Semi-analytical Computations Based on TKSL

The paper deals with semi-analytical computations and gives the examples of absolutely exact solutions that can be obtained using numerical solutions of differential equations. Numerical solutions of differential equations based on the Taylor series are implemented in a simulation language TKSL. Polynomials functions, Finite integrals, Fourier Series and Exponential functions are but a few examples of successful application areas. The main idea behind the Modern Taylor Series Method is an automatic integration method order setting, i.e. using as many Taylor series terms for computing as needed to achieve the required accuracy. The Modern Taylor Series Method used in the computations increases the method order automatically, i.e. the values of the Taylor series terms are computed for increasing integer values of p until adding the next term does not improve the accuracy of the solution.