A Wavelet Collocation Method for the Numerical Solution of Partial Differential Equations

Abstract

We describe a wavelet collocation method for the numerical solution of partial differential equations which is based on the use of the autocorrelation functions of Daubechie's compactly supported wavelets. For such a method we discuss the application of wavelet based preconditioning techniques along with the treatment of boundary conditions, and we show the results of some numerical tests for several 1-and 2-dimensional model problems.

Wavelet Galerkin Methods for Aerosol Dynamic Equations in Atmospheric Environment

“…Other methods to construct scaling and wavelet basis functions on [0,1] based on Daubechies' functions can be found in [3,26]. Moreover, V j (Ω) satisfies the following properties (see [12,37]): (i) Approximation property: Let j ≥ J 0 and for any u ∈ H s (Ω) H 1 E (Ω), there exists a constant C 1 independent of j and u such that…”

Numerical solution of parabolic partial differential equations using adaptive gird Haar wavelet collocation method

“…For instance J = 2 ⇒ N = 8, then we have grid space with interval length . Now, we get the numerical solution on these points [10,11,15], whose method of solution and numerical implementation are given by the following sections.…”

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Wavelet Collocation Method for Optimal Control Problems

“…The wavelet applications in dealing with dynamic system problems, especially in solving partial differential equations with two-point boundary value constraints have been discussed in many papers [2][3][4][5]. By transforming differential equations into algebraic equations, the solution may be found by determining the corresponding coefficients that satisfy the algebraic equations.…”