scholarly journals Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Jan Čermák ◽  
Tomáš Kisela ◽  
Luděk Nechvátal
Keyword(s):  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Linear Difference Equation ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Fractional Difference ◽  
Uniqueness Of The Solution ◽  
Discrete Analogues ◽  
Fractional Type ◽  
Special Time

This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus andq-difference calculus. Some of our results are new also in these particular discrete settings.

scholarly journals Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Uniqueness Of Solution ◽  
Fractional Order Differential Equations ◽  
Banach Contraction ◽  
Uniqueness Of The Solution

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations(Dα-ρtDβ)x(t)=f(t,x(t),Dγx(t)),t∈(0,1)with boundary conditionsx(0)=x0,  x(1)=x1or satisfying the initial conditionsx(0)=0,  x′(0)=1, whereDαdenotes Caputo fractional derivative,ρis constant,1<α<2,and0<β+γ≤α. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions onf.

scholarly journals SOME FURTHER EXTENSIONS CONSIDERING DISCRETE PROPORTIONAL FRACTIONAL OPERATORS

Fractals ◽  
2021 ◽  
pp. 2240026
Author(s):  
SAIMA RASHID ◽  
SOBIA SULTANA ◽  
YELIZ KARACA ◽  
AASMA KHALID ◽  
YU-MING CHU
Keyword(s):  
Fractional Calculus ◽  
Existence And Uniqueness ◽  
Complex Dynamics ◽  
Fractional Operator ◽  
Fractional Operators ◽  
Discrete Fractional Calculus ◽  
Dynamic Features ◽  
Fractional Difference ◽  
Fractional Systems ◽  
Special Cases

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete [Formula: see text]-proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete [Formula: see text]-proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla [Formula: see text] fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

The Two-Order Quasi-Linear Singular Perturbed Problems with Infinite Initial Conditions

2013 ◽  
Vol 353-356 ◽  
pp. 3248-3250
Author(s):  
Han Xu
Keyword(s):  
Asymptotic Analysis ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Uniqueness Of The Solution

In this paper a kind of quasi-linear singular perturbed problems with infinite initial conditions is investigated. The existence and uniqueness of the solution are proved.Therefore the asymptotic analysis of the solution can be obtained.

scholarly journals STABILITY BY HOMOGENIZATION OF THERMOVISCOPLASTIC PROBLEMS

2010 ◽  
Vol 20 (09) ◽  
pp. 1591-1616 ◽  
Author(s):  
NICOLAS CHARALAMBAKIS ◽  
FRANCOIS MURAT
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Strain Rate ◽  
Existence And Uniqueness ◽  
Body Force ◽  
Initial Conditions ◽  
Rate Sensitivity ◽  
Effective Coefficients ◽  
Uniqueness Of The Solution ◽  
Theoretical Results

In this paper, we study the homogenization of the system of partial differential equations describing the quasistatic shearing of heterogeneous thermoviscoplastic materials. We first present the existence and uniqueness of the solution of the above system. We then define "stable by homogenization" models as the models where the equations in both the heterogeneous problems and the homogenized one are of the same form. Finally we show that the model with non-oscillating strain-rate sensitivity which is submitted to steady boundary shearing and body force, is stable by homogenization. In this model, the homogenized (effective) coefficients depend on the initial conditions and on the boundary shearing and body force. Those theoretical results are illustrated by one numerical example.

scholarly journals Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations

1996 ◽  
Vol 19 (3) ◽  
pp. 481-494 ◽  
Author(s):  
Pierluigi Colli ◽  
Angelo Favini
Keyword(s):  
Hilbert Space ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Maximal Monotone ◽  
Time Discretization ◽  
Cauchy Problems ◽  
Initial Value ◽  
Selfadjoint Operators ◽  
Convergence Results ◽  
Uniqueness Of The Solution

In this paper we deal with the equationL(d2u/dt2)+B(du/dt)+Au∋f, whereLandAare linear positive selfadjoint operators in a Hilbert spaceHand from a Hilbert spaceV⊂Hto its dual spaceV′, respectively, andBis a maximal monotone operator fromVtoV′. By assuming some coerciveness onL+BandA, we state the existence and uniqueness of the solution for the corresponding initial value problem. An approximation via finite differences in time is provided and convergence results along with error estimates are presented.

scholarly journals On Sumudu Transform Method in Discrete Fractional Calculus

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Fahd Jarad ◽  
Kenan Taş
Keyword(s):  
Fractional Calculus ◽  
Time Scale ◽  
Initial Value Problems ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Fractional Difference ◽  
Transform Method ◽  
Sumudu Transform ◽  
Definition Of ◽  
General Time

In this paper, starting from the definition of the Sumudu transform on a general time scale, we define the generalized discrete Sumudu transform and present some of its basic properties. We obtain the discrete Sumudu transform of Taylor monomials, fractional sums, and fractional differences. We apply this transform to solve some fractional difference initial value problems.

Solvability of coupled FBSDEs with diagonally quadratic generators

2017 ◽  
Vol 17 (06) ◽  
pp. 1750043 ◽  
Author(s):  
Peng Luo ◽  
Ludovic Tangpi
Keyword(s):  
Existence And Uniqueness ◽  
Small Time ◽  
Coupled Systems ◽  
Time Interval ◽  
Well Posedness ◽  
Initial Value ◽  
Small Time Interval ◽  
Uniqueness Of The Solution ◽  
Backward Sdes

We study the well-posedness for multi-dimensional and coupled systems of forward–backward SDEs when the generator can be separated into a quadratic and a subquadratic part. We obtain the existence and uniqueness of the solution on a small time interval. Moreover, the continuity and differentiability with respect to the initial value are presented.

scholarly journals On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

2021 ◽  
Vol 6 (10) ◽  
pp. 10920-10946
Author(s):  
Saima Rashid ◽  
◽  
Fahd Jarad ◽  
Khadijah M. Abualnaja ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fractional Derivative ◽  
Integrodifferential Equation ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Equivalent Form ◽  
Initial Value ◽  
Uniqueness Of The Solution ◽  
Generalized Lipschitz Condition ◽  
Fuzzy Framework

<abstract><p>This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional ($ \mathcal{GPF} $) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-$ \mathcal{GPF} $ operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations $ (\mathcal{FVFIE}s) $ via generalized fuzzy Hilfer-$ \mathcal{GPF} $ Hukuhara differentiability ($ \mathcal{HD} $) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy $ \mathcal{FVFIE}s $ which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.</p></abstract>

scholarly journals Exponential decay of solutions with \(L^{p}\)-norm for a class to semilinear wave equation with damping and source terms

2020 ◽  
Vol 4 (2) ◽  
pp. 123-131
Author(s):  
Amar Ouaoua ◽  
◽  
Messaoud Maouni ◽  
Aya Khaldi ◽  
◽  
...  
Keyword(s):  
Wave Equation ◽  
Lyapunov Function ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Local Solution ◽  
Initial Value ◽  
Local Existence And Uniqueness ◽  
Decay Of Solutions ◽  
Uniqueness Of The Solution ◽  
Damping And Source Terms

In this paper, we consider an initial value problem related to a class of hyperbolic equation in a bounded domain is studied. We prove local existence and uniqueness of the solution by using the Faedo-Galerkin method and that the local solution is global in time. We also prove that the solutions with some conditions exponentially decay. The key tool in the proof is an idea of Haraux and Zuazua with is based on the construction of a suitable Lyapunov function.

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