scholarly journals Optimal Homotopy Asymptotic Method-Least Square for Solving Nonlinear Fractional-Order Gradient-Based Dynamic System from an Optimization Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Oluwaseun Olumide Okundalaye ◽  
Wan Ainun Mior Othman ◽  
Nallasamy Kumaresan
Keyword(s):  
Dynamic System ◽  
Fractional Order ◽  
Asymptotic Method ◽  
Optimization Problems ◽  
Least Square ◽  
Convergence Control ◽  
Gradient Based ◽  
Fractional Differential ◽  
Optimal Homotopy Asymptotic Method ◽  
Convergence Control Parameter

In this paper, we consider an approximate analytical method of optimal homotopy asymptotic method-least square (OHAM-LS) to obtain a solution of nonlinear fractional-order gradient-based dynamic system (FOGBDS) generated from nonlinear programming (NLP) optimization problems. The problem is formulated in a class of nonlinear fractional differential equations, (FDEs) and the solutions of the equations, modelled with a conformable fractional derivative (CFD) of the steepest descent approach, are considered to find the minimizing point of the problem. The formulation extends the integer solution of optimization problems to an arbitrary-order solution. We exhibit that OHAM-LS enables us to determine the convergence domain of the series solution obtained by initiating convergence-control parameter Cj′s. Three illustrative examples were included to show the effectiveness and importance of the proposed techniques.

Mathematical study of fractional-order biological population model using optimal homotopy asymptotic method

2016 ◽  
Vol 09 (06) ◽  
pp. 1650081 ◽  
Author(s):  
S. Sarwar ◽  
M. A. Zahid ◽  
S. Iqbal
Keyword(s):  
Fractional Order ◽  
Asymptotic Method ◽  
Population Model ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Population Models ◽  
Mathematical Study ◽  
Third Order ◽  
Biological Population ◽  
Optimal Homotopy Asymptotic Method

In this paper, we study the fractional-order biological population models (FBPMs) with Malthusian, Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method (OHAM) for partial differential equations (PDEs) is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional-order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.

scholarly journals On the Application of Homotopy Analysis Method to Fractional Differential Equations

2021 ◽  
pp. 1-18
Author(s):  
Khalid Suliman Aboodh ◽  
Abu baker Ahmed
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
Fractional Differential ◽  
Convergence Control Parameter

In this paper, an attempt has been made to obtain the solution of linear and nonlinear fractional differential equations by applying an analytic technique, namely the homotopy analysis method (HAM). The fractional derivatives are described by Caputo’s sense. By this method, the solution considered as the sum of an infinite series, which converges rapidly to exact solution with the help of the nonzero convergence control parameter ℏ. Some examples are given to show the efficiently and accurate of this method. The solutions obtained by this method has been compared with exact solution. Also our graphical represented of the solutions have been given by using MATLAB software.

scholarly journals Optimum solutions of space fractional order diffusion equation

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 329-339
Author(s):  
Sajjad Ali ◽  
Kamal Shah ◽  
Yongjin Li ◽  
Muhammad Arif
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Science And Engineering ◽  
Approximate Analytical Solutions ◽  
Special Cases ◽  
Fractional Differential ◽  
Optimal Homotopy Asymptotic Method ◽  
Excellent Accuracy

The present paper is concerned with the implementation of optimal homotopy asymptotic method to handle the approximate analytical solutions of fractional order partial differential equations. Fractional differential equations have great importance regarding distinct fields of science and engineering. Approximate solutions of space fractional order diffusion model and its various special cases are handled using the innovative proposed method. The space fractional derivatives are described in the Caputo sense. The results obtained by the proposed method are compared with various methods. The proposed method demonstrates excellent accuracy and strength over various methods.

scholarly journals Dynamical Study of Fokker-Planck Equations by Using Optimal Homotopy Asymptotic Method

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 264 ◽  
Author(s):  
H. Younas ◽  
Muhammad Mustahsan ◽  
Tareq Manzoor ◽  
Nadeem Salamat ◽  
S. Iqbal
Keyword(s):  
Convergence Rate ◽  
Exact Solutions ◽  
Fractional Order ◽  
Asymptotic Method ◽  
Dynamical Study ◽  
Optimal Homotopy Asymptotic Method ◽  
Relative Errors ◽  
Fokker Planck Equations ◽  
Fokker Planck

In this article, Optimal Homotopy Asymptotic Method (OHAM) is used to approximate results of time-fractional order Fokker-Planck equations. In this work, 3rd order results obtained through OHAM are compared with the exact solutions. It was observed that results from OHAM have better convergence rate for time-fractional order Fokker-Planck equations. The solutions are plotted and the relative errors are tabulated.

scholarly journals Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM

2016 ◽  
Vol 6 (2) ◽  
pp. 75-83 ◽  
Author(s):  
Firat Evirgen
Keyword(s):  
Nonlinear Programming ◽  
Fractional Order ◽  
Optimization Problems ◽  
Variational Iteration Method ◽  
Optimal Solutions ◽  
Order Model ◽  
Fractional Order Differential Equations ◽  
Runge Kutta Method ◽  
Fractional Order Model ◽  
Gradient Based

In this paper, a class of Nonlinear Programming problem is modeled with gradient based system of fractional order differential equations in Caputo's sense. To see the overlap between the equilibrium point of the fractional order dynamic system and theoptimal solution of the NLP problem in a longer timespan the Multistage Variational ?teration Method isapplied. The comparisons among the multistage variational iteration method, the variationaliteration method and the fourth order Runge-Kutta method in fractional and integer order showthat fractional order model and techniques can be seen as an effective and reliable tool for finding optimal solutions of Nonlinear Programming problems.

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