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The numerical solution of weakly singular integral equations based on the meshless product integration (MPI) method with error analysis

Authors
Journal
Applied Numerical Mathematics
0168-9274
Publisher
Elsevier
Volume
81
Identifiers
DOI: 10.1016/j.apnum.2014.02.013
Keywords
  • Fredholm Integral Equation
  • Weakly Singular Kernel
  • Radial Basis Function (Rbf)
  • Meshless Method
  • Product Integration Method
  • Error Analysis
Disciplines
  • Computer Science
  • Mathematics

Abstract

Abstract This article investigates a numerical scheme based on the radial basis functions (RBFs) for solving weakly singular Fredholm integral equations by combining the product integration and collocation methods. A set of scattered points over the domain of integration is utilized to approximate the unknown function by using the RBFs. Since the proposed scheme does not require any background mesh for its approximations and numerical integrations unlike other product integration methods, it is called the meshless product integration (MPI) method. The method can be easily implemented and its algorithm is simple and effective to solve weakly singular integral equations. This approach reduces the solution of weakly singular integral equations to the solution of linear systems of algebraic equations. The error analysis of the proposed method is provided. The validity and efficiency of the new technique are demonstrated through several tests.

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