Artificial neural network approach to the fuzzy Abel integral equation problem

Abstract

Artificial neural networks (ANNs) have a large appeal to many researchers due to their feature to simulate and solve different kinds of problems that do not have algorithmic solutions. The outlined in this paper is an efficient and robust collocation method based on the ANNs and Bernstein polynomials intended for the fuzzy Abel integral equation problem. To do this, first truncated Bernstein-series polynomial of the solution function is substituted in parametric form of the given fuzzy problem. Then an architecture of ANNs namely the feed-back neural nets is designed to determine values for the unknown coefficients. Eventually, the proposed method is implemented on some numerical examples, and also is compared with an usual and classical technique.

Solving the First Kind Fuzzy Integral Equations Using a Hybrid Regularization Method and Bernstein Polynomials

“…Now we consider the examples of Abel fuzzy integral equations. Operation of hybridal method will be compared with examples in (Jafarian & Measoomy, 2014). Jafarian et al, in (Jafarian & Measoomy, 2014) solved the Abel fuzzy integral equations by multilayer neural networks.…”

“…Finally we achieve to the sufficient approximation of solutions ( u and u ) for For comparing the results of hybrid method with Jafarian et al, method (Jafarian & Measoomy, 2014) we illustrate the solutions in following table: Table 3. The absolute error of numerical results for example 5.3…”

“…Another type of the first kind equations is the Abel integral equation which is presented in Fuzzy form. Jafarian et al (Jafarian & Measoomy, 2014), solved Abel fuzzy integral equations by multi-layers neural network and "back-propagation learning". Here, first the regularization method was used for solving the non-fuzzy integral equation and integral equation of the first kind.…”

“…Example 5.3 consider the following fuzzy Abel integral equation (Jafarian & Measoomy, 2014) Fα (x), f (x) X www.ccsenet.org/mas Modern Applied Science Vol. 10, No.…”