scholarly journals Operator Ordering and Solution of Pseudo-Evolutionary Equations

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 35
Author(s):  
Nicolas Behr ◽  
Giuseppe Dattoli ◽  
Ambra Lattanzi
Keyword(s):  
Differential Equations ◽  
Central Importance ◽  
Initial Value ◽  
Evolutionary Equations ◽  
Explicit Formulation ◽  
Operator Ordering ◽  
Promising Tool ◽  
Dyson Series ◽  
Fractional Differential

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.

scholarly journals Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1093
Author(s):  
Daniel Cao Labora
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Initial Value ◽  
Major Question ◽  
Initial Values ◽  
Fractional Differential

One major question in Fractional Calculus is to better understand the role of the initial values in fractional differential equations. In this sense, there is no consensus about what is the reasonable fractional abstraction of the idea of “initial value problem”. This work provides an answer to this question. The techniques that are used involve known results concerning Volterra integral equations, and the spaces of summable fractional differentiability introduced by Samko et al. In a few words, we study the natural consequences in fractional differential equations of the already existing results involving existence and uniqueness for their integral analogues, in terms of the Riemann–Liouville fractional integral. In particular, we show that a fractional differential equation of a certain order with Riemann–Liouville derivatives demands, in principle, less initial values than the ceiling of the order to have a uniquely determined solution, in contrast to a widely extended opinion. We compute explicitly the amount of necessary initial values and the orders of differentiability where these conditions need to be imposed.

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Małgorzata Klimek ◽  
Marek Błasik
Keyword(s):  
Differential Equations ◽  
Arbitrary Order ◽  
Continuous Solution ◽  
Initial Value ◽  
Uniqueness Results ◽  
Banach Theorem ◽  
Arbitrary Partition ◽  
Particular Solution ◽  
Stationary Function ◽  
Fractional Differential

AbstractTwo-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.

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 The Existence and Uniqueness of Solutions and Lyapunov-Type Inequality for CFR Fractional Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Xia Wang ◽  
Run Xu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Type Inequality ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Maximum Value ◽  
Fractional Differential ◽  
Lyapunov Type Inequality

In this paper, we research CFR fractional differential equations with the derivative of order 3<α<4. We prove existence and uniqueness theorems for CFR-type initial value problem. By Green’s function and its corresponding maximum value, we obtain the Lyapunov-type inequality of corresponding equations. As for application, we study the eigenvalue problem in the sense of CFR.

scholarly journals Computer Simulation and Iterative Algorithm for Approximate Solving of Initial Value Problem for Riemann-Liouville Fractional Delay Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 477
Author(s):  
Snezhana Hristova ◽  
Kremena Stefanova ◽  
Angel Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Finite Interval ◽  
Initial Value ◽  
Monotone Iterative ◽  
Practical Usefulness ◽  
Fractional Delay ◽  
Fractional Differential ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated.

scholarly journals Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative

2019 ◽  
Vol 52 (1) ◽  
pp. 437-450 ◽  
Author(s):  
Mouffak Benchohra ◽  
Soufyane Bouriah ◽  
Juan J. Nieto
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
The Stability ◽  
The Banach Contraction Principle ◽  
Implicit Fractional Differential Equations

AbstractIn this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

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.

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