Numerical solution of Helmholtz equation by the modified Hopfield finite difference techniques
Mehdi Dehghan, Mojtaba Nourian, Mohammad Bagher Menhaj · Numerical Methods for Partial Differential Equations · 2008
Abstract One property of the Hopfield neural networks is the monotone minimization of energy as time proceeds. In this article, this property is applied to minimize the energy functions obtained by finite difference techniques of the Helmholtz‐equation. The mathematical representation and correlation between finite difference techniques and modified Hopfield neural networks of the Helmholtz equation are presented. Significant advantages of the above method are its parallel, robust, easy programming nature, and ability of direct hardware implementation. Results of numerical simulations are described and analyzed to demonstrate the method. The results obtained using the proposed method show a very good agreement with theoretical and numerical solutions. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009