Toward solving fractional differential equations via solving ordinary differential equations

2022 ◽  
Vol 41 (1) ◽  
Author(s):  
Ahmed F. Abdel Jalil ◽  
Ayad R. Khudair
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Differential

scholarly journals On the solution of two-sided fractional ordinary differential equations of Caputo type

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Ma. Elena Hernández-Hernández ◽  
Vassili N. Kolokoltsov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Well Posedness ◽  
Solutions To Equations ◽  
Fractional Differential ◽  
The Right ◽  
Stochastic Representations ◽  
Exit Problems ◽  
Special Case

AbstractThis paper provides well-posedness results and stochastic representations for the solutions to equations involving both the right- and the left-sided generalized operators of Caputo type. As a special case, these results show the interplay between two-sided fractional differential equations and two-sided exit problems for certain Lévy processes.

scholarly journals Solution of Linear Fuzzy Fractional Differential Equations Using Fuzzy Natural Transform

2021 ◽  
pp. 41-65
Author(s):  
Hameeda Oda Al-Humedi ◽  
Shaimaa Abdul-Hussein Kadhim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Wide Range ◽  
Real Functions ◽  
Fractional Models ◽  
Fractional Differential ◽  
Fuzzy Fractional Differential Equations

The purpose of this paper is to apply the fuzzy natural transform (FNT) for solving linear fuzzy fractional ordinary differential equations (FFODEs) involving fuzzy Caputo’s H-difference with Mittag-Leffler laws. It is followed by proposing new results on the property of FNT for fuzzy Caputo’s H-difference. An algorithm was then applied to find the solutions of linear FFODEs as fuzzy real functions. More specifically, we first obtained four forms of solutions when the FFODEs is of order α∈(0,1], then eight systems of solutions when the FFODEs is of order α∈(1,2] and finally, all of these solutions are plotted using MATLAB. In fact, the proposed approach is an effective and practical to solve a wide range of fractional models.

scholarly journals General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative

2017 ◽  
Vol 6 (2) ◽  
pp. 49 ◽  
Author(s):  
Zainab Ayati ◽  
Jafar Biaar ◽  
Mousa Ilei
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Riccati Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Conformable Fractional Derivative ◽  
Numerical Examples ◽  
Fractional Differential

This paper is aimed to develop two well-known nonlinear ordinary differential equations, Bernoulli and Riccati equations to fractional form. General solution to fractional differential equations are detected, based on conformable fractional derivative. For each equation, numerical examples are presented to illustrate the proposed approach.  

scholarly journals On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Haci Mehmet Baskonus ◽  
Hasan Bulut
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Two Dimensional ◽  
Fractional Differential ◽  
Linear And Nonlinear

AbstractIn this paper, we apply the Fractional Adams-Bashforth-Moulton Method for obtaining the numerical solutions of some linear and nonlinear fractional ordinary differential equations. Then, we construct a table including numerical results for both fractional differential equations. Then, we draw two dimensional surfaces of numerical solutions and analytical solutions by considering the suitable values of parameters. Finally, we use the L

Initialization of Fractional-Order Operators and Fractional Differential Equations

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Laplace Transforms ◽  
Time Varying ◽  
Fractional Operators ◽  
Fractional Differential ◽  
Derivatives Of

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. The new transforms unify the initialization of systems of fractional and ordinary differential equations. The paper provides background on past work in the area and determines the Laplace transforms for the initialized fractional integral and fractional derivatives of any (real) order. An application provides insight and demonstrates the theory.

scholarly journals Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner ◽  
Adem C. Cevikel
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Fractional Complex Transform ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Fractional Differential

The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.

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.

scholarly journals Multivalue Collocation Methods for Ordinary and Fractional Differential Equations

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 185
Author(s):  
Angelamaria Cardone ◽  
Dajana Conte ◽  
Raffaele D’Ambrosio ◽  
Beatrice Paternoster
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Experiments ◽  
Collocation Methods ◽  
Spline Collocation ◽  
Fractional Differential ◽  
Convergence And Stability Analysis

The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods.

scholarly journals ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

2016 ◽  
Vol 21 (5) ◽  
pp. 610-629 ◽  
Author(s):  
Mohammed D. Kassim ◽  
Khaled M. Furati ◽  
Nasser-Eddine Tatar
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Higher Order ◽  
Asymptotic Behavior Of Solutions ◽  
Power Type ◽  
Fractional Damping ◽  
Integer Order ◽  
Fractional Differential

It is known that, under certain conditions, solutions of some ordinary differential equations of first, second or even higher order are asymptotic to polynomials as time goes to infinity. We generalize and extend some of the existing results to differential equations of non-integer order. Reasonable conditions and appropriate underlying spaces are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach power type functions as time goes to infinity. The case of fractional differential problems with fractional damping is also considered. Our results are obtained by using generalized versions of GronwallBellman inequality and appropriate desingularization techniques.

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