Orthogonal Franklin system and orthogonal system of finite functions in numerical methods of boundary problems solving

V. L. Leontiev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 2, pp. 398-404.

Résumé

Possibilities of classical trigonometric Fourier series are substantially limited in 2-D and 3-D boundary value problems. Boundary conditions of such problems for areas with curvilinear boundaries often fails when using the classical Fourier series. The solution of this problem is the use of orthogonal finite functions. However, orthogonal Haar basis functions are not continuous. The orthogonal Daubechies wavelets have compact supports, but is not written in analytical form and have low smoothness. Continuous finite Schauder–Faber functions are not orthogonal. Orthogonal Franklin continuous functions are not finite. The connection of the orthogonal Franklin functions with a sequence of grid groups of piecewise linear orthogonal finite basis functions (OFF) is established here. The Fourier-OFF series on the basis of such continuous OFF is formed. Such series allows to execute boundary conditions of Dirichlet's type on curvilinear boundaries in integral performances of boundary value problems. A similar problem is connected with a satisfaction of Neumann boundary conditions and also is eliminated in the integral mixed performances of boundary value problems. Fourier-OFF series increases the effectiveness of mixed numerical methods for boundary value problems solving.

Détail

Keywords: orthogonal system of functions, orthogonal finite functions, Fourier series, mixed numerical methods for boundary problems solving.

Bibliographie

[1] Leontiev V. L., “Orthogonal Franklin system and orthogonal system of finite functions in numerical methods of boundary problems solving”, The 4nd International Conference “Mathematical Physics and its Applications”, Book of Abstracts and Conference Materials, eds. I. V. Volovich; V. P. Radchenko, Samara State Technical Univ., Samara, 2014, 228–229 (In Russian)

[2] Haar A., “Zur Theorie der orthogonalen Funktionensysteme”, Math. Ann., 69:3 (1910), 331–371 | DOI | MR | Zbl

[3] Daubechles I., “Orthonormal Bases of Compactly Supported Wavelets”, Commun. Pure Appl. Math., 41:7 (1988), 909–996 ; Daubechles I., “Orthonormal Bases of Compactly Supported Wavelets”, Fundamental Papers in Wavelet Theory, Princeton University Press, Princeton, 2009, 564–652 ; | DOI | MR | DOI | DOI

[4] Faber G., “Über die Orthogonalfunktionen des Herrn Haar”, Jahresbericht der Deutschen Mathematiker-Vereinigung, 19 (1910), 104–112 | Zbl

[5] Shauder J., “Eine Eigenschaft des Haarschen Orthogonalsystems”, Math. Z., 28:1 (1928), 317–320 | DOI | MR

[6] Franklin P., “A set of continuous orthogonal functions”, Math. Ann., 100:1 (1928), 522–529 ; Franklin P., “A set of continuous orthogonal functions”, Fundamental Papers in Wavelet Theory, Princeton University Press, Princeton, 2009, 189–196 ; | DOI | MR | Zbl | DOI | DOI

[7] Ul'yanov P. L., “Haar series”, Sov. Math., Dokl., 4 (1963), 437–440 | Zbl

[8] Ul'janov P. L., “On Haar series”, Mat. Sb. (N.S.), 63:105 (1964), 356–391 (In Russian) | MR | Zbl

[9] Schipp F., Simon P., “Investigation of Haar and Franklin series in Hardy spaces”, Anal. Math., 8:1 (1982), 47–56 | DOI | MR | Zbl

[10] Gevorkyan G. G., “Absolute and conditional convergence of series in Franklin systems”, Math. Notes, 45:3 (1989), 200–210 | DOI | MR | Zbl

[11] Wojtaszczyk P., Woźniakowski K., “Orthonormal polynomial bases in function spaces”, Israel J. Math., 75:2/3 (1991), 167–191 | DOI | MR | Zbl

[12] Kashin B. S., Saakyan A. A., Ortogonal'nye riady [Orthogonal series], AFTs, Moscow, 1999, 550 pp. (In Russian) | MR | Zbl

[13] Chen W., Cai Z., Qi D., “A New Class of Orthogonal Spline Moments and Its Application”, J. Inf. Comput. Sci., 10:14 (2013), 4563–4571 | DOI

[14] Leontiev V. L., Ortogonal'nye finitnye funktsii i chislennye metody [Orthogonal compactly supported functions and numerical methods], Ulyanovsk State Univ., Ulyanovsk, 2003, 178 pp. (In Russian) | MR

[15] Leont'ev V. L., “A variational-grid method involving orthogonal finite functions for solving problems of natural vibrations of 3D elastic solids”, Mech. Solids, 37:3 (2002), 101–109 | MR

[16] Leontiev V. L., “Orthogonal splines and variational-grid method”, Mat. Model., 14:3 (2002), 117–127 (In Russian) | MR | Zbl

[17] Leont'ev V. L., Lukashanets N. Ch., “Grid bases of orthogonal compactly supported functions”, Comput. Math. Math. Phys., 39:7 (1999), 1116–1126 | MR | Zbl

[18] Krasil'nikov A. R.; Leontiev V. L., “On the variation-grid method of the plate theory”, Mat. Model., 17:3 (2005), 23–34 (In Russian) | MR | Zbl

[19] Leontiev V. L., Rikov E. A., “Integral transforms associated with orthogonal finite functions in the spectral analysis of signals”, Mat. Model., 18:7 (2006), 93–100 (In Russian) | MR | Zbl

[20] Leontiev V. L., Mikhaylov I. S., “On the Building the Potential of the Atomic Interaction Based on Orthogonal Finite Functions”, Nano- i mikrosistemnaia tekhnika, 2011, no. 9, 48–50 (In Russian)