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Numerical solution of hypersingular equation using recursive wavelet on invariant set. (English) Zbl 1201.65215

Summary: We construct the Chebyshev recursive wavelets on a unit interval of the first kind, the second kind and their corresponding weight functions. We apply the wavelet collocation method to solve the natural boundary integral equation of the harmonic equation on the lower half-plane numerically. It is convenient and accurate to generate the stiffness matrix. Two numerical examples are presented. It is shown that the stiffness matrix is highly sparse when the order of the stiffness matrix becomes large. The current method allows choosing an appropriate weight function to increase the convergence rate and accuracy of the numerical results.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
65T60 Numerical methods for wavelets
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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[1] Feng, K., Differential versus integral equation and finite versus infinite Element, Math. Numer. Sinica., 2, 1, 100-105 (1980) · Zbl 0466.65064
[4] Hadamard, J., Lectures on Cauchy’s Problems in Linear Partial Differential Equations (1923), Yale University Press: Yale University Press New Haven · JFM 49.0725.04
[5] Yu, D., Mathematical Theory of Natural Boundary Element Method (1993), Science Press: Science Press Beijing, (in Chinese)
[6] Yu, D., Natural Boundary Integral Method and its Applications (2002), Kluwer academic publishers.: Kluwer academic publishers. Dordrecht
[7] Yu, D.; Zhao, L., Natural boundary integral method and related numerical methods, Eng. Anal. Bound. Elen., 28, 937-944 (2004) · Zbl 1074.65143
[8] Tong, M. S.; Chew, W. C., Evaluation of singular Fourier coefficients in solving electromagnetic scattering by body of revolution, Radio Science, 43, RS4003 (2008)
[9] Tong, M. S.; Chew, W. C., Super-hyper singularity treatment for solving 3D electrical field integration equations, Microwave Opt. Technol. Lett., 49, 6, 1383-1388 (2007)
[10] Xu, J.; Shann, W., Galerkin-wavelet methods for two-point boundary value problems, Numer. Math. Anal., 24, 246-262 (1992)
[11] Ren, J., Wavelet methods for boundary integral equations, Commun. Numer. Meth. Eng., 13, 373-385 (1997) · Zbl 0878.65100
[12] Chuev, G. N.; Fedorov, M. V., Wavelet algorithm for solving integral equations of molecular liquids. A test for the reference interaction site model, J. Comput. Chem., 25, 1369-1377 (2004)
[13] Tong, M. S., Multiscalet basis in Galerkin’s method for solving three dimensional electromagnetic integral equations, Int. J. Numer. Model., 21, 235-251 (2008) · Zbl 1154.78003
[14] Chen, W.; Lin, W., Galerin trigonometric wavelet methods for the natural boundary integral equations, Appl. Math. Comput., 121, 75-92 (2002)
[15] Shen, Y.; Lin, W., The natural integral equations of plane elasticity problem and its wavelet methods, Appl. Math. Comput., 150, 417-438 (2004) · Zbl 1059.74061
[16] Shan, Z.; Du, Q., Trigonometric wavelet method for some elliptic boundary value problems, J. Math. Anal. Appl., 344, 1105-1119 (2008) · Zbl 1149.65092
[17] Quak, E.; Weyrich, N., Wavelets on the interval, (Singh, S. P., Approximation Theory, Wavelets and Application (1995), Kluwer academic publishers.: Kluwer academic publishers. Dordrecht), 247-283 · Zbl 0922.42019
[18] Micchelli, C. A.; Xu, Y., Using the matrix refinement equation for the construction of wavelets on invariant sets, Appl. Comput. Harmon. Anal., 1, 391-401 (1994) · Zbl 0815.42019
[19] Micchelli, C. A.; Xu, Y., Reconstruction and decomposition algorithms for biorthogonal multiwavelets, Multidimens. Systems Signal Process., 8, 31-69 (1997) · Zbl 0872.42010
[20] Chen, Z.; Micchelli, C. A.; Xu, Y., The Petrov-Galerkin methods for second kind integral equations II: multiwavelet scheme, Adv. Comput. Math., 7, 199-233 (1997) · Zbl 0915.65134
[21] Chen, Z.; Micchelli, C. A.; Xu, Y., Discrete wavelet Petrov-Galerkin methods, Adv. Comput. Math., 16, 1-28 (2002) · Zbl 0998.65120
[22] Shen, Y.; Lin, W., Collocation method for the natural boundary integral equation, Appl. Math. Lett., 19, 1278-1285 (2006) · Zbl 1176.65146
[23] Hsiao, G. C.; Rathsfeld, A., Wavelet collocation methods for a first kind boundary integral equation in acoustic scattering, Adv. Comput. Math., 17, 281-308 (2002) · Zbl 0999.65138
[24] Kaya, A. C.; Erdogan, F., On the solution of integral equations with strongly singular kernels, Quart. Appl. Math., 45, 1, 105-122 (1987) · Zbl 0631.65139
[25] Chan, Y. S.; Fannjiang, A. C.; Paulino, G. H., Integral equations with hypersingular kernels – theory and applications to fracture mechanics, Int. J. Eng. Sci., 41, 683-720 (2003) · Zbl 1211.74184
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