Numerical Solution Of Volterra Integral Equations Based On Maximum Entropy Method

Integral equations are widely used in many fields,such as physics,population problems,fluid mechanics and so on.However,because the analytical solution of the integral equations is usually difficult to find,it is particularly important to design algorithm to find the numerical solution of the integral equations.This work aims to study the numerical methods for solving the second type of linear Volterra integral equations and the second type of linear Fredholm integral equations.For the second type of linear Volterra integral equations,based on the principle of maxi-mum entropy and piecewise linear basis functions,a numerical method for approximating the non-negative solution of Volterra integral equations is proposed.The convergence of the new method is proved and the convergence rate in~1-norm and~?-norm are evaluated.Numeri-cal experiments show that this method can solve the Volterra integral equations effectively and the convergence rate is confirmed.So far,most of the numerical methods can only solve Fredholm integral equations of dimension not more than 3.However,in the fields of engineering and so on,solving high-dimensional integral equations is often encountered.In this work,the neural network method is used to compute the numerical solution of the high dimensional Fredholm integral equations.Numerical experiments show that the neural network is an effective method for solving high-dimensional problems.