scholarly journals A Numerical Method for Solving Fractional Differential Equations by Using Neural Network

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Haidong Qu ◽  
Xuan Liu
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Initial Value ◽  
The Neural Networks ◽  
Fractional Differential

We present a new method for solving the fractional differential equations of initial value problems by using neural networks which are constructed from cosine basis functions with adjustable parameters. By training the neural networks repeatedly the numerical solutions for the fractional differential equations were obtained. Moreover, the technique is still applicable for the coupled differential equations of fractional order. The computer graphics and numerical solutions show that the proposed method is very effective.

scholarly journals Numerical Simulations for Solving Fuzzy Fractional Differential Equations by Max-Min Improved Euler Methods

2015 ◽  
Vol 7 (1) ◽  
pp. 53-83 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Oyoon Abdul Razzaq ◽  
Fatima Riaz
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Value ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Euler Methods ◽  
Fuzzy Fractional Differential Equations

Abstract In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial value problems of fuzzy fractional differential equations (FFDEs). Two modified fractional Euler type methods have been proposed and investigated to obtain numerical solutions of linear and nonlinear FFDEs. The proposed algorithms are tested on various illustrative examples. Exact values are also simulated to compare and discuss the closeness and accuracy of approximations so obtained. Comparatively, tables and graphs results reveal the complete reliability, efficiency and accuracy of the proposed methods.

Cosine Radial Basis Function Neural Networks for Solving Fractional Differential Equations

2017 ◽  
Vol 9 (3) ◽  
pp. 667-679 ◽  
Author(s):  
Haidong Qu
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Radial Basis Function ◽  
Fractional Differential Equations ◽  
Basis Function ◽  
Numerical Solutions ◽  
Langevin Equations ◽  
Initial Value ◽  
Radial Basis ◽  
Fractional Differential

AbstractIn this paper, we first apply cosine radial basis function neural networks to solve the fractional differential equations with initial value problems or boundary value problems. In the examples, we successfully obtained the numerical solutions for the fractional Riccati equations and fractional Langevin equations. The computer graphics and numerical solutions show that this method is very effective.

On the initial value problems for the Caputo–Fabrizio impulsive fractional differential equations

2020 ◽  
pp. 2150073
Author(s):  
Mohamed I. Abbas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Initial Value ◽  
Impulsive Fractional Differential Equations ◽  
Uniqueness Results ◽  
Fractional Differential

This paper is devoted to initial value problems for impulsive fractional differential equations of Caputo–Fabrizio type fractional derivative. By means of Banach’s fixed point theorem and Schaefer’s fixed point theorem, the existence and uniqueness results are obtained. Finally, an example is given to illustrate one of the main results.

scholarly journals A Spectral Deferred Correction Method for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jia Xin ◽  
Jianfei Huang ◽  
Weijia Zhao ◽  
Jiang Zhu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Caputo Derivative ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Correction Method ◽  
Initial Value ◽  
Deferred Correction ◽  
Fractional Differential ◽  
Spectral Deferred Correction

A spectral deferred correction method is presented for the initial value problems of fractional differential equations (FDEs) with Caputo derivative. This method is constructed based on the residual function and the error equation deduced from Volterra integral equations equivalent to the FDEs. The proposed method allows that one can use a relatively few nodes to obtain the high accuracy numerical solutions of FDEs without the penalty of a huge computational cost due to the nonlocality of Caputo derivative. Finally, preliminary numerical experiments are given to verify the efficiency and accuracy of this method.

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