Multiple positive solutions of a singular fractional boundary value problem.
Multiplicity results for a class of fractional boundary value problems
We prove the existence of at least three solutions to the following fractional boundary value problem: ⎧ , a.e. t ∈ [0, T], ⎨ ⎩ u (0) = u (T) = 0, where and are the left and right Riemann-Liouville fractional integrals of order 0 ≤ σ < 1 respectively. The approach is based on a recent three critical points theorem of Ricceri [B. Ricceri, A further refinement of a three critical points theorem, Nonlinear Anal. 74 (2011), 7446-7454].
New class of boundary value problem for nonlinear fractional differential equations involving Erdélyi-Kober derivative
In this paper, we introduce a new class of boundary value problem for nonlinear fractional differential equations involving the Erdélyi-Kober differential operator on an infinite interval. Existence and uniqueness results for a positive solution of the given problem are obtained by using the Banach contraction principle, the Leray-Schauder nonlinear alternative, and Guo-Krasnosel'skii fixed point theorem in a special Banach space. To that end, some examples are presented to illustrate the usefulness...
New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions.
New existence results for nonlinear fractional differential equations with three-point integral boundary conditions.
Nonlinear delay integrodifferential systems with Caputo fractional derivative in infinite-dimensional spaces
This paper is mainly concerned with existence of mild solutions and optimal controls for nonlinear delay integrodifferential systems with Caputo fractional derivative in infinite-dimensional spaces. We do not assume that the relevant strongly continuous semigroup is compact.
Nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in Banach spaces
We consider a nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in a Banach space. The existence of at least one solution is proved by using the set-valued analog of Mönch fixed point theorem associated with the technique of measures of noncompactness.
Nonlinear fractional differential inclusions with anti-periodic type integral boundary conditions
This article studies a boundary value problem of nonlinear fractional differential inclusions with anti-periodic type integral boundary conditions. Some existence results are obtained via fixed point theorems. The cases of convex-valued and nonconvex-valued right hand sides are considered. Several new results appear as a special case of the results of this paper.
Nonlinear Implicit Hadamard’s Fractional Differential Equationswith Delay in Banach Space
In this paper, we establish sufficient conditions for the existence of solutions for nonlinear Hadamard-type implicit fractional differential equations with finite delay. The proof of the main results is based on the measure of noncompactness and the Darbo’s and Mönch’s fixed point theorems. An example is included to show the applicability of our results.
Nonlinear Time-Fractional Differential Equations in Combustion Science
MSC 2010: 34A08 (main), 34G20, 80A25The application of Fractional Calculus in combustion science to model the evolution in time of the radius of an isolated premixed flame ball is highlighted. Literature equations for premixed flame ball radius are rederived by a new method that strongly simplifies previous ones. These equations are nonlinear time-fractional differential equations of order 1/2 with a Gaussian underlying diffusion process. Extending the analysis to self-similar anomalous diffusion...
Nonlocal boundary value problem for impulsive differential equations of fractional order.
Nontrivial solutions for fractional -difference boundary value problems.
Note on a discretization of a linear fractional differential equation
The paper discusses basics of calculus of backward fractional differences and sums. We state their definitions, basic properties and consider a special two-term linear fractional difference equation. We construct a family of functions to obtain its solution.
On a nonlinear fractional order differential inclusion.
On a nonlocal problem for fractional integrodifferential inclusions in Banach spaces
This paper investigates a class of fractional functional integrodifferential inclusions with nonlocal conditions in Banach spaces. The existence of mild solutions of these inclusions is determined under mixed continuity and Carathéodory conditions by using strongly continuous operator semigroups and Bohnenblust-Karlin's fixed point theorem.
On asymptotics of discrete Mittag-Leffler function
The (modified) two-parametric Mittag-Leffler function plays an essential role in solving the so-called fractional differential equations. Its asymptotics is known (at least for a subset of its domain and special choices of the parameters). The aim of the paper is to introduce a discrete analogue of this function as a solution of a certain two-term linear fractional difference equation (involving both the Riemann-Liouville as well as the Caputo fractional -difference operators) and describe its...
On contraction principle applied to nonlinear fractional differential equations with derivatives of order α ∈ (0,1)
One-term and multi-term fractional differential equations with a basic derivative of order α ∈ (0,1) are solved. The existence and uniqueness of the solution is proved by using the fixed point theorem and the equivalent norms designed for a given value of parameters and function space. The explicit form of the solution obeying the set of initial conditions is given.
On efficient method for system of fractional differential equations.
On Euler methods for Caputo fractional differential equations
Numerical methods for fractional differential equations have specific properties with respect to the ones for ordinary differential equations. The paper discusses Euler methods for Caputo differential equation initial value problem. The common properties of the methods are stated and demonstrated by several numerical experiments. Python codes are available to researchers for numerical simulations.
On solutions of some fractional -point boundary value problems at resonance.