference methods is impossible due to instability thereof. As the difference schemes are characterized by simplicity and performance, of considerable interest is development of special stable schemes. The article is devoted to development of stable difference schemes of potential fields' extension. Materials and methods. Development of difference schemes and potential fields' extension are based on optimal methods of approximation of potential fields belonging to the function classes $Q_{r,\gamma}(\Omega, M), B_{r,\gamma}(\Omega, M)$, where $\Omega$ is the area into which a field is extended. The nodes of local splines, being the optimal methods of approx.imation of function classes $Q_{r,\gamma}(\Omega, M), B_{r,\gamma}(\Omega, M)$, act as the nodes of the difference schemes. Results. The authors developed the stable difference schemes being the effective method of potential fields' extension. Conclusions. The researchers proved the possibility of potential fields' extension by means of the difference methods.