Fast Expansion Method Research Articles

Application of the Fast Expansion Method in Space–Related Problems

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.

Fast Expansion Method for Evaluating Definite Integrals with a Variable Upper Limit and a Composite or Implicitly Defined Integrand

Fast Expansion Method for Evaluating Definite Integrals with a Variable Upper Limit and a Composite or Implicitly Defined Integrand

Using of fast expansions in the construction of two-dimensional exact solutions of the Poisson equation

Using the fast expansion method, we obtained several exact solutions of the boundary value problem for the Poisson equation in a rectangular domain. We have given graphs of exact solutions corresponding to different boundary conditions and different types of the Poisson equation free term. We showed the influence of the type of boundary conditions and the right-hand side on the type of exact solution. We obtained a solution to the problem of membrane deflections under the action of variable load. We have given graphs of the stress components, from the analysis of which it follows that the greatest stress is in the middle of the rectangular membrane long sides.

Solution of a Nonlinear Heat Conduction Equation for a Curvilinear Region with Dirichlet Conditions by the Fast-Expansion Method

The analytical solution of the nonlinear heat conduction problem for a curvilinear region is obtained with the use of the fast-expansion method together with the method of extension of boundaries and pointwise technique of computing Fourier coefficients.

Investigation of Contact Thermal Resistance in a Finite Cylinder with an Internal Source by the Fast Expansion Method and the Problem of Consistency of Boundary Conditions

By the fast expansion method, the authors have obtained the approximate solution of the problem in analytical form, which holds true at all points of the cylinder up to the boundary. From an analysis of the solution, it follows that the broken thermal contact between annular regions gives rise to a weak thermal resistance. Beginning with four annular regions, the thermal resistance remains constant, in practice. This result was obtained for the first time. When no more than three terms in a Fourier series are used, the maximum residual of the differential heat-conduction equation is 10–2.

Application of the fast expansion method for spacecraft trajectory calculation

This paper deals with the problems on the spacecraft motion in the Earth’s atmosphere taking into account the Earth’s atmospheric drag as well as on the extra-atmospheric part of the trajectory, where resistance is absent. An analytical solution of these problems by the method of fast expansions is proposed.

Method of fast expansions for solving nonlinear differential equations

A method is proposed for constructing fast converging Fourier series with the help of a special boundary function Mq. The convergence rate of the series is determined by the order q of Mq, which makes it possible to use a small number of series terms. The general theory of constructing fast expansions is described, the error of the partial sum of a series is estimated, and an example of a non- linear integrodifferential problem is considered. Due to its remarkable properties, the fast expansion method can be effectively used in applications.