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Axioms 2017, 6(1), 4; doi:10.3390/axioms6010004

Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients

Department of Mathematics and Didactics of Mathematics, Technical University in Liberec, Studentská 2, 461 17 Liberec, Czech Republic
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Received: 12 January 2017 / Accepted: 20 February 2017 / Published: 22 February 2017
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
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Abstract

We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efficiency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black–Scholes equation with a quadratic volatility. View Full-Text
Keywords: Riesz basis; wavelet; spline; interval; differential equation; sparse matrix; Black–Scholes equation Riesz basis; wavelet; spline; interval; differential equation; sparse matrix; Black–Scholes equation
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Černá, D.; Finĕk, V. Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients. Axioms 2017, 6, 4.

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