EXISTENCE AND CONTINUATION THEOREMS OF RIEMANN–LIOUVILLE TYPE FRACTIONAL DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250077 ◽  
Author(s):  
CHUNHAI KOU ◽  
HUACHENG ZHOU ◽  
CHANGPIN LI
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Global Existence ◽  
Local Existence ◽  
Existence Results ◽  
Liouville Type ◽  
Local Existence Theorem ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Theoretical Results

In this paper we study the existence and continuation of solution to the general fractional differential equation (FDE) with Riemann–Liouville derivative. If no confusion appears, we call FDE for brevity. We firstly establish a new local existence theorem. Then, we derive the continuation theorems for the general FDE, which can be regarded as a generalization of the continuation theorems of the ordinary differential equation (ODE). Such continuation theorems for FDE which are first obtained are different from those for the classical ODE. With the help of continuation theorems derived in this paper, several global existence results for FDE are constructed. Some illustrative examples are also given to verify the theoretical results.

scholarly journals Application of topological degree method in quantitative behavior of fractional differential equations

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 421-432
Author(s):  
Rahman ur ◽  
Saeed Ahmad ◽  
Fazal Haq
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Existence Results ◽  
Research Article ◽  
Fractional Differential ◽  
Topological Degree Method

In the present manuscript we incorporate fractional order Caputo derivative to study a class of non-integer order differential equation. For existence and uniqueness of solution some results from fixed point theory is on our disposal. The method used for exploring these existence results is topological degree method and some auxiliary conditions are developed for stability analysis. For further elaboration an illustrative example is provided in the last part of the research article.

Lie symmetries analysis and conservation laws for the fractional Calogero–Degasperis–Ibragimov–Shabat equation

2018 ◽  
Vol 15 (07) ◽  
pp. 1850110 ◽  
Author(s):  
S. Sahoo ◽  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Operator ◽  
Conservation Laws ◽  
Symmetry Analysis ◽  
Lie Symmetries ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Conservation Theorem

The present paper includes the study of symmetry analysis and conservation laws of the time-fractional Calogero–Degasperis–Ibragimov–Shabat (CDIS) equation. The Erdélyi–Kober fractional differential operator has been used here for reduction of time fractional CDIS equation into fractional ordinary differential equation. Also, the new conservation theorem has been used for the analysis of the conservation laws. Furthermore, the new conserved vectors have been constructed for time fractional CDIS equation by means of the new conservation theorem with formal Lagrangian.

Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient

2020 ◽  
Vol 27 (4) ◽  
pp. 593-603 ◽  
Author(s):  
Kemal Özen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Adjoint Problem ◽  
Linear Ordinary Differential Equation ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
Order Linear ◽  
Theoretical Results

AbstractIn this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

Resonance and non-resonance in terms of average values. II

2001 ◽  
Vol 131 (5) ◽  
pp. 1217-1235
Author(s):  
M. N. Nkashama ◽  
S. B. Robinson
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Term ◽  
Spectral Gap ◽  
Elliptic Boundary ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Elliptic Boundary Value ◽  
Semilinear Elliptic ◽  
The Given

We prove existence results for semilinear elliptic boundary-value problems in both the resonance and non-resonance cases. What sets our results apart is that we impose sufficient conditions for solvability in terms of the (asymptotic) average values of the nonlinearities, thus allowing the nonlinear term to have significant oscillations outside the given spectral gap as long as it remains within the interval on the average in some sense. This work generalizes the results of a previous paper, which dealt exclusively with the ordinary differential equation (ODE) case and relied on ODE techniques.

Solitary Wave and other Solutions of Nonlinear Space-Time Fractional Differential Equation Systems

2018 ◽  
pp. 515-522
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Expansion Method ◽  
Travelling Wave ◽  
Boussinesq System ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Fractional System ◽  
Partial Fractional Differential Equation

In this study, we have successfully found some travelling wave solutions of the variant Boussinesq system and fractional system of two-dimensional Burgers' equations of fractional order by using the -expansion method. These exact solutions contain hyperbolic, trigonometric and rational function solutions. The fractional complex transform is generally used to convert a partial fractional differential equation (FDEs) with modified Riemann-Liouville derivative into ordinary differential equation. We showed that the considered transform and method are very reliable, efficient and powerful in solving wide classes of other nonlinear fractional order equations and systems.

scholarly journals Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Yanning Wang ◽  
Jianwen Zhou ◽  
Yongkun Li
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Differential Equation ◽  
Time Scales ◽  
Fractional Derivatives ◽  
Point Theory ◽  
Existence Results ◽  
Recent Approach ◽  
Equation Boundary ◽  
Fractional Differential

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap-Laplacian conformable fractional differential equation boundary value problem on time scaleT:  Tα(Tαup-2Tα(u))(t)=∇F(σ(t),u(σ(t))),Δ-a.e.  t∈a,bTκ2,u(a)-u(b)=0,Tα(u)(a)-Tα(u)(b)=0,whereTα(u)(t)denotes the conformable fractional derivative ofuof orderαatt,σis the forward jump operator,a,b∈T,  0<a<b,  p>1, andF:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.

Existence results for some impulsive partial functional fractional differential equations

2019 ◽  
Vol 13 (04) ◽  
pp. 2050074 ◽  
Author(s):  
Hadda Hammouche ◽  
Mouna Lemkeddem ◽  
Kaddour Guerbati ◽  
Khalil Ezzinbi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Differential Equations ◽  
Mild Solutions ◽  
Existence Results ◽  
Completely Continuous ◽  
Fractional Differential ◽  
Functional Differential ◽  
Fractional Functional

In this paper, we study the existence of mild solutions of impulsive evolution fractional functional differential equation of order [Formula: see text] involving a Lipschitz condition on term [Formula: see text]. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators due to Burton and Kirk.

Nonlocal Differential Equations with Convolution Coefficients and Applications to Fractional Calculus

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christopher S. Goodrich
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Differential Equations ◽  
Fractional Integral ◽  
Fixed Point Theory ◽  
Conjugate Problem ◽  
Point Theory ◽  
Liouville Type ◽  
Model Case ◽  
Fractional Differential

Abstract The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is - A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) ⁢ u ′′ ⁢ ( t ) = λ ⁢ f ⁢ ( t , u ⁢ ( t ) ) , t ∈ ( 0 , 1 ) , q ≥ 1 , -A((b*u^{q})(1))u^{\prime\prime}(t)=\lambda f(t,u(t)),\quad t\in(0,1),\,q\geq 1, is considered. Due to the coefficient A ⁢ ( ( b * u q ) ⁢ ( 1 ) ) {A((b*u^{q})(1))} appearing in the differential equation, the equation has a coefficient containing a convolution term. By choosing the kernel b in various ways, specific nonlocal coefficients can be recovered such as nonlocal coefficients equivalent to a fractional integral of Riemann–Liouville type. The results rely on the use of a nonstandard order cone together with topological fixed point theory. Applications to fractional differential equations are given, including a problem related to the ( n - 1 , 1 ) {(n-1,1)} -conjugate problem.

scholarly journals Oscillation criteria for nonlinear fractional differential equation with damping term

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 119-128 ◽  
Author(s):  
Mustafa Bayram ◽  
Hakan Adiguzel ◽  
Aydin Secer
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Riccati Transformation ◽  
Damping Term ◽  
Nonlinear Fractional Differential Equation ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Oscillation Of Solutions

AbstractIn this paper, we study the oscillation of solutions to a non-linear fractional differential equation with damping term. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By using a variable transformation, a generalized Riccati transformation, inequalities, and integration average techniquewe establish new oscillation criteria for the fractional differential equation. Several illustrative examples are also given.

scholarly journals Studies on BVPs for IFDEs involved with the Riemann-Liouville type fractional derivatives

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Differential Equation ◽  
Boundary Value ◽  
Resonance Case ◽  
Liouville Type ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
At Resonance ◽  
Fractional Differential

AbstractIn this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory. Examples are given to illustrate main results. This paper is motivated by [Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance, Electron. J. Qual. Theory Differ. Equ. 89(2011), 1-19], [Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance, J, Appl, Math, Comput. 39(2012) 421-443] and [Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Probl. 2013, 2013: 80].

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