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K-State Today

March 10, 2015

Mathematics Colloquium Lecture March 10

By David Pedergnana

Yuri Antipov, Louisiana State University, will present "Singular Integral Equations in a Segment with Two Fixed Singularities and Applications" at 2:30 p.m. Tuesday, March 10, in 122 Cardwell Hall.

Abstract: A singular integral equation in a segment whose kernel is a sum of the Cauchy kernel and a function with fixed singularities at the ending points is analyzed. The class of solutions is the set of H"older functions bounded at the ends. The equation reduces to a vector Hilbert problem for a half-disc and then to a vector Riemann-Hilbert problem on the real axis with a piece-wise constant matrix coefficient. A condition of solvability and a closed-form solution are derived. For the Chebyshev polynomials of the first kind in the right-hand side the solution of the integral equation is expressed in terms of some polynomials. Based on this spectral relation an approximate solution to the complete singular integral equation with two fixed singularities is obtained. Applications to discontinuous boundary value problems for the harmonic and biharmonic operators arising in fracture mechanics and plate theory are discussed.

Sponsored by a Simons Grant.