On the convergence problem of one-dimensional hypersingular integral equations (Q474346)

Summary: We develop the expansion method of singular integral equation (SIE) for hypersingular integral equation (HSIE). Relating the hypersingular integrals to Cauchy principal-value integrals, we interpolate the kernel and the density functions to the truncated Chebyshev series of the second kind. The corresponding convergence results for the functions \(f\in C^l([-1,1])\) and \(K(t,x)\in C^l([-1,1]\times [-1,1])\), \(l\geq 1\) are derived in an appropriate \(L_2[-1,1]\) norm to the true solution of the weight function. Numerical examples are also presented to validate the theoretical findings.