Numerical Solution of Green’s Function for Solving Inhomogeneous Boundary Value Problems with Trigonometric Functions by New Technique

ABSTRACT

A numerical technique is presented for solving integration operator of Green’s function. The approach is based on Hermite trigonometric scaling function on [0,2π], which is constructed for Hermite interpolation. The operational matrices of derivative for trigonometric scaling function are presented and utilized to reduce the solution of the problem. One test problem is presented and errors plots show the efficiency of the proposed technique for the studied problem.

A numerical technique is presented for solving integration operator of Green’s function. The approach is based on Hermite trigonometric scaling function on [0,2π], which is constructed for Hermite interpolation. The operational matrices of derivative for trigonometric scaling function are presented and utilized to reduce the solution of the problem. One test problem is presented and errors plots show the efficiency of the proposed technique for the studied problem.

KEYWORDS

Numerical Technique, Differential Equation, Green’s Function, Hermite Trigonometric Scaling, Wavelet, Error Estimate

Numerical Technique, Differential Equation, Green’s Function, Hermite Trigonometric Scaling, Wavelet, Error Estimate

Cite this paper

Safdari, H. and Aghdam, Y. (2015) Numerical Solution of Green’s Function for Solving Inhomogeneous Boundary Value Problems with Trigonometric Functions by New Technique.*Applied Mathematics*, **6**, 764-772. doi: 10.4236/am.2015.65072.

Safdari, H. and Aghdam, Y. (2015) Numerical Solution of Green’s Function for Solving Inhomogeneous Boundary Value Problems with Trigonometric Functions by New Technique.

References

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http://dx.doi.org/10.1023/A:1018992413504

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http://dx.doi.org/10.1007/BF02571732

[3] Huybrechs, D., Simoens, J. and Vandewalle, S. (2004) A Note on Wave Number Dependence of Wavelet Matrix Compression for Integral Equations with Oscillatory Kernel. Journal of Computational and Applied Mathematics, 172, 233-246.

http://dx.doi.org/10.1016/j.cam.2004.02.006

[4] Huybrechs, D. and Vandewalle, S. (2000) A Two-Dimensional Wavelet Packet Transform for Matrix Compression of Integral Equations with Highly Oscillatory Kernel. Journal of Computational and Applied Mathematics, 197, 218-232.

http://dx.doi.org/10.1016/j.cam.2005.11.001

[5] Von petersdorff, T. and Schwab, C. (1996) Wavelet Approximation for First Kind Integral Equations on Polygons. Numerische Mathematik, 74, 479-516.

http://dx.doi.org/10.1007/s002110050226

[6] Beylkin, G., Coifman, R. and Rokhlin, V. (1991) Fast Wavelet Transforms and Numerical Algorithms. Communications on Pure and Applied Mathematics, 44, 141-183.

http://dx.doi.org/10.1002/cpa.3160440202

[7] Greengard, L. and Rokhlin, V. (1987) A Fast Algorithm for Particle Simulation. Journal of Computational Physics, 73, 325-348.

http://dx.doi.org/10.1016/0021-9991(87)90140-9

[8] Hackbusch, W. and Nowak, Z.P. (1984) On the Fast Matrix Multiplication in the Boundary Element Method by Panel Clustering. Numerische Mathematik, 54, 463-491.

http://dx.doi.org/10.1007/BF01396324

[9] Chui, C.K. and Mhaskar, H.N. (1993) On Trigonometric Wavelets. Constructive Approximation, 9, 167-190.

http://dx.doi.org/10.1007/BF01198002

[10] Prestin, J. (2001) Trigonometric Wavelets. In: Jain, P.K., et al., Eds., Wavelet and Allied Topics, Narosa Publishing House, New Delhi, 183-217.

[11] Themistoclakis, W. (1999) Trigonometric Wavelet Interpolation in Besov Spaces. Facta Universitatis, Series: Mathematics and Informatics, 14, 49-70.

[12] Quak, E. (1996) Trigonometric Wavelets for Hermite Interpolation. Mathematics of Computation, 683-722.

[13] Chen, W.S. and Lin, W. (1997) Hadamard Singular Integral Equations and Its Hermite Wavelet Methods. Proceedings of the 5th International Colloquiumon Finite Dimensional Complex Analysis, Beijing, 13-22.

[14] Chen, W.S. and Lin, W. (2002) Trigonometric Hermite Wavelet and Natural Integral Equations for Stockes Problem. International Conference on Wavelet Analysis and Its Applications, Guangzhou, 73-86.

[15] Lakestani, M. and Saray, B.N. (2010) Numerical Solution of Telegraph Equation Using Interpolating Scaling Functions. Computer & Mathematics with Application, 60, 1964-1972.

http://dx.doi.org/10.1016/j.camwa.2010.07.030

[1] Chen, H. and Peng, S. (1999) A Quasi-Wavelet Algorithm for Second Boundary Integral Equations. Advances in Computational Mathematics, 11, 355-375.

http://dx.doi.org/10.1023/A:1018992413504

[2] Dahmen, W., Prossdorf, S. and Scheider, R. (1994) Wavelet Approximation Methods for Pseudodifferential Equations: I Stability and Convergence. Mathematische Zeitschrift, 215, 583-620.

http://dx.doi.org/10.1007/BF02571732

[3] Huybrechs, D., Simoens, J. and Vandewalle, S. (2004) A Note on Wave Number Dependence of Wavelet Matrix Compression for Integral Equations with Oscillatory Kernel. Journal of Computational and Applied Mathematics, 172, 233-246.

http://dx.doi.org/10.1016/j.cam.2004.02.006

[4] Huybrechs, D. and Vandewalle, S. (2000) A Two-Dimensional Wavelet Packet Transform for Matrix Compression of Integral Equations with Highly Oscillatory Kernel. Journal of Computational and Applied Mathematics, 197, 218-232.

http://dx.doi.org/10.1016/j.cam.2005.11.001

[5] Von petersdorff, T. and Schwab, C. (1996) Wavelet Approximation for First Kind Integral Equations on Polygons. Numerische Mathematik, 74, 479-516.

http://dx.doi.org/10.1007/s002110050226

[6] Beylkin, G., Coifman, R. and Rokhlin, V. (1991) Fast Wavelet Transforms and Numerical Algorithms. Communications on Pure and Applied Mathematics, 44, 141-183.

http://dx.doi.org/10.1002/cpa.3160440202

[7] Greengard, L. and Rokhlin, V. (1987) A Fast Algorithm for Particle Simulation. Journal of Computational Physics, 73, 325-348.

http://dx.doi.org/10.1016/0021-9991(87)90140-9

[8] Hackbusch, W. and Nowak, Z.P. (1984) On the Fast Matrix Multiplication in the Boundary Element Method by Panel Clustering. Numerische Mathematik, 54, 463-491.

http://dx.doi.org/10.1007/BF01396324

[9] Chui, C.K. and Mhaskar, H.N. (1993) On Trigonometric Wavelets. Constructive Approximation, 9, 167-190.

http://dx.doi.org/10.1007/BF01198002

[10] Prestin, J. (2001) Trigonometric Wavelets. In: Jain, P.K., et al., Eds., Wavelet and Allied Topics, Narosa Publishing House, New Delhi, 183-217.

[11] Themistoclakis, W. (1999) Trigonometric Wavelet Interpolation in Besov Spaces. Facta Universitatis, Series: Mathematics and Informatics, 14, 49-70.

[12] Quak, E. (1996) Trigonometric Wavelets for Hermite Interpolation. Mathematics of Computation, 683-722.

[13] Chen, W.S. and Lin, W. (1997) Hadamard Singular Integral Equations and Its Hermite Wavelet Methods. Proceedings of the 5th International Colloquiumon Finite Dimensional Complex Analysis, Beijing, 13-22.

[14] Chen, W.S. and Lin, W. (2002) Trigonometric Hermite Wavelet and Natural Integral Equations for Stockes Problem. International Conference on Wavelet Analysis and Its Applications, Guangzhou, 73-86.

[15] Lakestani, M. and Saray, B.N. (2010) Numerical Solution of Telegraph Equation Using Interpolating Scaling Functions. Computer & Mathematics with Application, 60, 1964-1972.

http://dx.doi.org/10.1016/j.camwa.2010.07.030