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Solving Finite Part Singular Integral Equations Using Orthogonal Polynomials

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Abstract

In this paper, we use the Gauss–Jacobi quadrature formula to get an approximate solution for a finite part singular integral equation. The zeros of Jacobi polynomials and Jacobi functions of the second kind are used to construct a square system of algebraic equations which yield an interpolating function for the regular part of the solution of the singular integral equation. In a special case, another approach consists in converting the finite part singular integral equation to a Fredholm integral equation of the second kind using the analytical solution of a simple finite part singular integral equation. To simplify the calculation of kernel and right-hand side of the obtained Fredholm equation, we use the new polynomials \(J_n(x)\), which admit a recurrence relation. The equivalent form of the integral equation in the last case is used to investigate the error bound of the approximated solution.

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Authors and Affiliations

  1. Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran

    Samad Ahdiaghdam

  2. Department of Applied Mathematics, University of Tabriz, Tabriz, Iran

    Sedaghat Shahmorad

Authors
  1. Samad Ahdiaghdam
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  2. Sedaghat Shahmorad
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Correspondence to Samad Ahdiaghdam.

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Communicated by Mahmoud Hadizadeh.

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Ahdiaghdam, S., Shahmorad, S. Solving Finite Part Singular Integral Equations Using Orthogonal Polynomials. Bull. Iran. Math. Soc. 46, 799–814 (2020). https://doi.org/10.1007/s41980-019-00293-5

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  • DOI: https://doi.org/10.1007/s41980-019-00293-5

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