5. Conclusions

boundary condition handling simpler. Another problem that can arise is that many wavelets have no closed-form description, for example, Daubechies wavelets that are described as the result of the iteration of operator O in (39). This can make their application to PDE more difficult, for example, when computing scalar products

Finally, another common issue is that most of the known wavelets are defined

multiresolution analysis by a one-dimensional one is the separable (or tensor product or Kronecker product) approach that create a multidimensional function by the

This kind of approach, however, produces "cube-like" wavelets, and their application to FEM schemes based on triangular elements can be difficult.

In this section we briefly summarize some interesting wavelet-based schemes that can be found in the literature. As said above, wavelets and scaling functions can be used as basis in the approximation used in collocation, weighted residuals, and

Wavelets in Galerkin and weighted residual methods bring the advantage of their multiresolution and localization properties while, however, suffering from difficulties in handling complex boundary conditions. Moreover, nonlinear equations can turn out to be difficult to handle. Nevertheless, there have been many successful examples in the application to elliptic, hyperbolic, and parabolic PDE [11–27]. Wavelet-based collocation methods, where wavelet functions are used as shape functions, also registered some success. The advantage of collocation methods is that they are more easily applicable in nonlinear cases [28, 29] and irregular boundary conditions [30]. A collocation method based on second-generation wave-

Much more popular seems to be the application of wavelets to FEM techniques. In this case wavelets or scaling functions are used as shape functions instead of the more traditional polynomials. Daubechies wavelets are particularly popular most probably because of their compact support property. Also of interest is the fact that Daubechies' wavelets can have any number of null moments, making possible the perfect interpolation of polynomials. Some examples of successful application Daubechies wavelets to PDE (mostly mechanical problems) are [32–36]. Of special interest is the proposal of Mitra [37] where wavelet-based FEM is used to transform a wave propagation problem into ordinary differential equations that are succes-

Another popular solution for wavelet-based FEM is the wavelets based on spline spaces. Although spline bases cannot have both compact support and orthogonality, in differential equations, as explained above, we gladly give up on orthogonality if we can get compact support and smoothness. Another important advantage of splines is that a simple closed-form expression is known. Examples of spline applications can be found in [38–40]. Of special interest is the application of Hermite cubic splines (HCS), a kind of multiwavelet [41] that shows promise in handling in a numerically robust way boundary conditions. The HCS is a multiwavelet with four smooth (twice differentiable) components defined on interval 0½ � ; 1 . Some examples of application can be found in [42–44]. A problem with the application of wavelets

let and lifting is applied to a nonlinear vibration problem in [30, 31].

ϕ3Dð Þ¼ x; y; z ϕð Þ x ϕð Þy ϕð Þz (61)

on a one-dimensional domain, while many physical systems are on a multidimensional domain. The easiest way to create a multidimensional

involved in weighted residual and other methods.

Wavelet Transform and Complexity

product of several one-dimensional ones, e.g.,

4.1 Some schemes from the literature

other methods.

sively solved.

50

This chapter introduced the reader to the field of applying wavelets to the numerical solution of differential equations. Both wavelets and differential equations are research fields with many applications, contributions, and results. Their combination gives rise to wide varieties of methods, each one suited for specific applications. By looking at the literature, we can see that wavelets can be a very powerful tool for solving PDE especially because of their multiresolution nature that allows to optimize the level of detail where it is needed. Wavelets, however, are not a silver bullet for all problems either, since they can have some characteristics (multidimensional construction via tensor product, nonexistence of a closed-form expression, difficulty in handling some boundary conditions, etc.) that can make their application not trivial in some cases. We can say that this is a field where, more than ever, no single solution fits all and that every practitioner needs to find the solution specific for the problem at hand using knowledge in both fields and some ingenuity.
