Fully discretized collocation methods for nonlinear singular Volterra integral equations
The authors consider a nonlinear weakly singular Volterra integral equation as \[ F^4(t) = -\frac{1}{2\sqrt{t}}\int\limits^t_0 \frac{F'(s)}{(t^{3/2}-s^{3/2})^{1/3}}ds, ~\, t > 0. \tag{1} \] A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if fully discretized collocation methods based on graded meshes are used. A theoretical proof is given and some numerical results are presented which illustrate the performance of the methods.
nonlinear weakly singular Volterra integral equation
scientific article; zbMATH DE number 6155443