Polynomial spline collocation methods for the nonlinear Basset equation (Q1263270)

The Basset equation is a weakly singular nonlinear Volterra integro- differential equation, which models the motion of a small particle in a turbulent fluid: \[ y'(t)=f(t,y(t))+\int^{t}_{0}(t-s)^{- \alpha}k(t,s,y'(s))ds,\quad t\in [0,T]. \] Due to the structure of the kernel smooth data f leads to a solution y which is not smooth at \(t=0\). Therefore standard algorithms like product integration methods exhibit poor convergence rates. The authors prove convergence results for spline approximations on nonuniform grids. While quasi-uniform grids don't give the desired convergence rates, using graded grids results in high order approximations. The grading exponent is chosen according to the smoothness of f and k. Numerical tests confirm the predicted convergence rates.