ABSTRACT
This chapter considers numerically hardly solvable equations by conventional methods. From a purely mathematical point of view these equations are properly stated. The derivative and integral are the most important notations of the mathematical analysis. Numerical Computations is a separate great topic of applied mathematics arisen in ancient times and extensively developed from the second generation of computers in 1947. Symbolic computations became popular at the end of 20th century with development of the Computer Algebra Systems. An iterative process to find higher order approximations has been constructed. The main problem consists of its convergence. The definite integral is introduced as the area enclosed between the graph of the continuous positive function and the x-axis making use of the Riemann sum and the principle of transition continuous ↔ discrete.
