scholarly journals APPLICATIONS OF BI-FRAMELET SYSTEMS FOR SOLVING FRACTIONAL ORDER DIFFERENTIAL EQUATIONS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040051 ◽  
Author(s):  
MUTAZ MOHAMMAD ◽  
CARLO CATTANI
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sparse Matrices ◽  
Computational Method ◽  
Extension Principle ◽  
Vanishing Moments ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique extension principle. We use this system for solving some types of FDEs by solving a series of important examples of FDEs related to many mathematical applications. The quasi-affine bi-framelet-based methods for numerical FDEs show the advantages of using sparse matrices and its accuracy in numerical analysis.

scholarly journals Solutions for Impulsive Fractional Differential Equations via Variational Methods

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Peiluan Li ◽  
Hui Wang ◽  
Zheqing Li
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Critical Points ◽  
Existence Results ◽  
Fractional Order Differential Equations ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin and Willem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.

scholarly journals Non-instantaneous impulses in Caputo fractional differential equations

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Real World ◽  
Existence Results ◽  
Advantages And Disadvantages ◽  
Fractional Order Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

AbstractRecent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of the solutions. Both approaches are compared and their advantages and disadvantages are illustrated with examples. Also some existence results are derived.

scholarly journals Regarding on the exact solutions for the nonlinear fractional differential equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 478-482 ◽  
Author(s):  
Melike Kaplan ◽  
Murat Koparan ◽  
Ahmet Bekir
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Simple Equation ◽  
Wave Transformation ◽  
Nonlinear Ordinary Differential Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

AbstractIn this work, we have considered the modified simple equation (MSE) method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW) and the modified equal width (mEW) equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

scholarly journals NUMERICAL SOLUTION FOR TIME-FRACTIONAL MURRAY REACTION-DIFFUSION EQUATIONS VIA REDUCED DIFFERENTIAL TRANSFORM METHOD

2021 ◽  
Author(s):  
Muhammed Yiğider ◽  
Serkan Okur
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Differential Transform Method ◽  
Reaction Diffusion Equations ◽  
Transform Method ◽  
Differential Transform ◽  
Fractional Differential ◽  
Reduced Differential Transform Method ◽  
Time Fractional Differential Equations

In this study, solutions of time-fractional differential equations that emerge from science and engineering have been investigated by employing reduced differential transform method. Initially, the definition of the derivatives with fractional order and their important features are given. Afterwards, by employing the Caputo derivative, reduced differential transform method has been introduced. Finally, the numerical solutions of the fractional order Murray equation have been obtained by utilizing reduced differential transform method and results have been compared through graphs and tables. Keywords: Time-fractional differential equations, Reduced differential transform methods, Murray equations, Caputo fractional derivative.

scholarly journals On Stability of Nonautonomous Perturbed Semilinear Fractional Differential Systems of Order α∈(1,2)

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammed M. Matar ◽  
Esmail S. Abu Skhail
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Protected Area ◽  
Marine Protected Area ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential ◽  
Stability Results

We study the Mittag-Leffler and class-K function stability of fractional differential equations with order α∈(1,2). We also investigate the comparison between two systems with Caputo and Riemann-Liouville derivatives. Two examples related to fractional-order Hopfield neural networks with constant external inputs and a marine protected area model are introduced to illustrate the applicability of stability results.

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