A Numerical Method for Boundary Value Problems for a Homogeneous Equation with the Squared Laplace Operator with the Use of Interpolating Wavelets
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 198-204
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We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces $h_{\infty}(K)$ of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.
Keywords:
biharmonic function, boundary value problems, interpolating wavelets, multiresolution analysis (MRA).
@article{TIMM_2019_25_2_a18,
author = {Yu. N. Subbotin and N. I. Chernykh},
title = {A {Numerical} {Method} for {Boundary} {Value} {Problems} for a {Homogeneous} {Equation} with the {Squared} {Laplace} {Operator} with the {Use} of {Interpolating} {Wavelets}},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {198--204},
publisher = {mathdoc},
volume = {25},
number = {2},
year = {2019},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2019_25_2_a18/}
}
TY - JOUR AU - Yu. N. Subbotin AU - N. I. Chernykh TI - A Numerical Method for Boundary Value Problems for a Homogeneous Equation with the Squared Laplace Operator with the Use of Interpolating Wavelets JO - Trudy Instituta matematiki i mehaniki PY - 2019 SP - 198 EP - 204 VL - 25 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TIMM_2019_25_2_a18/ LA - ru ID - TIMM_2019_25_2_a18 ER -
%0 Journal Article %A Yu. N. Subbotin %A N. I. Chernykh %T A Numerical Method for Boundary Value Problems for a Homogeneous Equation with the Squared Laplace Operator with the Use of Interpolating Wavelets %J Trudy Instituta matematiki i mehaniki %D 2019 %P 198-204 %V 25 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/TIMM_2019_25_2_a18/ %G ru %F TIMM_2019_25_2_a18
Yu. N. Subbotin; N. I. Chernykh. A Numerical Method for Boundary Value Problems for a Homogeneous Equation with the Squared Laplace Operator with the Use of Interpolating Wavelets. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 198-204. http://geodesic.mathdoc.fr/item/TIMM_2019_25_2_a18/