# Visiting Lecturer Program (273)

### Published at: 2017-05-20

Speaker: Dr. Sonia Seyed Allaei

Postdoc
Department of Mathematics (CUMUC)
Coimbra University, Coimbra, Portugal

Title: Nonlinear singular VIEs; Analytical and numerical study and its possible applications for colon cancer modelling

We consider a class of nonlinear second-kind Volterra integral equations, whose kernels have an Abel-type singularity, which arise in applications such as: boundary value problems of heat transfer between solids and gases, theory of superfluidity and the temperature distribution on the surface of a projectile moving through a laminar layer (Lighthill’s equation). A unique continuous (global) solution is shown to exist on $[0, \infty)$. The typical nonsmooth properties of the exact solutions to these equations cause a drop in the global convergence orders of numerical methods with uniform meshes. We introduce a numerical method where an initial integral over a small interval is calculated analytically, by using a convergent series solution, and optimal convergence orders are attained. The application of the Jacobi (spectral) collocation method is also investigated, after a transformation of variables on the original equation allowing a new integral equation with smoother solution, to which the method is applied. Error bounds in the $L_\infty$ and $L_2$ norms are obtained and numerical examples illustrate the exponential convergence order of the method.