scholarly journals Generalized iterative method for the solution of linear and nonlinear fractional differential equations with composite fractional derivative operator

2020 ◽  
Vol 5 (4) ◽  
pp. 2888-2898 ◽  
Author(s):  
Krunal B. Kachhia ◽  
◽  
Jyotindra C. Prajapati ◽  
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Derivative Operator ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
Linear And Nonlinear

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.

Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function

2021 ◽  
Vol 24 (4) ◽  
pp. 1220-1230
Author(s):  
Mohammed Al-Refai
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Maximum Principles ◽  
Initial Value ◽  
Derivative Operator ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Two Parameters

Abstract In this paper, we formulate and prove two maximum principles to nonlinear fractional differential equations. We consider a fractional derivative operator with Mittag-Leffler function of two parameters in the kernel. These maximum principles are used to establish a pre-norm estimate of solutions, and to derive certain uniqueness and positivity results to related linear and nonlinear fractional initial value problems.

On Hyers–Ulam stability of fractional differential equations with Prabhakar derivatives

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 37-46 ◽  
Author(s):  
Mohammad Hossein Derakhshan ◽  
Alireza Ansari
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Ulam Stability ◽  
Nonlinear Fractional Differential Equations ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
The Stability

AbstractIn this article, we study the Hyers–Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. By using the Laplace transform, we show that the introduced fractional differential equations with the Prabhakar fractional derivative is Hyers–Ulam stable. The results generalize the stability of ordinary and fractional differential equations in the Riemann–Liouville sense.

Systems of Nonlinear Fractional Differential Equations

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Unique Solution ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
The Right ◽  
Linear Fractional Differential Equations

AbstractUsing the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives $D^{q}_{T}x$ and $D^{q}_{T}y$. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.

scholarly journals Lie symmetry analysis for the solution of first-order linear and nonlinear fractional differential equations

2018 ◽  
Vol 7 (2) ◽  
pp. 37 ◽  
Author(s):  
Mousa Ilie ◽  
Jafar Biazar ◽  
Zainab Ayati
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Symmetry ◽  
Order Linear ◽  
Nonlinear Fractional Differential Equations ◽  
First Order ◽  
Fractional Equations ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Fractional Equation

Obtaining analytical or numerical solution of fractional differential equations is one of the troublesome and challenging issues among mathematicians and engineers, specifically in recent years. The purpose of this paper is to solve linear and nonlinear fractional differential equations such as first order linear fractional equation, Bernoulli, and Riccati fractional equations by using Lie Symmetry method, based on conformable fractional derivative. For each equation, some numerical examples are presented to illustrate the proposed approach.  

Monotone iterative method for a class of nonlinear fractional differential equations

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Guotao Wang ◽  
Dumitru Baleanu ◽  
Lihong Zhang
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Fractional Differential Equations ◽  
Monotone Iterative Technique ◽  
Monotone Iterative Method ◽  
Iterative Technique ◽  
Nonlinear Fractional Differential Equations ◽  
Monotone Iterative ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

AbstractBy applying the monotone iterative technique and the method of lower and upper solutions, this paper investigates the existence of extremal solutions for a class of nonlinear fractional differential equations, which involve the Riemann-Liouville fractional derivative D q x(t). A new comparison theorem is also build. At last, an example is given to illustrate our main results.

scholarly journals Monotonicity, Concavity, and Convexity of Fractional Derivative of Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Xian-Feng Zhou ◽  
Song Liu ◽  
Zhixin Zhang ◽  
Wei Jiang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential

The monotonicity of the solutions of a class of nonlinear fractional differential equations is studied first, and the existing results were extended. Then we discuss monotonicity, concavity, and convexity of fractional derivative of some functions and derive corresponding criteria. Several examples are provided to illustrate the applications of our results.

Sign in / Sign up

Export Citation Format

Share Document