Fractional differential equations with arbitrary singularities

Journal of Mathematical Analysis and Applications, 2007

In this paper, sufficient conditions for the existence and uniqueness of solution are studied for a class of initial value problem of fractional order, involving the Caputo-type derivative of a hypergeometric fractional operator applying fixed point theory. Examples are also provided to illustrate the results.

Analysis in Theory and Applications, 2017

In this article, we study on the existence of solution for a singularities of a system of nonlinear fractional differential equations (FDE). We construct a formal power series solution for our considering FDE and prove convergence of formal solutions under conditions. We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function.

Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2021

In this study, some classes of Riemann-Liouville fractional differential equations with right-hand side functions having a singularity with respect to their first variable and with a nonhomogeneous initial condition are considered. First, it is briefly stated that under which conditions the existence of a local continuous solution of this initial value problem occurs. Later, uniqueness theorems were developed in types of Krasnosel’skii-Krein, Kooi, Roger and Banaś-Rivero, respectively. These theorems improve the previously obtained results, and for their proofs pre-existing techniques are enriched by the tools of Lebesgue spaces.

Filomat, 2017

The aim of this work is to study a class of boundary value problem including a fractional order differential equation. Sufficient and necessary conditions will be presented for the existence and uniqueness of solution of this fractional boundary value problem.

Fractional Calculus and Applied Analysis

We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order of growth of $\alpha$-entire solutions is given. An analog of the Frobenius method for systems with regular singularity is developed. For a model example of an equation with a kind of an irregular singularity, a series for a formal solution is shown to be convergent for $t>0$ (if $\alpha$ is an irrational number poorly approximated by rational ones) but divergent in the distribution sense.

The aim of this paper is to develop a monotone iterative technique by introducing upper and lower solutions to Riemann-Liouville fractional differential equations with deviating arguments and integral boundary conditions. As an application of this technique, existence and uniqueness results are obtained.

Axioms, 2020

A coupled system of singular fractional differential equations involving Riemann–Liouville integral and Caputo derivative is considered in this paper. The question of existence and uniqueness of solutions is studied using Banach contraction principle. Furthermore, the question of existence of at least one solution is discussed. At the end, an illustrative example is given in details.

The author (B. Math. Anal. App. 6(4)(2014):1-15), introduced a new fractional derivative, \[{}^\rho \mathcal{D}_a^\alpha f (x) = \frac{\rho^{\alpha-n+1}}{\Gamma({n-\alpha})} \, \bigg(x^{1-\rho} \,\frac{d}{dx}\bigg)^n \int^x_a \frac{\tau^{\rho-1} f(\tau)}{(x^\rho - \tau^\rho)^{\alpha-n+1}}\, d\tau %\big({}^\rho \mathcal{D}^\alpha_{a+}f\big)(x) = \] which generalizes two familiar fractional derivatives, namely, the Riemann-Liouville and the Hadamard fractional derivatives to a single form. In this paper, we derive the existence and uniqueness results for a generalized fractional differential equation governed by the fractional derivative in question.

2011

Abstract: In this article, 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 conditions x(0) = x0, x(1) = x1 or satisfying the initial conditions x(0) = 0, x′(0) = 1 where D denotes Caputo fractional derivative, ρ is constant, 1 < α < 2 and 0 < β + γ ≤ 1. Schaurder’s fixed point Theorem is the main tool used here to establish the existence. We use Banach contraction principle to show the uniqueness of the solution under certain conditions on f .

Applied Mathematics and Computation, 2013

In this paper, by using the fibering map and the Nehari manifold, we prove the existence and multiple results of solutions for the following fractional differential equation: t D α T (0 D α t u) = λh(t)|u| p-2 u + b(t)|u| q-2 u, t ∈ [0, T], u(0) = u(T) = 0, where α ∈ (1 2 , 1), 0 < p < 2, q > 2, λ > 0 and h(t), b(t) are sign-changing continuous functions.