Singular Nonlinear Fractional Differential Equations Research Articles

A study for a higher order Riemann-Liouville fractional differential equation with weakly singularity

<abstract><p>In this paper, we study an initial value problem with a weakly singular nonlinear fractional differential equation of higher order. First, we establish the existence of global solutions to the problem within the appropriate function space. We then introduce a generalized Riemann-Liouville mean value theorem. Using this theorem, we prove the Nagumo-type uniqueness theorem for the stated problem. Moreover, we give two examples to illustrate the applicability of the existence and uniqueness theorems.</p></abstract>

Artificial neural network for solving the nonlinear singular fractional differential equations

This paper proposes an artificial neural network (ANN) architecture for solving nonlinear fractional differential equations. The proposed ANN algorithm is based on a truncated power series expansion to substitute the unknown functions in the equations in this approach. Then, a set of algebraic equations is resolved using the ANN technique in an iterative minimization process. Finally, numerical examples are provided to demonstrate the usefulness of the ANN architectures. The results verify that the suggested ANN architecture achieves high accuracy and good stability.

Solutions for a category of singular nonlinear fractional differential equations subject to integral boundary conditions

We concentrate on a category of singular boundary value problems of fractional differential equations with integral boundary conditions, in which the nonlinear function f is singular at t=0, 1. We use Banach’s fixed-point theorem and Hölder’s inequality to verify the existence and uniqueness of a solution. Moreover, also we prove the existence of solutions by Krasnoselskii’s and Schaefer’s fixed point theorems.

Existence of positive solutions for a singular nonlinear semipositone fractional differential equations with parametric dependence

We examine the existence and multiplicity of positive solutions for a class of nonlinear semipositone fractional differential equations involving integral boundary conditions. The results are obtained in terms of different intervals of the parameters by means of the Leray-Schauder and Guo-Krasnoselskii fixed point theorems. Examples are included to verify our main results.

Comment for “Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel”

Abstract In a published paper of Journal of Chaos, Solitons and Fractals, some miss prints were found. One is about the calculation of the solution and also about the expression of the solution using Green’s function of the fractional differential equation studied in this paper; The others are for the properties for Green’s function, these miss prints affected the deriving of the main results in Khan’s paper, some corrections for Khan’s paper should be needed. In this paper, we make some corrections and give the correct proof proceeding for the results, a new example is given to validate part of the proven results.

Existence and Uniqueness of Positive Solutions for Singular Nonlinear Fractional Differential Equation via Mixed Monotone Operator Method

In this article, we discuss the existence and uniqueness of positive solution for a class of singular fractional differential equations, where the nonlinear term contains fractional derivative and an operator. By applying the fixed point theorem in cone, we get the existence and uniqueness of positive solutions for the fractional differential equation. Moreover, we give an example to demonstrate our main result.

Positive solutions of singular Hadamard-type fractional differential equations with infinite-point boundary conditions or integral boundary conditions

We establish the existence of positive solutions to a class of a singular nonlinear Hadamard-type fractional differential equations with infinite-point boundary conditions (BCs) or integral BCs. Our analysis is based on Leray–Schauder type continuation. Several examples are given to illustrate our results.

Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel

In this paper we are established the existence of positive solutions (EPS) and the Hyers-Ulam (HU) stability of a general class of nonlinear Atangana-Baleanu-Caputo (ABC) fractional differential equations (FDEs) with singularity and nonlinear p-Laplacian operator in Banach’s space. To find the solution for the EPS, we use the Guo-Krasnoselskii theorem. The fractional differential equation is converted into an alternative integral structure using the Atangana-Baleanu fractional integral operator. Also, HU-stability is analyzed. We include an example with specific parameters and assumptions to show the results of the proposal.

New unique existence criteria for higher-order nonlinear singular fractional differential equations

In this paper, a nonlinear three-point boundary value problem of higher-order singular fractional differential equations is discussed. By applying the properties of Green function and some fixed point theorems for sum-type operator on cone, some new criteria on the existence and uniqueness of solutions are obtained. Moreover, two iterative sequences are given for uniformly approximating the positive solution, which are important for practical application. At last, we give two examples to illustrate the main results.

Existence of positive solutions for a singular nonlinear fractional differential equation with integral boundary conditions involving fractional derivatives

In this article, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, some properties of the first eigenvalue of a fractional differential equation are obtained. Based on these properties and through the fixed point index theory, the singular nonlinear fractional differential equations with Riemann–Stieltjes integral boundary conditions involving fractional derivatives are considered under some appropriate conditions, and the nonlinearity is allowed to be singular in regard to not only time variable but also space variable and it includes fractional derivatives. The existence of positive solutions for boundary conditions involving fractional derivatives is established. Finally, an example is given to demonstrate the validity of our main results.

Nontrivial Solutions of Higher-Order Nonlinear Singular Fractional Differential Equations with Fractional Multi-point Boundary Conditions

Abstract This paper deals with the existence and multiplicity of nontrivial solutions of fractional multi-point boundary value problems for higher-order nonlinear singular fractional differential equations with sign-changing nonlinear term. The main tool used in the proof is topological degree theory. Some examples are given to illustrate our main results.

Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

In this article, we study the existence of iterative positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables. By using the properties of the Green function and the fixed point theorem of mixed monotone operators in cones we obtain some results on the existence and uniqueness of positive solutions. We also construct successively some sequences for approximating the unique solution. Our results include the multipoint boundary problems and integral boundary problems as special cases, and we also extend and improve many known results including singular and non-singular cases.

Numerical study of nonlinear singular fractional differential equations arising in biology by operational matrix of shifted Legendre polynomials

In this paper, we present a new computational method for solving nonlinear singular boundary value problems of fractional order arising in biology. To this end, we apply the operational matrices of derivatives of shifted Legendre polynomials to reduce such problems to a system of nonlinear algebraic equations. To demonstrate the validity and applicability of the presented method, we present some numerical examples.

Positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

In this article, we study the existence of positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions. Using the properties of the Green function and the fixed point theory in cones, we obtain some results on the existence of positive solutions. Our results extend and improve many known results including singular and nonsingular cases.

Positive solutions for a system of higher-order singular nonlinear fractional differential equations with nonlocal boundary conditions

The paper deals with the existence and multiplicity of positive solutions for a system of higher-order singular nonlinear fractional differential equations with nonlocal boundary conditions. The main tool used in the proof is fixed point index theory in cone. Some limit type conditions for ensuring the existence of positive solutions are 0+ u(x) + h1(x) f1(x, u(x), v(x) = 0, D b 0+ v(x) + h2(x) f2(x, u(x), v(x)) = 0,

Existence and nonexistence of positive solutions for a system of nonlinear singular fractional differential equations

Existence and nonexistence of positive solutions for a system of nonlinear singular fractional differential equations

Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition

In this paper, we investigate a boundary value problem for singular fractional differential equations with a fractional derivative condition. The existence and uniqueness of solutions are obtained by means of the fixed point theorem. Some examples are presented to illustrate our main results.

Positive properties of Green’s function for focal-type BVPs of singular nonlinear fractional differential equations and its application

In this paper, we consider the properties of Green’s function for the singular nonlinear fractional differential equation boundary value problem

Uniqueness and existence of positive solutions for a multi-point boundary value problem of singular fractional differential equations

In this paper, we study the uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem , , , , where is a real number, is the standard Riemann-Liouville differentiation and , with . Our analysis relies on a fixed-point theorem in partially ordered set. As an application, an example is presented to illustrate the main result. MSC:26A33, 34B15, 34K37.

Existence Results for a Coupled System of Nonlinear Singular Fractional Differential Equations with Impulse Effects

A boundary value problem for the singular fractional differential system with impulse effects is presented. By applying Schauder's fixed point theorem in a suitably Banach space, we obtain the existence of at least one solution for this problem. Two examples are presented to illustrate the main theorem.