A Fully Discrete Galerkin Method for Abel-type Integral Equations
This work advances numerical methods for solving Abel-type integral equations, which are important in applied mathematics and physics, but the contribution is incremental as it extends existing Galerkin techniques.
The paper develops a fully discrete Galerkin method for Abel-type integral equations, providing stability and quasi-optimal convergence estimates in fractional-order Sobolev norms, with numerical experiments confirming theoretical sharpness.
In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in ractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation.