Abstract
The method of solving volterra integral equation by using numerical solution is a simple operation but to require many memory space to compute and save the operation. The importance of this equation appeares new direction to solve the equation by using new methods to avoid obstacles. One of these methods employ neural network for obtaining the solution.
 This paper presents a proposed method by using cascade-forward neural network to simulate volterra integral equations solutions. This method depends on training cascade-forward neural network by inputs which represent the mean of volterra integral equations solutions, the target of cascade-forward neural network is to get the desired output of this network. Cascade-forward neural network is trained multi times to obtain the desired output, the training of cascade-forward neural network model terminal when there is no enhancement in result. The model combines all training cascade-forward neural network to obtain the best result. This method proved its successful in training and testing cascade-forward neural network for obtaining the desired output of numerical solution of volterra integral equation for multi intervals. Cascade-forward neural network model measured by calculating MSE to compute the degree of error at each training time.
Highlights
Volterra Integral Equation (VIE) represents as a special state in mathematics science and various methods are introduced to solve it, base of this equation depends on it a model of several science field such as in physics, engineering, biology etc
This paper shows the efficiency of Cascade-Forward Neural Network (CFNN) in obtain the target solution of VIE by measuring the performance of CFNN model to test their work, mean squared error (MSE) used to measure the quality of neural network, the MSE of training CFNN for four training attempts appear in Table (1)
CFNN would have been built to simulate VIE solutions, CFNN represents an appropriate network to solve integral equation because its structures is nonlinear and nonparametric, makes it more flexible to get the predictions of time series
Summary
Volterra Integral Equation (VIE) represents as a special state in mathematics science and various methods are introduced to solve it, base of this equation depends on it a model of several science field such as in physics, engineering, biology etc. VIE solved by multi numerical methods which are based as a linear or nonlinear converting system of integral equation that have direct or iterative methods solutions. The development in parallel processing affect all fields of knowledge, one of them is mathematics science methods, there are multi parallel techniques employed to take the typical solutions, examples of these are: genetic learning systems, artificial neural network, simulated annealing systems, associative memories, and fuzzy learning systems [2]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
