Wavelet-Galerkin Method for Second-Order Integro-differential Equations on Product Domains

Abstract

AbstractThis chapter is concerned with the study of the wavelet-Galerkin method for the numerical solution of the second-order partial integro-differential equations on the product domains. Prescribed boundary conditions are of Dirichlet or Neumann type on each facet of the domain. The variational formulation is derived, and the existence and uniqueness of the weak solution are discussed. Multi-dimensional wavelet bases satisfying boundary conditions are constructed by the tensor product of wavelet bases on the interval using isotropic and anisotropic approaches. The constructed wavelet bases are used in the Galerkin method to find the numerical solution of the integro-differential equations. The convergence of the method is proven, and error estimates are derived. The advantage of the method consists in the uniform boundedness of the condition numbers of discretization matrices and in the fact that these matrices exhibit an exponential decay of their elements away from the main diagonal. Based on the decay estimates, we propose a compression strategy for an approximation of the discretization matrices by sparse or quasi-sparse matrices. Numerical experiments are presented to confirm the theoretical results and illustrate the efficiency and applicability of the method.KeywordsWaveletSplineRiesz basisBoundary conditionsCondition numberIntegro-differential equationError estimateMathematics Subject Classification (2010)47G2065N1265T60