The Super-convergence Methods For Solving First Kind Weakly Singular Volterra Integral Equations

It is the purpose of this thesis to study the mechanical quadrature method and itssuper-convergence for solving first kind weakly singular Volterra integral equations. It iswell known that since the equation is ill-posed together with the singularity of the kerneland the non-smoothness of the solution at the initial point, so far there has no maturealgorithm although many works have been done on the equation. This thesis presentsome new algorithms for solving this kind of equations. These algorithms have high ac-curacy, low complexity and require low storage, and at the same time, they can increasefurther accuracy by using extrapolation or combinatory techniques. These algorithmshave a posteriori estimates and are adaptive, i.e., they can examine whether the requiredaccuracy has been attained and adjust accordingly. Above all, these algorithms for thefirst kind Volterra integral equation will reduce the computational complexity, increasethe accuracy, improve the antinoise ability and ease the realization evidently.This thesis discusses the extrapolation and combination algorithms for solving Volterraintegral equation, especially, first kind weakly singular Volterra integral equations, the-oretically and methodologically. Because first kind weakly singular Volterra integralequations are ill-posed problems with singular kernels, and the solution of the equationsare usually non-smooth, even singular, these algorithms are valuable theoretically andpractically. This thesis synthesizes the following methods for solving first kind Volterraintegral equations for the first time.(1) To avoid the ill-posedness of problem, the first kind weakly singular Volterraequation is transformed to the second kind Volterra integral equation with a continuouskernel and a smooth right-hand side term expressed by weakly singular integrals.