A New Method For Solving Integral Equations And Its Applications

Inverse problems exist in various fields, in geophysics, digital image processing, life sciences, material sciences, remote sensing technology and many other fields, many of the inverse problem can be come down to the Fredholm integral equation of the first kind. Fixed solution problems of ordinary differential equations and the partial differential equations can be turned into the equivalent integral equations, numerical methods of inverse problem for partial differential equations, often derived Fredholm integral equation of the first kind. Because such problems have extensive and important application background, and the theory has obvious novelty and challenge, thus attracting a lot of domestic and foreign scholars to research on it.Inverse problems are often ill-conditioned, so they pose difficulties to the solution of the problem, if ordinary methods are used, unreasonable answer will be accquired. So various methods are proposed by scholars to solve these problems, such as pulse spectrum method, the best perturbation method, Monte Carlo method, optimized and regularization method. One of the most universal, in theory, the most comprehensive and effective method is regularization method (or strategy), which was put forward by Tikhonov and Phillips respectively in the early 1960s, and subsequently fully develpoed.Based on the Phillips regularization method theoretical framework, the Fredholm integral equations of the first kind derived from inverse problems are studied, the main contents are summarized below:1. A multiple constrained regularization method is presented, the accuracy and stability the solution are improved by adding multiple constraints.2. The application of the multiple constrained regularization method to the identification of structural parameters is discussed, and the selection of regularization parameters is studied. Simulation results show that the method is feasible, and effective to the identification of various structural parameters.3. The idea of multiple constraints is applied to image restoration technology, and the self-adaptive Van Cittert iterative format with multiple constraints is introduced. Simulation results show that the method is effective in the control of noise during restoration, and the ill-conditioned nature of the problem is solved.