The Regularization For The First Kind Of Fredholm Integtal Equation

Mathematics and computer science have been widely used for complicatedproblems in different fields, such as engineering, physics, and mechanics, etc. Manyscience and engineering problems can be described by solving the first kind ofFredholm integral equations. However, for this kind of integral equations, the moresmooth the integral kernels are, the more unstable the equations are. Thus, how to getstable numerical solutions of this kind of integral equations has become one of the hotissues in recent computational mathematics.First, the related concepts on the first kind of Fredholm integral equations andill-posed problems, and the discretization and regularization theories of ill-posedproblems are introduced. In addition, several kinds of continuous regularizationmethods are also given, for instance, singular value decomposition method, Landweberiteration method, and Fridman iteration method. These theories are proved by using thecollectively compact theory.Second, the interpolation methods by Sidi quadrature formula are used toapproximate the solutions of the integral equations, and two well known projectionmethods are presented, i.e., Galerkin method and the least square method. Nevertheless,the stability of solutions reduced with the increasing of discrete dimensions. Hence, thediscrete-regularization method is prosed for solving the integral equations, that is,discretizing the integral equations and then conducting the regularization of the discretesolutions. The convergence and stability of numerical solutions are proved. Thenumerical examples show the efficiency of our methods.Third, quasi solution method and minimum norm solution method are given, theyare Tikhonov regularizations. From numerical results, we can see that the minimumnorm solution method is more stable than the quasi solution method.Finally,This paper proposes a novel choice of regularization matrix for Tikhonovregularization and exploits Generaliazation Cross Validation method(GCV) to gainregularization parameter. This new method have smaller errors in comparison withtraditional Tikhonov regularization and singular value decomposition method. Numerical results show that our method is effective for ill-posed problems.