# Approximate solution of a system of singular integral equations of the first kind by using Chebyshev polynomials

Document Type : Research Article

Authors

2 University of Tabriz

Abstract

The aim of the present work is to introduce a method based on the Chebyshev polynomials for numerical solution of a system of Cauchy type singular integral equations of the first kind on a finite segment. Moreover, an estimation error is computed for the approximate solution. Numerical resultsdemonstrate the effectiveness of the proposed method.

Keywords

#### References

1. Abdolkawi, M., Solution of Cauchy type singular integral equations of the first kind by using differential transform method, Appl. Math, Model. 39 (2015), 2107–2118.
2. Ahdiaghdam, S., Shahmorad S. and Ivaz, K., Approximate solution of dual integral equations using Chebyshev polynomials, Int. J. Comput. Math. 94(3) (2017), 493–502.
3. Chakrabarti, A. and Berghe, G.V., Approximate solution of singular integral equations, Appl. Math. Lett., 17 (2004), 533–559.
4. Erdogan, F., Gupta, G.D. and Cook, T.S., Mechanics of fracture, Noord hoff International Publishing, Leyden, Netherlands, 1973.
5. Eshkuvatov, Z.K., Nik Long, N.M.A. and Abdulkawi, M., Approximatesolution of singular integral equations of the first kind with Cauchy kernel, Appl. Math. Lett., 22 (2009), 651–657.
6. Karczmarek, P., Pylak, D. and Sheshko, M.A., Application of Jacobi polynomials to approximate solution of a singular integral equation with Cauchy kernel, Appl. Math. Comput., 181 (2006), 694–707.
7. Kashfi, M. and Shahmorad, S., Approximate solution of a singular integral Cauchy-kernel equation of the first kind, Comput. Methods Appl. Math. 10(4) (2010), 345–353.
8. Liu, D., Zhang, X. and Wu, J., A collocation scheme for a certain Cauchy singular integral equation based on the superconvergence analysis, Appl. Math. Comput. 219 (2013), 5198–5209.
9. Mandal, B.N. and Chakrabarti, A., Applied singular integral equations, Taylor and Francis Group, CRC Press, 2011.
10. Mason, J.C. and Handscomb, D.C., Chebyshev polynomials, Chapman and Hall, CRC Press, 2003.
11. Miel, G., On the Galerkin and collocation methods for a Cauchy singular integral equation, SIAM J. Numer. Anal. 23(1) (1986), 135–143.
12. Muskhelishvili, N.I. and RADOK, J.R.M., Singular integral equations, Wolters-Noordhoff Publishing Groningen, Netherlands, 1958.
13. Setia, A., Numerical solution of various cases of Cauchy type singular integral equation, Appl. Math. Comput. 230 (2014), 200–207.
14. Sharma, V., Setia, A. and Agarwal, R., Numerical solution for system of Cauchy type singular integral equations with its error analysis in complex plane, Appl. Math. Comput., 328 (2018), 338–352.
15. Tsalamengas, J.L., A direct method to quadrature rules for a certain class of singular integrals with logarithmic, Cauchy, or Hadamard-type singularities, Int. J. Numer. Model. 25 (2012), 512–524.
16. Turhan, I., Oguz, H. and Yusufoglu, E., Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind, Int. J. Comput. Math. 90(5) (2013), 944–954.