1. Introduction

Partial differential equations (PDEs) are used commonplace in science and in engineering to model the behavior of physical systems. Because of their importance, many numerical techniques for their solutions have been developed: finite difference methods (FDMs), finite element methods (FEMs), spectral methods, Ritz/ Galerkin approach, meshless approaches, and so on. The main characteristic of PDEs is that the "unknown" is a function, that is, an object with an infinite number of degrees of freedom. Because of this, it is usually impossible (even in principle) to get an exact solution by numerical means. The objective of every technique for PDE solving is to get a good approximation of the solution with limited computational resources (CPU time, memory, etc.).

The first step of every PDE solution algorithm is to discretize the PDE, that is, to approximate it with a finite-dimensional problem that can be solved by numerical means. A popular discretization technique is to discretize the space where the solution is searched by restricting the problem to a finite-dimensional vector space, that is, by writing the solution as the linear combination of several base functions. If the original PDE was linear, discretization will map it to a linear problem (typically a linear system or an eigenvalue problem). Even techniques such as FDM (that discretizes the domain) can be often reformulated as suitable discretization of the function space.

Intuitively, better approximations of the solution require finer discretization (the exact meaning of finer depends on the specific approach), especially if the solution has some regions of large variability. Since finer discretization implies larger problems (and, therefore, higher computational efforts), it is of interest to be able to change locally the discretization resolution to the solution variability, possibly in an adaptive way.

on regions of ∂Ω. Typical examples of physical systems giving rise to this type of problem are systems in steady state (e.g., temperature distribution, potential distribution, steady flows, and so on). Typically equilibrium problems are elliptic, that is, D is an elliptic operator (a generalization of the Laplacian).

• In a propagation problem, we are interested in modeling the time evolution of a physical system. The PDE can still be written as in (1)–(3), but the domain can typically be written as <sup>Ω</sup> <sup>¼</sup> IR≥<sup>0</sup> � <sup>W</sup>, where <sup>W</sup> <sup>⊆</sup>IRd�<sup>1</sup> and the first coordinate represents time. Boundary conditions for t ¼ 0 are known as initial conditions. Example physical problems are heat or wave propagation. Propagation

• Finally, in eigenvalue problems we are interested in finding u and λ that satisfy

A wide class of eigenvalue problems is represented by Sturm-Liouville problems

where the apostrophe denotes derivation, the unknowns are λ, and function is

The field of numerical solution of differential equations is very wide, and many techniques have been developed. Nevertheless, a categorization in few large classes is possible. An important step in every solution algorithm is mapping the differential equation into a discrete version with only a finite number of degrees of freedom. A first distinction can be done between techniques that achieve this objective by

The most known technique based on a domain discretization is the FDM where the unknown function is sampled in a finite number of points p1, p2, …, pN ∈ Ω and the derivatives are approximated with finite differences. By writing the differential equation for every p, with the derivative approximated as finite differences, one obtains a system of N equations in N unknowns that can be solved with known techniques. If the original PDE was linear, the discretized system will be

FDM is maybe the simplest approach and the most intuitive, and it can work quite well for simple problems and geometries. Moreover, in the linear case, since any approximation of a derivative in p will consider only few points around p, the matrix of the discretized linear system will be very sparse, allowing for a reduction in the computational effort. The application of FDM techniques becomes difficult,

y∈ Hð Þ ½ � a; b , while p, q, and w, all belonging to Hð Þ ½ � a; b , are known. Sturm-Liouville problems include Bessel differential equations (obtained by writing Laplace, Helmholtz, or Schrodinger equation in cylindrical coordinates) and Lagrange differential equation (obtained working in spherical coordinates).

Du ¼ λu (4)

þ q � y ¼ �λw � y (5)

problems are typically hyperbolic or parabolic.

Wavelets for Differential Equations and Numerical Operator Calculus

DOI: http://dx.doi.org/10.5772/intechopen.82820

p � y<sup>0</sup> ½ �<sup>0</sup>

that can be written as

2.1 Solution of differential equations

2.1.1 Domain discretization

linear too.

37

discretizing the domain Ω or the function space Hð Þ Ω .

albeit possible, in the case of complex problems.

This need for different resolutions in different regions is the idea that links PDE with multiresolution analysis. The birth of multiresolution analysis goes back to 1990 with the works of Mallat [1] and Meyer [2]. Since then there have been a large number of papers ranging from very theoretical ones to application [3]. In a multiresolution analysis, a space of signals (most commonly L<sup>2</sup> ð Þ IR , but not only) is represented as a nesting of spaces with different levels of "resolution." This allows to write a signal as the sum of a "low-resolution" version plus some higherresolution "details."

Because of the ability of changing the resolution used to observe the signal (by adding or removing details), the multiresolution analysis is sometimes described as a mathematical zoom. This fact inspired many applications, including numerical solution of PDEs where they sound promising, especially for those problems that contains localized phenomena (e.g., shockwaves) or intra-scale interaction (e.g., turbulence).

The objective of this chapter is to introduce the reader to the application of wavelets to PDE solutions. This chapter can be ideally divided in three parts: in the first part, we recall briefly the main concepts about PDE and the main algorithms for solving PDE; successively we do a brief recall of multiresolution analysis and wavelets including also multiwavelets and second-generation wavelets that find often application in PDE solutions; and finally, we will illustrate few techniques that can be found in the literature.

### 1.1 Notation

Ω ⊆ IR<sup>d</sup> is the domain where the functions of interest are defined. The boundary of <sup>Ω</sup> is partitioned as follows: <sup>∂</sup><sup>Ω</sup> <sup>¼</sup> <sup>Ω</sup><sup>D</sup> <sup>∪</sup> <sup>Ω</sup>N, <sup>Ω</sup><sup>D</sup> <sup>∩</sup> <sup>Ω</sup><sup>N</sup> <sup>¼</sup> <sup>∅</sup>, where <sup>Ω</sup><sup>D</sup> or <sup>Ω</sup><sup>N</sup> can be empty.

<sup>H</sup>ð Þ <sup>Ω</sup> will denote a space of functions <sup>u</sup> : <sup>Ω</sup> ! IR defined on <sup>Ω</sup> <sup>⊆</sup>IR<sup>d</sup>.

IR≥<sup>0</sup> is the set of nonnegative reals.
