A feedforward neural network based on Legendre polynomial for solving linear Fredholm integro-differential equations

Abstract

This paper presents a method for solving linear Fredholm integro-differential equations using a feedforward neural network based on Legendre polynomials. Firstly, the Legendre polynomials are used to approximate the unknown function and the kernel function in the equations. Secondly, The roots of Legendre polynomials are obtained by using Newton iteration method to obtain Gaussian integration points, and the obtained Gaussian integration points are used as the input nodes of the neural network, and the corresponding weight is learned by the gradient descent method to obtain an approximate solution. Finally, through the numerical value Case analysis to verify the effectiveness of the method.

KEYWORDS:

• Fredholm integro-differential equations
• feedforward neural network
• Legendre polynomials
• Gaussian integration points
• learning algorithm

2000 AMS SUBJECT CLASSIFICATIONS:

• 33C45
• 65D15
• 45J05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11701133].