Novel determination of differential-equation solutions: universal approximation method

• Published in: Journal of computational and applied mathematics Vol. 146; no. 2; pp. 443 - 457
• Main Author: Leephakpreeda, Thananchai
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.09.2002; Elsevier
• Subjects: Applied sciences; Artificial intelligence; Computer science; control theory; systems; Connectionism. Neural networks; Exact sciences and technology; Finite difference; Finite element; Fuzzy linguistic model; Mathematics; Neural network model; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use; Solving differential equations; Universal approximation; Finite difference; Neural network model; Fuzzy linguistic model; Universal approximation; Solving differential equations; Finite element; Finite element method; Numerical computation; Error estimation; Differential equation; Optimization method; Equation resolution; Continuous function; Neural network; Boundary condition; Linguistic model; Finite difference method
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/S0377-0427(02)00397-7

In a conventional approach to numerical computation, finite difference and finite element methods are usually implemented to determine the solution of a set of differential equations (DEs). This paper presents a novel approach to solve DEs by applying the universal approximation method through an artificial intelligence utility in a simple way. In this proposed method, neural network model (NNM) and fuzzy linguistic model (FLM) are applied as universal approximators for any nonlinear continuous functions. With this outstanding capability, the solutions of DEs can be approximated by the appropriate NNM or FLM within an arbitrary accuracy. The adjustable parameters of such NNM and FLM are determined by implementing the optimization algorithm. This systematic search yields sub-optimal adjustable parameters of NNM and FLM with the satisfactory conditions and with the minimum residual errors of the governing equations subject to the constraints of boundary conditions of DEs. The simulation results are investigated for the viability of efficiently determining the solutions of the ordinary and partial nonlinear DEs.