scholarly journals Solving Fractional Differential Equations by Using Triangle Neural Network

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Feng Gao ◽  
Yumin Dong ◽  
Chunmei Chi
Keyword(s):  
Neural Network ◽  
Numerical Methods ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Gradient Descent ◽  
Quadratic Optimization ◽  
Quadratic Optimization Problem ◽  
The Neural Network ◽  
Fractional Differential

In this paper, numerical methods for solving fractional differential equations by using a triangle neural network are proposed. The fractional derivative is considered Caputo type. The fractional derivative of the triangle neural network is analyzed first. Then, based on the technique of minimizing the loss function of the neural network, the proposed numerical methods reduce the fractional differential equation into a gradient descent problem or the quadratic optimization problem. By using the gradient descent process or the quadratic optimization process, the numerical solution to the FDEs can be obtained. The efficiency and accuracy of the presented methods are shown by some numerical examples. Numerical tests show that this approach is easy to implement and accurate when applied to many types of FDEs.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

scholarly journals Solving Linear and Nonlinear Fractional Differential Equations Using Spline Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Adel Al-Rabtah ◽  
Shaher Momani ◽  
Mohamed A. Ramadan
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Spline Functions ◽  
Exact Analytical Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Good Agreement

Suitable spline functions of polynomial form are derived and used to solve linear and nonlinear fractional differential equations. The proposed method is applicable for0<α≤1andα≥1, whereαdenotes the order of the fractional derivative in the Caputo sense. The results obtained are in good agreement with the exact analytical solutions and the numerical results presented elsewhere. Results also show that the technique introduced here is robust and easy to apply.

On the Application of Fractional Derivatives to the Study of Memristor Dynamics

2020 ◽  
pp. 342-363
Author(s):  
Rawid Banchuin
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Phase Portraits ◽  
Fractional Differential

In this chapter, the authors report their work on the application of fractional derivative to the study of the memristor dynamic where the effects of the parasitic fractional elements of the memristor have been studied. The fractional differential equations of the memristor and the memristor-based circuits under the effects of the parasitic fractional elements have been formulated and solved both analytically and numerically. Such effects of the parasitic fractional elements have been studied via the simulations based on the obtained solutions where many interesting results have been proposed in the work. For example, it has been found that the parasitic fractional elements cause both charge and flux decay of the memristor and the impasse point breaking of the phase portraits between flux and charge of the memristor-based circuits similarly to the conventional parasitic elements. The effects of the order and the nonlinearity of the parasitic fractional elements have also been reported.

scholarly journals A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications

2019 ◽  
Vol 22 (1) ◽  
pp. 27-59 ◽  
Author(s):  
HongGuang Sun ◽  
Ailian Chang ◽  
Yong Zhang ◽  
Wen Chen
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Relevant Literature ◽  
Physical Models ◽  
Variable Order ◽  
Mathematical Tool ◽  
Fractional Differential ◽  
Mathematical Foundations ◽  
Selection Of

Abstract Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

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