The Second Category Of Nonlinear Iterative Galerkin Fredholm Integral Equation Method

With the rapid development of mathematics and computer science, the scholars ofcomputational mathematics, applied mathematics, computer science, physics andengineering are more and more attention to the integral equation. Many problems onengineering, mechanics, physics and some other projects can be transformed tosolve for the integral equation. Now, the study of linear integral equation has tended tomature. And engineering problems become more complex. The nonlinear integralequation can be a better description of practical problems in practical application.Therefore, the study of the nonlinear integral equation has become popular. The highprecision numerical algorithms of the nonlinear integral equation become a focus ofcomputational science. It has high value in mathematics, physics and engineeringcalculations. I mainly discussed the nature, theory and high precision numerical solutionof the nonlinear Fredholm integral equation in this paper.First, we introduce some commonly used numerical solution of nonlinearequations, such as Newton’s method and the iterative Newton method. We introduce theadvantages and disadvantages of these algorithms and adaptation range.Secondly, we study several commonly method for solving Fredholm integralequation, such as projection algorithm for solving the integral equation. We introducethe collocation method in the projection method, and study the theoretical basisand general framework of the collocation method, and point out that the convergenceand error of collocation method. We give several numerical examples by collocationmethod, and get the corresponding results. Thus we obtain the feasibility and merits ofcollocation method.Finally, we study the nonlinear integral equation of the Galerkin method and theiterative Galerkin method. We select several sets of standardized orthogonal Legendrebasis functions as an orthogonal basis functions which is needed in the Galerkin methodto discrete the nonlinear Fredholm integral equation. Thus the original nonlinearintegral equations are discrete into a set of nonlinear algebraic equations. Nonlinearintegral equations can be converted into nonlinear equations to be solved. In addition, we also analyze the nonlinear Fredholm integral equation of the iterative nature of theGalerkin method in this paper. And we give several numerical examples by Galerkinmethod and the iterative Galerkin method. And we verify the accuracy of the theory ofthe Galerkin method by calculating the numerical example.