Numerical Simulation Using GEM for the Optimization Problem as a System of FDEs

Mohamed Adel,Mohamed M Khader

doi:10.4236/am.2017.812126

Mohamed Adel, Mohamed M Khader

Open Access

https://doi.org/10.4236/am.2017.812126

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Journal: Applied mathematics	Publication Date: Jan 1, 2017
Citations: 1	License type: CC BY 4.0

Affiliation: Cairo University, Benha University

Abstract

In this paper, we introduce a numerical treatment using generalized Euler method (GEM) for the non-linear programming problem which is governed by a system of fractional differential equations (FDEs). The appeared fractional derivatives in these equations are in the Caputo sense. We compare our numerical solutions with those numerical solutions using RK4 method. The obtained numerical results of the optimization problem model show the simplicity and the efficiency of the proposed scheme.

Highlights

Fractional differential equations (FDEs) have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others
We introduce a numerical treatment using generalized Euler method (GEM) for the non-linear programming problem which is governed by a system of fractional differential equations (FDEs)
The penalty function techniques are classical methods for solving non-linear programming (NLP) problem [6]. In this type of methods the optimization problem is formulated as a system of FDEs so that the equilibrium point of this system converges to the local minimum of the optimization problem ([7] [8] [9])

Summary

Introduction

Fractional differential equations (FDEs) have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. Most FDEs do not have exact solutions, so approximate and numerical techniques ([2], [3]), must be used. Optimization theory is aimed to find out the optimal solution of problems which are defined mathematically from a model arise in wide range of scientific and engineering disciplines. The penalty function techniques are classical methods for solving non-linear programming (NLP) problem [6]. In this type of methods the optimization problem is formulated as a system of FDEs so that the equilibrium point of this system converges to the local minimum of the optimization problem ([7] [8] [9])

Mathematical Formulation

Generalized Euler Method

Numerical Simulation

Conclusion