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A wavelet Petrov-Galerkin method for solving integro-differential equations. (English) Zbl 1170.65337

Summary: We solve integro-differential equation by using the Alpert multiwavelets as basis functions. We also use the orthogonality of the basis of the trial and test spaces in the Petrov-Galerkin method. The computations are reduced because of orthogonality. Thus the final system that we get from discretizing the integro-differential equation has a very small dimension and enough accuracy. We compare the results with [M. Lakestani, M. Razzaghi and M. Dehghan, Math. Probl. Eng. 2006 (2006)] and [A. Ayad, Stud. Univ. Babeş-Bolyai, Math. 41, No. 3, 1–8 (1996; Zbl 1009.65082)] which used a system of much larger dimension and got less accurate results. In [Z. Chen and Y. Xu, SIAM J. Numer. Anal. 35, No. 1, 406–434 (1998; Zbl 0911.65143)], convergence of Petrov-Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree.

MSC:

65R20 Numerical methods for integral equations
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[1] DOI: 10.1137/0524016 · Zbl 0764.42017
[2] DOI: 10.1017/CBO9780511626340
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