The Tikhonov Regularization Method For FREDHOLM Equations Of The First Kind With Applications

The Fredholm equation of the first kind is widely encountered in the fields ofstructural mechanics, image processing, geological prospecting etc. Due to the ill-posedness, it is both theoretical and practical importance to develop effective methodsto solve the problem. To date, the Tikhonov regularization method is very effective insolving ill-posed problems. The basic idea is, based upon certain a priori informationon the exact solution, to transform the original problem into some feasible well-posedoptimization problems to get the regularization solutions. Then one can obtain a stablesolution method for the original problem.In this thesis, such a method is applied to solve the Fredholm equation of thefirst kind, where many kinds of existing strategies for choosing regularization param-eters and different scales of partitions for discretizing the integral equation are consid-ered, to show extensively and thoroughly the numerical performance of the method.Furthermore, to illustrate the computational performance for functions with differentsmoothness, the method is used to solve problems which have differentiable, contin-uous but not differentiable, and discontinuous solutions, respectively. The method issuccessfully applied to solve two Fredholm equations of the first kind arising fromgeological prospecting.