Wavelet applications to the Petrov--Galerkin method for Hammerstein equations

The purpose of this paper is two-fold. First, we develop the Petrov-Galerkin method and the iterated Petrov-Galerkin method for a class of nonlinear Hammerstein equations. Alpert [SIAM J. Math. Anal. 24 (1993) 246] established a class of wavelet basis and applied it to approximate solutions of the Fredholm second kind integral equations by the Galerkin method. He then demonstrated an advantage of a wavelet basis application to such equations by showing that the corresponding linear system is sparse. The second purpose of this paper is to study how this advantage of the sparsity can be extended to nonlinear Hammerstein equations.

[1]  Yuesheng Xu,et al.  Fast Collocation Methods for Second Kind Integral Equations , 2002, SIAM J. Numer. Anal..

[2]  E. Atkinson.  THE NUMERICAL SOLUTION OF ANONLINEAR BOUNDARY INTEGRALEQUATION ON SMOOTH SURFACESKendall , 1994 .

[3]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[4]  Kendall E. Atkinson,et al.  The numerical solution of a non-linear boundary integral equation on smooth surfaces , 1994 .

[5]  Hideaki Kaneko,et al.  Superconvergence of the iterated Galerkin methods for Hammerstein equations , 1996 .

[6]  G. M. Vainikko.  Galerkin's perturbation method and the general theory of approximate methods for non-linear equations☆ , 1967 .

[7]  Reinhold Schneider,et al.  Multiwavelets for Second-Kind Integral Equations , 1997 .

[8]  Yuesheng Xu,et al.  The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes , 1997, Adv. Comput. Math..

[9]  Zhongying Chen,et al.  The Petrov--Galerkin and Iterated Petrov--Galerkin Methods for Second-Kind Integral Equations , 1998 .

[10]  Hideaki Kaneko,et al.  Superconvergence of the iterated collocation methods for Hammerstein equations , 1997 .

[11]  B. Alpert.  A class of bases in L 2 for the sparse representations of integral operators , 1993 .

[12]  L. Delves,et al.  Computational methods for integral equations: Frontmatter , 1985 .

[13]  Christoph Schwab,et al.  Wavelet approximations for first kind boundary integral equations on polygons , 1996 .

[14]  K AlpertBradley.  A class of bases in L2 for the sparse representations of integral operators , 1993 .

[15]  Ian H. Sloan,et al.  A new collocation-type method for Hammerstein integral equations , 1987 .

[16]  C. Micchelli,et al.  Using the Matrix Refinement Equation for the Construction of Wavelets on Invariant Sets , 1994 .