Study On Numerical Method For Fredholm Integral Equations Of The Sconed Kind

With the development of science and technology,many scholars exhibit a s-trong interest in mathematical and physical problems.The basic theory of the mathematical model of physical problems is the calculus.Differential and integral equations are always used to model the practical physics.Fredholm integral equations of the second kind is an important part of integral equations,which plays a very important role in solving mathematical and physical problems.In this thesis,two methods for the numerical solutions of the second kind Fredholm integral equations will be given.The main novelties are summarized as follows:(1)A new numerical method for a nonlinear Fredholm integral equation of the second kind is developed.Applying the ideas of the Taylor-series expansion and piecewise approximation,a discretization format for the nonlinear integral equation of Hammerstein type is made.By introducing two parameters,the convergence and error estimate of the approximation solution are given.The special case of linear Fredholm integral equation of the second kind is further analyzed.Some numerical results are carried out to show the effectiveness of the proposed method and compare with some existing ones.(2)An optimization method for numerically solving Fredholm integral equations of the second kind is proposed.Numerical integration is an important topic in scientific computing and engineering applications.In the typical methods for numerically solving integrations,the quadrature points should be always given previously.In the present study,it is considered that the quadrature points are to be determined.An optimization problem is constructed by minimizing the error estimate between the exact and the approximate solutions.The quadrature points are obtained by solving the constructed optimization according to the algorithm of particle swarm optimization(PSO).It is found that some typical formulae for solving numerical integrations can be retrieved.The proposed method is further extended to numerically solve Fredholm integral equations of the second kind.Numerical results are carried out to show the advantages of the proposed method by comparing with the existing ones.The observation shows that the approximate solution using un-equidistant points could be more accurate than that using equidistant points.The novel idea for solving numerical integrations and integral equations is effective and it can be achieved through a feasible approach.The above studies enriched the methods for numerical solutions Fredholm integral equation of the second kind.The propoed methods have great significance for solving mathematical physical problems.