Methods of evaluation and scale of functional spaces

Linear equations of mathematical physics with constant coefficients have fed calculational mathematics since the 18th century. The area of nonlinear equations with variable coefficients arose due to gas-hydrodynamic problems in the 20th century. Now, one of the methods of research for properties of solutions of such equations and, accordingly, applied problems is the use of calculations on modern computers. The capacities of computers and their efficiency have increased in the 21st century and allowed progress to be made in solving applied problems, except for cases of methodical errors in calculations. One of the basic sources of such methodical errors is the “uncontrollable” machine accuracy of calculations. One of the methods of solving such numerical problems is a suitable localization of the problem and the choice of an adequate basis of the necessary functional space. Below, we state new results in this area of mathematical and applied research.