### Journals Information

**
Mathematics and Statistics Vol. 10(2), pp. 320 - 328 DOI: 10.13189/ms.2022.100206 Reprint (PDF) (467Kb) **

## Application of the Fast Expansion Method in Space–Related Problems

**Mikhail Ivanovich Popov ^{1}, Aleksey Vasilyevich Skrypnikov ^{1}, Vyacheslav Gennadievich Kozlov ^{2}^{,*}, Alexey Viktorovich Chernyshov ^{2}, Alexander Danilovich Chernyshov ^{1}, Sergey Yurievich Sablin ^{1}, Vladimir Valentinovich Nikitin ^{1}, Roman Alexandrovich Druzhinin ^{2}**

^{1}Voronezh State University of Engineering Technologies, Voronezh, Russian Federation

^{2}Voronezh State Agrarian University named after Emperor Peter the Great, Voronezh, Russian Federation

**ABSTRACT**

In the paper, numerical and approximate analytical solutions for the problem of the motion of a spacecraft from a starting point to a final point during a certain time are obtained. The unpowered and powered portions of the flight are considered. For a numerical solution, a finite-difference scheme of the second order of accuracy is constructed. The space-related problem considered in the study is essentially nonlinear, which necessitates the use of trigonometric interpolation methods to replace the task of calculating the Fourier coefficients with the integral formulas by solving the interpolation system. One of the simplest options for trigonometric sine interpolation on a semi-closed segment [–a, a), where the right end is not included in the general system of interpolation points, is considered. In order to maintain the conditions of orthogonality of sines, an even number of 2M calculation points is uniformly applied to the segment. The sine interpolation theorem is proved and a compact formula is given for calculating the interpolation coefficients. A general theory of fast sine expansion is given. It is shown that in this case, the Fourier coefficients decrease much faster with the increase in serial number compared to the Fourier coefficients in the classical case. This property allows reducing the number of terms taken into account in the Fourier series, as well as the amount of computer calculations, and increasing the accuracy of calculations. The analysis of the obtained solutions is carried out, and their comparison with the exact solution of the test problem is proposed. With the same calculation error, the time spent on a computer using the fast expansion method is hundreds of times less than the time spent on classical finite-difference method.

**KEYWORDS**

Spacecraft, Fast Expansion Method, Finite-Difference Scheme

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Mikhail Ivanovich Popov , Aleksey Vasilyevich Skrypnikov , Vyacheslav Gennadievich Kozlov , Alexey Viktorovich Chernyshov , Alexander Danilovich Chernyshov , Sergey Yurievich Sablin , Vladimir Valentinovich Nikitin , Roman Alexandrovich Druzhinin , "Application of the Fast Expansion Method in Space–Related Problems," Mathematics and Statistics, Vol. 10, No. 2, pp. 320 - 328, 2022. DOI: 10.13189/ms.2022.100206.

(b). APA Format:

Mikhail Ivanovich Popov , Aleksey Vasilyevich Skrypnikov , Vyacheslav Gennadievich Kozlov , Alexey Viktorovich Chernyshov , Alexander Danilovich Chernyshov , Sergey Yurievich Sablin , Vladimir Valentinovich Nikitin , Roman Alexandrovich Druzhinin (2022). Application of the Fast Expansion Method in Space–Related Problems. Mathematics and Statistics, 10(2), 320 - 328. DOI: 10.13189/ms.2022.100206.