Solution Method Of Hermite Neural Network For Fredholm-Volterra Integro-Differential Equations

The existence of analytical solutions of integro-differential equations and the exploration of efficient numerical solutions have always been hot topics.In recent years,with the continuous development of neural network models,the basis function neural network has been widely used to solve the numerical solutions of integro-differential equations because of its better approximation ability and network structure.Hermite neural network is a kind of basis function neural network.Its recursion relation of orthogonal basis function is simple,and the independent variable has no scope limit.In this paper,Hermite neural network is used to solve integer order and fractional order FredholmVolterra integro-differential equation by combining the properties of integro-differential equation and the advantages of Hermite orthogonal basis function.The first chapter introduces the partial numerical methods for solving integral and fractional order integro-differential equations,summarizes the research status of the neural network for solving integral and fractional order integro-differential equations at domestic and overseas.In the second chapter,the definition and related properties of fractional order integro-differential equation and the definition of Hermite orthogonal polynomials are simply explained,and the structure and algorithm steps of Hermite neural network model are elaborated in detail.In the third chapter,Hermite orthogonal polynomials are taken as the excitation function of hidden layer neurons,and a 1 × m × 1 Hermite function link neural network is established,which is applied to solve a class of nonlinear FredholmVolterra integro-differential equations.The Hermite neural network solution formats of the different activation functions constructed are derived,and the convergence of the algorithm is proved theoretically.Numerical examples are used to compare the numerical approximation effects of the Hermite neural network solution to the nonlinear Fredholm-Volterra integro-differential equation when the activation function is different.In the fourth chapter,Hermite neural network method is used to solve the fractional Fredholm-Volterra integro-differential equation,and the corresponding numerical solution format is given.Gradient descent method is used to adjust the network weight,and the convergence of the algorithm is analyzed theoretically.Through numerical solution and comparison with the existing algorithms,verify the rationality of Hermite neural network and the effectiveness of algorithm.The fifth chapter summarizes the work of this paper and the prospect of the later research work.