Approximate Solution ofnth-Order Fuzzy Linear Differential Equations

Xiaobin Guo,Dequan Shang

doi:10.1155/2013/406240

Abstract

The approximate solution ofnth-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.

Highlights

Fuzzy differential equations (FDEs), which are utilized for the purpose of the modeling problems in science and engineering, have been studied by many researchers
Considering that the case of the order of the fuzzy differential equation and the number of basic functions in assumed solution are not always equal, we obtain the approximate solution of the original equations (1) and (2) by calculating the minimal norm least squares solution of crisp system of linear equations
In order to solve (1) and (2), we need to consider the system of linear equations (8)

Summary

Introduction

Fuzzy differential equations (FDEs), which are utilized for the purpose of the modeling problems in science and engineering, have been studied by many researchers. For an nth-order linear differential equation y(n) + an−1 (t) y(n−1) + ⋅ ⋅ ⋅ + a1 (t) y󸀠 + a0 (t) y = g (t) (1) with fuzzy initial conditions y (t0) = ̃b0, y󸀠 (t0) = ̃b1, . The nth-order linear differential equation with fuzzy initial conditions is further investigated. It shows that the result obtained in this paper is an extension of Allahviranloo’s conclusions. Considering that the case of the order of the fuzzy differential equation and the number of basic functions in assumed solution are not always equal, we obtain the approximate solution of the original equations (1) and (2) by calculating the minimal norm least squares solution of crisp system of linear equations.

Fuzzy Number

Basic Results

Numerical Examples

Conclusion