Abstract
Daubechies interval wavelet is used to solve numerically weakly singular Fredholm integral equations of the second kind. Utilizing the orthogonality of the wavelet basis, the integral equation is reduced into a linear system of equations. The vanishing moments of the wavelet make the wavelet coefficient matrices sparse, while the continuity of the derivative functions of basis overcomes naturally the singular problem of the integral solution. The uniform convergence of the approximate solution by the wavelet method is proved and the error bound is given. Finally, numerical example is presented to show the application of the wavelet method.
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References
Atkinson K. An Automatic Program for Linear Fredholm Integral Equations of the Second Kind [J]. ACM Trans Math Software (TOMS), 1976, 2:154–171.
Atkinson K. A Survey of Numerical Solution of Fredholm Integral Equations of the Second Kind [J]. SIAM J Numer Anal, 1976, 3:337–348.
Klerk J H, Eyre D, Venter L M. Lp-approximation Method for the Numerical Solution of Singular Equation [J]. Appl Math Comp, 1995, 72:285–300.
Kancko H, Xu Y. Numerical Solutions for Weakly Singular Fredholm Integral Equation of the Second Kind [J]. Appl Numer Math, 1991, 7:167–177.
Maleknejad K, Mirzaee F. Numerical Solution of Linear Fredholm Integral Equations System by Rationalized Haar Function Method [J]. International Journal of Computation and Mathematics, 2003, 80(11):1397–1405.
Zhong Xianchi, Li Xianfang. Taylor Expansion Method for Solving Fredholm Integral Equations of the Second Kind [J]. Mathematical Theory and Applications, 2004, 24(3):21–23 (Ch).
Maleknejad K, Derili H. The Collocation Method for Hammerstein Equations by Daubechies Wavelets [J]. Applied Mathematics and Computation, 2006,172:846–864.
Zhu T L, Lin W. Fast Wavelet Algorithm of the Poisson Integral [J]. Appl Math Comp, 1998, 96:1–13.
Beylkin G, Coifman R, Rokhlin V. Fast Wavelet Transform and Numerical Algorithms I [J]. Communications on Pure Applied Mathematics, 1991, 44:141–183.
Alpert B. A Class of Bases in L 2 for Sparse Representation of Integral Operators [J]. SIAM J Math Anal, 1993, 24:246–262.
Cohen A, Daubechies I, Vial P. Wavelets on the Interval and Fast Wavelet Transforms [J]. Appl Comput Harmonic Analysis, 1993,1:54–81.
Wang Gaofeng. Application of Wavelets on the Interval to Numerical Analysis of Integral Equations in Electromagnetic Scattering Problems [J]. International Journal for Numerical Methods in Engineering, 1997,40:1–13.
Chen H L, Peng S L. Solving Integral Equations with Logarithmic Kernel by Using the Periodic Quasi-Wavelet [J]. J Comput Math, 2000, 18:487–512.
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Foundation item: Supported by the National Natural Science Foundation of China (60572048) and the Natural Science Foundation of Guangdong Province (054006621)
Biography: TANG Xinjian (1959–), male, Ph. D. candidate, Associate professsor of Institute of Rock and Soil Mechanics, research direction: the theory and application of wavelet analysis.
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Tang, X., Pang, Z., Zhu, T. et al. Wavelet numerical solutions for weakly singular Fredholm integral equations of the second kind. Wuhan Univ. J. of Nat. Sci. 12, 437–441 (2007). https://doi.org/10.1007/s11859-006-0110-5
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DOI: https://doi.org/10.1007/s11859-006-0110-5