Solving Mixed Volterra - Fredholm Integral Equation (MVFIE) by Designing Neural Network

Al-Saif Et Al

doi:10.21123/bsj.2019.16.1.0116

Abstract

In this paper, we focus on designing feed forward neural network (FFNN) for solving Mixed Volterra – Fredholm Integral Equations (MVFIEs) of second kind in 2–dimensions. in our method, we present a multi – layers model consisting of a hidden layer which has five hidden units (neurons) and one linear output unit. Transfer function (Log – sigmoid) and training algorithm (Levenberg – Marquardt) are used as a sigmoid activation of each unit. A comparison between the results of numerical experiment and the analytic solution of some examples has been carried out in order to justify the efficiency and the accuracy of our method.

Highlights

Integral equation is one of the main branches of modern mathematics that appear in various applied areas including mechanics, physics and engineering ...etc
By using Hybrid Legendre Functions, Nemati et al introduced a numerical method for the (MVFIEs)(2)
Shahooth presented a Numerical Solution for Mixed (MVFIEs) of the second kind using Bernstein Polynomial Method (3), this method is used to obtain the system of algebraic equations from the integral equation

Summary

Introduction

Integral equation is one of the main branches of modern mathematics that appear in various applied areas including mechanics, physics and engineering ...etc. Levenberg–Marquardt Algorithm( LMA) (7) "The Levenberg-Marquardt algorithm is a variation of Newton’s method that designed for minimizing functions that are sums of squares of other nonlinear functions This is very well suited to neural network training where the performance index is the mean squared error. Where I , μ and J are identity unit matrix, the learing parameter and jacobian of m out- put error of the neural network with respect to n weights, respectively. (6) In our approach, the trial solution ut(x, y, p) employs a fast feed neural network (FFNN)and the parameters p correspond to the weight and biases of the neural architecture , we choose form for the trial function ut(x, y) ut(xi, yi, p) = G(x, y, N(x, y, p)) ... Table (4) gives the weight, bias, Epoch, time and performance of the designer network

Weight and Bias

Conclusion