On Bernstein Polynomials Method to the System of Abel Integral Equations

This paper deals with a new implementation of the Bernstein polynomials method to the numerical solution of a special kind of singular system. For this aim, first the truncated Bernstein series polynomials of the solution functions are substituted in the given problem. Using some properties of these polynomials, the solution of the problem is reduced to solve a linear system of algebraic equations. In order to confirm the reliability and accuracy of the proposed method, some weakly Abel integral equations systems with comparisons are solved in detail as numerical examples

Utilizing a new feed-back fuzzy neural network for solving a system of fuzzy equations

Abstract This paper intends to offer a new iterative method based on artificial neural networks for finding solution of a fuzzy equations system. Our proposed fuzzified neural network is a five-layer feedback neural network that corresponding connection weights to output layer are fuzzy numbers. This architecture of artificial neural networks, can get a real input vector and calculates its corresponding fuzzy output. In order to find the approximate solution of the fuzzy system that supposedly has a real solution, first a cost function is defined for the level sets of the fuzzy network and target output. Then a learning algorithm based on the gradient descent method is used to adjust the crisp input signals. The present method is illustrated by several examples with computer simulations

A Numerical Scheme to Solve Fuzzy Linear Volterra Integral Equations System

The current research attempts to offer a new method for solving fuzzy linear Volterra integral equations system. This method converts the given fuzzy system into a linear system in crisp case by using the Taylor expansion method. Now the solution of this system yields the unknown Taylor coefficients of the solution functions. The proposed method is illustrated by an example and also results are compared with the exact solution by using computer simulations

An application of artificial neural networks for solving fractional higher-order linear integro-differential equations

Abstract This ongoing work is vehemently dedicated to the investigation of a class of ordinary linear Volterra type integro-differential equations with fractional order in numerical mode. By replacing the unknown function by an appropriate multilayered feed-forward type neural structure, the fractional problem of such initial value is changed into a course of non-linear minimization equations, to some extent. Put differently, interest was sparked in structuring an optimized iterative first-order algorithm to estimate solutions for the origin fractional problem. On top of that, some computer simulation models exemplify the preciseness and well-functioning of the indicated iterative technique. The outstanding accomplished numerical outcomes conveniently reflect the productivity and competency of artificial neural network methods compared to customary approaches