Numerical resolution of nonlinear partial equations using the wavelet approach. (English) Zbl 0802.65100

Ruskai, Mary Beth (ed.) et al., Wavelets and their applications. Boston, MA etc.: Jones and Bartlett Publishers. 227-238 (1992).
The paper deals with the approximate solution of the initial-boundary value problem \(\partial U/\partial t+ LU+ G(U)= 0\), \(U(0,t)= U(1,t)\); \(U(x,0)= U_ 0\), \(t\geq 0\), \(0\leq x\leq 1\). Here, \(L\) is a linear elliptic operator with constant coefficients having a nonnegative symbol and \(G\) depends on \(U\) and its derivatives. An approximate solution is sought in a trial space consisting of wavelets.
The paper is divided into three sections. In the first one some properties of wavelets used in the sequel are formulated (concerning the definition and basic properties of wavelets the reader is referred to Y. Meyer, Ondelettes et opérateurs, I: Ondelettes (1990; Zbl 0694.41037)).
In the second section an algorithm for computing the approximate solution is presented (without any estimates of the error), and the last section contains some numerical tests in the case of the one-dimensional Burgers equation.
For the entire collection see [Zbl 0782.00087].


65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
35Q53 KdV equations (Korteweg-de Vries equations)


Zbl 0694.41037