# Fil'chakov F. P.

### The development of computational mathematics and mathematical modelling at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR

Fil'chakov F. P., Mitropolskiy Yu. A., Panchishin V. I.

Ukr. Mat. Zh. - 1969. - 21, № 2. - pp. 165–172

### Use of power series in solving linear and nonlinear ordinary differential equations and systems thereof

Ukr. Mat. Zh. - 1969. - 21, № 2. - pp. 230–237

### Percolation theory research carried out by the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR

Fil'chakov F. P., Lavrik V. I.

Ukr. Mat. Zh. - 1967. - 19, № 6. - pp. 97–99

### Construction of an extended Laurent series for infinitely connected uniperiodic regions

Ukr. Mat. Zh. - 1965. - 17, № 6. - pp. 80-90

### Conformal mapping of given regions by the trigonometric interpolation method. I

Ukr. Mat. Zh. - 1963. - 15, № 2. - pp. 158-172

A method is outlined for trigonometric interpolation which secures any given precision of the mapped function for a fairly wide class of regions encountered when solving practical problems. The contour may be given analytically, graphically oi merely by a discrete series of points.

### Asymptotic representation of the solution of a mixed problem for one class of integro-differential equations containing a small parameter

Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 299-307

An approximate method is considered for the conformal mapping of arbitrary simply connected univalent regions which may be means of some elementary function be mapped on a half-plane with an aperture of arbitrary shape cut in it (fig. 1). The mapped function is sought in the form of a section of series (H or (18). Using the method of least squares the problem in the case of series (1) is reduced to the solution of system (13) the coefficients of which are easily calculated by the recurrent formulae (10) and the formulae of numerical quadratures. In the case of series (IS) the problem is reduced to the solution of an analogous system of linear equations (26). Three examples are considered in which the results obtained are compared with exact mapping functions. In § 3 the described method is extended to the case of regions with a finite number of slits.

### Determination of the constants of the Christoffel-Schwarz integral by simulating on resistance paper

Ukr. Mat. Zh. - 1961. - 13, № 1. - pp. 72-79

A very simple procedure is described for the experimental determination of the constants of the Christoffel-Schwarz integral, which ensures sufficient precision for solving many technical problems.

### Hydrodynamic Calculation of Drained Aprons. II. Application of the Method of Successive Conformal Mappings

Ukr. Mat. Zh. - 1960. - 12, № 4. - pp. 439 - 462

In tbe first communication [10] an exact hydrodynamie solution is giveri-for the general case of a flat split apron floodbed with T=∞ and then,the most interesting: special cases are discussed. In the present research the earlier developed method of successive con" formal mapping [5; 7] is employed to obtain a very simple approximate solution fôr a fiat drained apron, which is êxtënded in §§ 2 and 3 to the case of a apron of arbitrary practical profile with n drainage openings w ith a finite or infinite depth of water-permeable soil. The application of an extremely simple graphoanalytical method permits solving these problems, in most of the cases encountered in practice, by graphic means in the course of 20—30 minutes, using a compass, ruler and previously constructed nomograms [6; 11]. Conditions (35), (46), (52) are indicated which should be observed in order to avoid large errors on using the graphoanalytical method. The Accuracy of the approximate formulae are illustrated by six examples, in which a comparison is made either with exact bydrodynamical solutions or with résults of simulation on resistance paper [4; 12]. A means of extending the earlier obtained results for anisotropic soil [7; III] and two-layei soil [9] to the case of drained aprons is indicated.

### Mikhail Alexeyevich Lavrentyev (on his sixtieth birthday)

Fil'chakov F. P., Mitropolskiy Yu. A., Shtokalo I. Z.

Ukr. Mat. Zh. - 1960. - 12, № 4. - pp. 490 - 491