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Galerkin-wavelet methods for two-point boundary value problems. (English) Zbl 0771.65050
Both the definition, basic properties and approximation properties of wavelets are described. The Galerkin method is applied to the numerical solution of the two-point boundary value problem \(-(q(x)u')'+r(x)u=f(x)\) for \(x\in (0,R)\) with Dirichlet (or mixed, or Neumann) condition, where the bases of a finite dimension solution subspace are formed by means of wavelet functions. Numerical examples are given.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
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[1] Axelsson O., Lindskog G. (1986): On the rate of convergence of the preconditioned conjugate gradient method. Numer. Math.48, 499-523 · Zbl 0564.65017 · doi:10.1007/BF01389448
[2] Calderón, A.P. (1964): Intermediate spaces and interpolation, the complex method. Studia Math.24, 113-190 · Zbl 0204.13703
[3] Ciarlet, P.G. (1978): The finite element methods for elliptic problems. North-Holland Amsterdam
[4] Coifman, R., Weiss, G. (1971): Analyse Harmonique non commutative sur certains espaces homogènes. Springer Berlin Heidelberg New York · Zbl 0224.43006
[5] Combes, J.M., Grossmann A. Tchamitchian, Ph. (eds.) (1990): Wavelets, time-frequency methods and phase space, 2nd ed. Springer, Berlin Heidelberg New York
[6] Cortina, E. Gomes, S.M. (1989): A wavelet based numerical method applied to free boundary problems. IPE Technical Report, Sao Jose dos Campos, Brasil
[7] Daubechies, I. (1988): Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math.41, 909-996 · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
[8] Daubechies, I., Lagarias, J.: Two-scale difference equations. I. Global regularity of solutions. II. Infinite matrix products, local regularity and fractals. AT&T Bell Laboratories, preprint · Zbl 0788.42013
[9] Duffin, R.J., Schaeffer, A.C. (1952): A class of nonharmonic Fourier series Trans. Am. Math. Soc.72, 341-366 · Zbl 0049.32401
[10] Glowinski, R., Lawton, W.M., Ravachol, M., Tenenbaum, E. (1990): Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. In: R. Glowinsky, A. Lichnewsky, eds., Computing Methods in Applied Sciences and Engineering. SIAM, Philadelphia, pp. 55-120 · Zbl 0799.65109
[11] Golub, G.H., Van Loan, C.F. (1988): Matrix computations, 2nd ed. Johns Hopkins University Press, Baltimore · Zbl 1118.65316
[12] Goupillaud, P., Grossmann, A., Morlet, J. (1984/85): Cycle-octave and related transforms in seismic signal analysis. Geoexploration23, 85-102 · doi:10.1016/0016-7142(84)90025-5
[13] Grossmann, A., Holschneider, M., Kronland-Martinet, R., Morlet, J. (1987): Detection of abrupt changes in sound signals with the help of wavelet transforms. In: Inverse problems: an interdisciplinary study. Academic Press, London Orlando, pp. 289-306
[14] Grossmann, A., Morlet, J. (1984): Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math.15, 723-736 · Zbl 0578.42007 · doi:10.1137/0515056
[15] Haar, A. (1910): Zur Theorie der orthogonalen Funktionensysteme. Math. Ann.69, 331-371 · JFM 41.0469.03 · doi:10.1007/BF01456326
[16] Heil, C. (1990): Wavelets and frames. In: Signal processing, pt. 1, 2nd ed., Springer, Berlin Heidelberg New York, pp. 147-160
[17] Kronland-Martinet, R., Morlet, J., Grossmann, A. (1992): Analysis of sound patterns through wavelet transforms. To appear · Zbl 0850.42006
[18] Latto, A., Tenenbaum, E. (1990): Les ondelettes á support compact et la solution numérique de l’équation de Burgers. Preprint, AWARE Cambridge · Zbl 0721.65073
[19] Liénard, J.S.: Speech analysis and reconstruction using short-time, elementary waveforms. LIMSI-CNRS, Orsay, France
[20] Mallat, S. (1989): Multiresolution approximations and wavelet orthonormal bases ofL 2?. Trans. Amer. Math. Soc.315, 69-87 · Zbl 0686.42018
[21] Mallat, S. (1989): Multifrequency channel decompositions of images and wavelet models. IEEE Trans. Acoust. Speech Signal Process37, 2091-2110 · doi:10.1109/29.45554
[22] Meyer, Y. (1985): Principe d’incertitude, bases hilbertiennes et algebres d’operateurs. Bourbaki Seminar, no. 662
[23] Meyer, Y. (1990): Ondelettes et opérateurs, I. Ondelettes. Hermann, Paris · Zbl 0694.41037
[24] Rodet, X. (1984): Time-domain formant-wave-function synthesis. Comput. Music J.3, 9-14 · doi:10.2307/3679809
[25] Strang, G. (1989): Wavelets and dilation equations: A brief introduction. SIAM Review31, 614-627 · Zbl 0683.42030 · doi:10.1137/1031128
[26] Stromberg, J. (1983): A modified Franklin system and higher-order systems of ? n as unconditional bases for Hardy spaces. In: Conference on harmonic analysis in honor of Antoni Zygmund. Wadsworth, Belmont, pp. 475-493
[27] Yserentant, H. (1986): On the multi-level splitting of finite element spaces. Numer. Math.49, 379-412 · Zbl 0608.65065 · doi:10.1007/BF01389538
[28] Zienciewicz, O.C., Kelly, D.W., Gago, J., Babu?ka, I. (1982): Hierarchical finite element approaches, error estimates and adaptive refinement. In: The mathematics of finite elements and applications IV. Academic Press, London, pp. 313-346
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