Solving Three Kinds Of Fractional Order Integro-Differential Equation By Euler Basis Neural Network

In this thesis,a numerical method based on Euler basis function neural network is presented for solving one-dimensional Fredholm-Volterra type integral-differential equation,one-dimensional complete fractional integral-differential equation and two-dimensional Fredholm type integral-partial differential equation.By means of modular segmentation and modular operation,the application range of the algorithm can be extended to solve fractional differential equations,fractional integral equations,fractional Fredholm type integral-partial differential equations,fractional Volterra type integral-partial differential equations,covering most linear fractional calculus equations.The first chapter mainly describes the research significance and background of fractional calculus equation,and explains and proves the preparatory knowledge and theorems used in the following paper.In Chapter 2,one-dimensional Fredholm-Volterra type integral differential equations are studied.By constructing calculus operator and discretization equation,the original equations are transformed into linear equations.After matrix processing,Euler basis function neural network is used to solve the equations iteratively.In addition,due to the modular segmentation in the pretreatment stage of the equation,the modules involved in the operation can be set according to the actual requirements in the operation stage,so that the algorithm is also suitable for solving fractional Fredholm type integral differential equation and fractional Volterra type integral differential equation.Finally,a numerical example is given to verify the feasibility and effectiveness of the proposed algorithm.In Chapter 3,one-dimensional complete fractional integral-differential equations are studied.The equation is composed of a differential module defined by Caputo fractional derivative and an integral module defined by Riemann-Liouville.The two modules construct operators respectively and then matrix them.Then loss function and Euler basis function neural network are constructed based on operator matrix,and momentum gradient descent method is used to solve the equation iteratively.Numerical experiment results show that the Euler basis function neural network algorithm has good numerical results in calculating such equations.In addition,the algorithm is also suitable for fractional differential equations and fractional integral equations by setting the modules involved in the operation.In Chapter 4,two dimensional fractional Fredholm type integral-partial differential equations are studied.Firstly,by using the idea of dimensionality reduction,the calculus operator in two-dimensional problems is transformed from tensor form to matrix form,so that the algorithm applicable to one-dimensional problems is extended to two-dimensional problems.Then,the original equations are transformed into algebraic equations by finite element discretization and configuration method,and the matrix processing is carried out.On this basis,an Euler neural network is constructed and the momentum gradient descent method is used to reduce the loss function value to obtain the numerical solution of the equation.Finally,a numerical example is given to demonstrate the good numerical effect of the proposed algorithm.In Chapter 5,the methods given in Chapter 2,3 and 4 are summarized and prospected.