Numerical solution of differential equations using Haar wavelets

doi:10.1016/j.matcom.2004.10.005

Mathematics and Computers in Simulation

Volume 68, Issue 2, 20 April 2005, Pages 127-143

https://doi.org/10.1016/j.matcom.2004.10.005 Get rights and content

Abstract

Haar wavelet techniques for the solution of ODE and PDE is discussed. Based on the Chen–Hsiao method [C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc.—Control Theory Appl. 144 (1997) 87–94; C.F. Chen, C.H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc. Control Theory Appl. 146 (1997) 213–219] a new approach—the segmentation method—is developed. Five test problems are solved. The results are compared with the result obtained by the Chen–Hsiao method and with the method of piecewise constant approximation [C.H. Hsiao, W.J. Wang, Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simulat. 57 (2001) 347–353; S. Goedecker, O. Ivanov, Solution of multiscale partial differential equations using wavelets, Comput. Phys. 12 (1998) 548–555].

Introduction

Solution of differential equations by the wavelet method has been discussed in many papers (see e.g. [1], [2], [3], [4], [5], [6], [7], [8]). For this purpose different approaches as Galerkin and collocation methods, FEM and BEM are used; in [3] the filter bank method and in [7] the Kalman filters are applied. In some papers adaptive methods, which give possibility for grid refinement, are proposed. In most papers the Daubechies wavelets are used, the Gaussian wavelet is applied in [2], [8]. Numerical difficulties appear in the treatment of nonlinearities, where integrals of products of wavelets and their derivatives must be computed. This can be done by introducing the connection coefficients [4], [9], but this method is applicable only for a narrow class of equations. An adaptive method for computing such integrals is also proposed [4].

The complexity of the wavelet solutions has induced some pessimistic estimates. Strong and Nguyen write in their text-book [10] “ $\dots$ the competition with other methods is severe. We do not necessarily predict that wavelets will win” (p. 394). Jameson [11] writes: “ $\dots$ nonlinearities etc., when treated in a wavelet subspace, are often unnecessarily complicated …. There appears to be no compelling reason to work with Galerkin-style coefficients in a wavelet method” (p. 1982). In order to mollify this criticism we would like to emphasize on the advantages of the wavelet method: using the wavelet method it is possible to detect singularities, local high frequencies, irregular structure and transient phenomena exhibited by the analyzed function.

Evidently attempts to simplify the wavelet approach for solving differential equations are welcome. One possibility for this is to make use of the Haar wavelet family.

In 1910 Alfred Haar introduced a function which presents an rectangular pulse pair (Fig. 1b). After that various generalizations and definitions were proposed (state-of-the art about Haar transforms can be found in [12]). In 1980s it turned out that the Haar function is in fact the Daubechies wavelet of order 1. This enabled to introduce the Haar wavelet, which is the simplest orthonormal wavelet with compact support. It should be mentioned that the Haar wavelet has an essential shortcoming: it is not continuous. In the points of discontinuity the derivatives do not exist, therefore it is not possible to apply the Haar wavelet directly for solving differential equations.

There are some possibilities to come out from this impasse. First the piecewise constant Haar function can be regularized with interpolation splines, this technique has been applied in several papers by Cattani (see e.g. [13], [14], [15]).

The second possibility is to make use of the integral method, by which the highest derivative appearing in the differential equation is expanded into the Haar series. This approximation is integrated while the boundary conditions are incorporated by using integration constants [2]. This approach has been realized for the Haar wavelets by Chen and Hsiao [16], [17]. The main idea of this technique is to convert a differential equation into an algebraic one; hence the solution procedure is either reduced or simplified accordingly. Later on this method was successfully applied for solving singular, bilinear and stiff systems [18], [19], [20]. For the Chen and Hsiao method (CHM) the choice of solution steps is essential: if the step is too small the coefficient matrix may be nearly singular and its inversion brings to instability of the solution. Nonlinearity of the system also complicates the solutions, since big systems of nonlinear equations must be solved.

Cattani observed that computational complexity can be reduced if the interval of integration is divided into some segments [15], this method he called the reduced Haar transform. The number of collocation points in each segment is considerably smaller as in the case of CHM; therefore we can hope that such solutions are more stable.

Further simplification of the solution can be obtained if in each segment only one collocation point is taken. It is assumed that the highest derivative is constant in each segment, therefore this method is called “piecewise constant approximation (PCA)”. This method is very convenient in the case of nonlinear differential equations, since for each segment only one nonlinear equation should be solved. This method has been applied by Goedecker and Ivanov [21] and also by Hsiao and Wang [20].

The main aim of this paper is to elaborate the method of segmentation (SM) and to apply it to some differential equations. To compare the efficiency of CHM, SM and PCA some test problems are solved.

Section snippets

Haar wavelet basis

The Haar wavelet family for $t \in [0, 1]$ is defined as follows: $h_{i} (t) = \{\begin{matrix} 1 & for & t \in [\frac{k}{m}, \frac{k + 0.5}{m}) \\ - 1 & for & t \in [\frac{k + 0.5}{m}, \frac{k + 1}{m}] \\ 0 & elsewhere \end{matrix} .$

Integer $m = 2^{j} (j = 0, 1, \dots, J)$ indicates the level of the wavelet; $k = 0, 1, \dots, m - 1$ is the translation parameter. Maximal level of resolution is J. The index i in (2.1) is calculated according the formula $i = m + k + 1$ ; in the case of minimal values $m = 1, k = 0$ we have $i = 2$ , maximal value of i is $i = 2 M = 2^{J + 1}$ . It is assumed that the value $i = 1$ corresponds to the scaling function for which $h_{1} \equiv 1$ in $[0, 1]$ and vanishes

Method of segmentation

In this section the ODE $\frac{d^{2} u}{{d t}^{2}} = F (t, u, \frac{d u}{d t}), t \in [0, T]$ is considered.

Instead of (3.1) the system of first-order equations $\frac{d u}{d t} = v, \frac{d v}{d t} = F (t, u, v)$ can be solved.

Let us divide the time interval into N segments; the length of the n-th segment is denoted by $d^{(n)}$ . If $t^{(n)}$ is the coordinate of the n-th dividing point, then $t^{(n + 1)} = t^{(n)} + d^{(n)}, n = 1, 2, \dots, N$ with $t^{(1)} = 0, t^{(N + 1)} = 1$ .

Consider the n-th segment. It is convenient to introduce the local time $τ = \frac{t - t^{(n)}}{d^{(n)}}$ and choose 2M collocation points $τ_{j} = \frac{1}{2 M} (j - \frac{1}{2}), j = 1, 2, \dots, 2 M .$

System

Second order linear ODE

Consider the linear differential equation $\frac{d^{2} u}{d t^{2}} + p^{*} \frac{d u}{d t} + q^{*} u = s^{*} f (t),$ where $p^{*}$ , $q^{*}$ , $s^{*}$ are constants and $f (t)$ is a prescribed function.

Carrying out the same operations which were needed for derivation of (3.5) the Eq. (4.1) gets the form $\dot{u} = d^{(n)} v, \dot{v} = - p v - q u + s ϕ (t),$ where $p = p^{*} d^{(n)}, q = q^{*} d^{(n)}, s = s^{*} d^{(n)}, ϕ (t) = f (t^{(n)} + d^{(n)} τ)$ .

Applying the substitution (3.6), dividing all terms with the matrix $H$ and taking into account the condition $Y = E H^{- 1}$ Eq. (4.2) can be brought into the form $\begin{matrix} a - d^{(n)} b P = d^{(n)} v_{n} Y \\ q a P + b (I + p P) = - (p v_{n} + q u_{n} \end{matrix}$

Nonlinear ODE solutions

The aim of this Section is to demonstrate how the Haar wavelets can be applied for solving nonlinear differential equations. For a test problem the Duffing equation $\frac{d^{2} u}{d t^{2}} + p^{*} \frac{d u}{d t} + q^{*} u + r^{*} u^{3} = s^{*} \cos ϖ t, t \in [0, T]$ with the initial conditions $u (0) = 0, {(d u / d t)}_{t = 0} = 1$ is taken. To apply CHM for the solution of this equation seems to be inconvenient since for calculating the wavelet coefficients big nonlinear systems must be solved; therefore here only SM and PCA are considered.

As in Section 4 the interval $t \in [0, T]$

Diffusion equation

In this section the diffusion (or heat) equation in the unit interval $\frac{\partial u}{\partial t} = A \frac{\partial^{2} u}{\partial x^{2}}, 0 \leq x \leq 1, 0 \leq t \leq 1$ with the initial condition $u (x, 0) = u_{0} (x), 0 \leq x \leq 1$ and boundary conditions $u (0, t) = u (1, t) = 0$ is considered.

This problem has been the touchstone for different solutions based on the wavelet method. Adapted methods were proposed in [5], [24]; in paper [5] the Deslauries–Dubuc interpolating-function for the Daubechies wavelets was applied, in [24] the Coiflet wavelets were used. In papers [15], [25] by Cattani Haar

Conclusions

Solution of ODE and PDE with the aid of Haar wavelets is discussed. A new technique which is called the segmentation method is proposed. Its applicability and efficiency is checked on five test problems. The achieved results are compared with the former wavelet solutions CHM and PCA. On the basis of this analysis following conclusions can be made.

The CHM is mathematically very simple, especially if the matrix representation is used. Since most elements of the matrices P and H are zero the

Acknowledgement

Financial support of the Estonian Science Foundation (grant no. 5240) is gratefully acknowledged.

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Numerical solution of differential equations using Haar wavelets

Abstract

Introduction

Section snippets

Haar wavelet basis

Method of segmentation

Second order linear ODE

Nonlinear ODE solutions

Diffusion equation

Conclusions

Acknowledgement

Comput. Methods Appl. Mech. Eng.

J. Comput. Phys.

Appl. Num. Math.

Appl. Num. Math.

Appl. Num. Math.

Adv. Eng. Software

IEE Proc. Control Theory Appl.

Math. Comput. Simulat.

Math. Comput. Simulat.

Math. Comput. Simulat.

Filter bank methods for hyperbolic PDEs

SIAM J. Num. Anal.