A new class of orthogonal polynomials for solving logarithmic singular integral equations

Abstract

In this note, we propose a new class of orthogonal polynomials (named Bachok–Hasham polynomials H1nk(x)), which is an extension of the Chebyshev polynomials. Eigenfunctions and corresponding eigenvalues are found for the homogeneous second kind of Logarithmic Singular Integral Equations (LogSIEs). For non-homogeneous LogSIEs truncated series of the first kind Bachok–Hasham polynomials are used to find approximate solution. It is found that first kind of Bachok–Hasham polynomials (H1nk(x)) are orthogonal with weight wk(x)=xk-11-x2k, where k is positive odd integer. Properties of first kind of Bachok–Hasham polynomials are also proved. Finally, two examples are presented to show the validity and accuracy of the proposed method.

Keywords

• Chebyshev polynomials
• New orthogonal basis functions
• Singular integral