Solution of Singular Integral Equations of the First Kind with Cauchy Kernel

Year 2019, Volume: 2 Issue: 1, 69 - 74, 22.03.2019

Subhabrata Mondal B.n. Mandal

https://doi.org/10.33434/cams.454740

Cited By: 5

Abstract

In this paper an analytic method is developed for solving Cauchy type singular integral equations of the first kind, over a finite interval. Chebyshev polynomials of the first kind, $T_n(x)$, second kind, $U_n(x)$, third kind, $V_n(x)$, and fourth kind, $W_n(x)$, corresponding to respective weight functions $W^{(1)}(x)=\frac{1}{\sqrt{1-x^2}},W^{(2)}(x)=\sqrt{1-x^2},W^{(3)}(x)=\sqrt{\frac{1+x}{1-x}},$ and $~ W^{(3)}(x)=\sqrt{\frac{1-x}{1+x}}, $ have been used to obtain the complete analytical solutions for four different cases.

Keywords

Singular integral equation, Cauchy Kernel, Weight function, Chebyshev polynomials, Weight function

References

• [1] N. I. Mushkelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953.
• [2] F.D. Gakhov, Boundary Value Problems, Addison-Wesley, 1966.
• [3] P. A. Martin, F. S. Rizzo, On boundary integral equations for crack problems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 421 (1989), 341-345.
• [4] S. Kim, Solving singular integral equations using Gaussian quadrature and overdetermined system, Appl. Math. Comput., 35 (1998), 63-71.
• [5] A. Chakrabarti, V. G. Berghe, Approximate solution of singular integral equations, Appl. Math. Lett., 17 (2004), 553-559.
• [6] M. M. Panja, B. N. Mandal, Solution of second kind integral equation with Cauchy type kernel using Daubechies scale function, J. Comput. Appl. Math., 241 (2013), 130-142.
• [7] M. Abdulkawi, Solution of Cauchy type singular integral equations of first kind by using differential transform method, Appl. Math. Model., 39 (2015), 2107-2118.
• [8] S. Mondal, B. N. Mandal, A note on the solution of a simple hypersingular integral equation, Glob. J. Pure Appl. Math., 13 (2017), 1959-1964.
• [9] J.C. Mason, Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms, J. Comput. Appl. Math., 49 (1993), 169-178.
• [10] Z. K. Eshkuvatov, N. M. A. Nik Long, M. Abdulkawi, Approximate solution of singular integral equations of the first kind with Cauchy kernel, Appl. Math. Lett., 22 (2009), 651-657.