# Application of dual-Chebyshev wavelets for the numerical solution of boundary integral equations with logarithmic singular kernels

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## Abstract

In this paper, the discrete Galerkin method based on dual-Chebyshev wavelets has been presented to approximate the solution of boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations occur as a reformulation of a boundary value problem of Laplace’s equation with linear Robin boundary conditions. The discrete Galerkin methods for solving logarithmic boundary integral equations with Chebyshev wavelets as a basis encounter difficulties for computing their singular integrals. To overcome this problem, we establish the dual-Chebyshev wavelets, such that they are orthonormal without any weight functions. This property adapts Chebyshev wavelets to discrete Galerkin method for solving logarithmic boundary integral equations. We obtain the error bound for the scheme and find that the convergence rate of the proposed method is of \(O(2^{-Mk})\). Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the new technique and confirm the theoretical error analysis.

## Keywords

Boundary integral equation Laplace’s equation Logarithmic singular kernel Dual-Chebyshev wavelet Discrete Galerkin method Error analysis## Mathematics Subject Classification

45A05 45E99 65L10 65T60## Notes

### Acknowledgements

The authors are very grateful to the reviewers for their valuable comments and suggestions which have improved the paper.

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