Fractional Differential Problem Research Articles

Halanay inequality involving Caputo-Hadamard fractional derivative and application

Abstract A Halanay inequality with distributed delay of non-convolution type is considered. We establish a decay of solutions as a Mittag-Leffler function composed with a logarithmic function. A general sufficient condition is found and a large class of admissible retardation kernels is provided. This needs the preparation of several lemmas on properties of the Hadamard derivative and some basic fractional differential problems with this kind of derivative. The obtained result is then applied to a Hopfield neural network system to discuss its stability.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR CAPUTO FRACTIONAL DIFFERENTIAL EQUATION WITH INFINITE-POINT BOUNDARY VALUE CONDITIONS

<abstract> In this paper, the existence of positive solutions for Caputo fractional differential equations boundary value problem with infinite points contained in the boundary value condition, and based on Green's function and its properties, the existence of multiple positive solutions are obtained by Avery-Peterson fixed point theorem. </abstract>

Existence Results for the Solution of the Hybrid Caputo–Hadamard Fractional Differential Problems Using Dhage’s Approach

In this work, we explore the existence results for the hybrid Caputo–Hadamard fractional boundary value problem (CH-FBVP). The inclusion version of the proposed BVP with a three-point hybrid Caputo–Hadamard terminal conditions is also considered and the related existence results are provided. To achieve these goals, we utilize the well-known fixed point theorems attributed to Dhage for both BVPs. Moreover, we present two numerical examples to validate our analytical findings.

Generalized Ulam–Hyers–Rassias Stability Results of Solution for Nonlinear Fractional Differential Problem with Boundary Conditions

The problem of existence and generalized Ulam–Hyers–Rassias stability results for fractional differential equation with boundary conditions on unbounded interval is considered. Based on Schauder’s fixed point theorem, the existence and generalized Ulam–Hyers–Rassias stability results are proved, and then some examples are given to illustrate our main results.

SHIFTED LEGENDRE FRACTIONAL PSEUDOSPECTRAL DIFFERENTIATION MATRICES FOR SOLVING FRACTIONAL DIFFERENTIAL PROBLEMS

A new differentiation technique, fractional pseudospectral shifted Legendre differentiation matrices (FSL D-matrices), was introduced. It depends on shifted Legendre polynomials (SLPs) as a base function. We take into consideration its extreme points and inner product. The technique was used to solve fractional ordinary differential equations (FODEs). Moreover, it extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The novel FSL D-matrices transformed these fractional differential problems (FDPs) into an algebraic system of equations. Also, an error and a convergence analysis for that technique were investigated. Finally, the correctness and efficiency of this technique were examined with test functions and several examples. All the results were compared with the results of other methods to ensure the investigated error analysis.

Nonexistence of Global Solutions for Fractional Differential Problems with Power Type Source Term

Nonexistence of Global Solutions for Fractional Differential Problems with Power Type Source Term

Existence of solution for Hilfer fractional differential problem with nonlocal boundary condition in Banach spaces

"This paper is devoted to study the existence of a solution to Hilfer fractional differential equation with nonlocal boundary condition in Banach spaces. We use the equivalent integral equation to study the considered Hilfer differential problem with nonlocal boundary condition. The Monch type fixed point theorem and the measure of the noncompactness technique are the main tools in this study. We demonstrate the existence of a solution with a suitable illustrative example."

On a sequential fractional differential problem with Riemann-Liouville integral conditions

In this paper, we are concerned with a new boundary value problem of sequential fractional type with nonlocal Riemann-Liouville integral conditions. Using the principle of Banach, so we present and we establish a new result for the uniqueness of integrals for our problem. Then, by Schaeffer fixed point theorem, an existence result is proved. At the end, some examples are illustrated and then discussed.

Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

Existence criteria via \u03b1\u2013\u03c8-contractive mappings of \u03c6-fractional differential nonlocal boundary value problems

In the existing study, we investigate the criteria of existence of solution for relatively new categories of φ-Caputo fractional differential equations and inclusions problems equipped with nonlocal φ-integral boundary conditions. In order to achieve the desired goal, we use α–ψ-contractive mappings and the theory of approximate endpoint. In the final stage, we exhibit some examples to provide the illustrations of our theoretical findings.

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were established for the classical integer-order operators and are then applied directly without having in mind the restrictions posed by their fractional-order versions. It was recently reported that the Cole–Hopf transformation can be used to convert the time-fractional nonlinear Burgers’ equation into the time-fractional linear heat equation. In this article, we show that, unlike integer-order differential equations, employing the Cole–Hopf transformation for reducing the nonlinear time-fractional Burgers’ equation into the time-fractional heat equation is wrong from two different perspectives. Indeed, such a reduction is accomplished using incorrect transcripts of the Leibniz or chain rules. Hence, providing numerical or analytical schemes based on the Cole–Hopf transformation leads to erroneous results for the nonlinear time-fractional Burgers’ equation. Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions derived in the literature. The transversality conditions were introduced as identical to those for the integer-order case, with a vanishing multiplier at the terminal of the interval. The correct condition should involve a constant-order, variable-order, or distributed-order fractional integral operator. We also deduce that if the control system is defined with a Caputo derivative, then the adjoint equations should be expressed in the Riemann–Liouville sense and vice versa. In fact, neglecting some terms in the integration by parts formulae, during the derivation of the optimality conditions, causes some confusion in the literature.

Notes on Krasnoselskii-type fixed-point theorems and their application to fractional hybrid differential problems

In this paper we prove a new version of Kransoselskii's fixed-point theorem under a ($\psi, \theta, \varphi$)-weak contraction condition. The theoretical result is applied to prove the existence of a solution of the following fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators orders of $0<\alpha<1$ and $\beta>0:$ \begin{equation}\nonumber \left\{\begin{array}{ll} D^{\alpha}[x(t)-f(t, x(t))]=g(t, x(t), I^{\beta}(x(t))), \,\,\, \text{a.e.} \,\,\, t\in J,\,\, \beta>0,\\ x(t_{0})=x_{0}, \end{array} \right. \end{equation} where $D^{\alpha}$ is the Riemann-Liouville fractional derivative order of $\alpha,$ $I^{\beta}$ is Riemann-Liouville fractional integral operator order of $\beta>0,$ $J=[t_{0}, t_{0}+a],$ for some fixed $t_{0}\in \mathbb{R},$ $a>0$ and the functions $f:J\times \mathbb{R}\rightarrow \mathbb{R}$ and $g:J\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ satisfy certain conditions. An example is also furnished to illustrate the hypotheses and the abstract result of this paper.

An anti-periodic singular fractional differential problem of Lane-Emden type

In this paper, we use both Riemann-Liouville integral and Caputo derivative to investigate a new nonlinear singular differential problem of Lane and Emden type. We note that for the studied singular problem, we will be concerned with some anti periodic conditions. So, we prove a first uniqueness result by application of Banach principle of contraction. Then, we prove a “second” existence result by application of Schaefer theorem. Two illustrative examples are discussed in details to show the applicability of the obtained results.

Existence and Stability for a Nonlinear Coupledp-Laplacian System of Fractional Differential Equations

In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of ordersθ1 and θ2and Riemann–Liouville derivative of ordersϱ1 and ϱ2with thep-Laplacian operator, wheren−1&lt;θ1,θ2,ϱ1,ϱ2≤n, andn≥3. With the help of two Green’s functionsGϱ1w,ℑ,Gϱ2w,ℑ, the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.

A Sequential Fractional Differential Problem of Pantograph Type: Existence,Uniqueness and Illustrations

In this study, a new class of sequential fractional differential problems of pantograph type is introduced. New existence and uniqueness criteria for the existence and uniqueness of solutions are discussed. Some existence results using Darbo's fixed point and measure of noncompactness are also studied. At the end, two illustrative examples are discussed.

Quintic B-spline polynomial for Solving Bagely-Torvik Fractional Differential Problems

In this paper, we use Quintic B-spline function to drive some error analysis of fractional derivatives and finding numerical solution for certain type fractional differential equations. It is know that a class of control points related to the modulus of continuity can be found by matrices operations, and discuss the error estimations for numerical solutions finding by using quantic B-spline to be in good agreement with the exact solutions. Results are shown graphically and illustrated the method by several examples.

Fundamental fractional exponential matrix: New computational formulae and electrical applications

It is well known that the fundamental fractional exponential matrix (FFEM) is closely related to the formal solution of the homogeneous and non-homogeneous linear time-conformable fractional dynamical differential system (LT-CFDDS) with delay in control and it also plays a central role in the solution of any other (matrix) fractional differential system (FDS). In this paper, we present several new theoretical results and computational formulae of the FFEM eAtαα for any arbitrary square matrix A. Moreover, some attractive and interesting special cases for FFEM are also derived and discussed. In addition, three important and interesting physical applications related to linear fractional electrical circuits (ECs) in 2 and 3-dimentions are considered and solved by means of FFEM. These applications are the fractional RC, RL and RLC-ECs. The method exists by converting the data of a given linear fractional EC into homogeneous or non-homogeneous LT-CFDDS for which solutions can be easily computed. For further and simplicity analysis, we provide an illustrative example for each problem and present the general exact solution (GES) in a simple form based on our new approach. Moreover, graphical results are created and discussed in order to ensure that the solutions of specific problems are stable at different values of α and assure that our suggestion technique is a simple, efficient, accurate, powerful analytic tool and can be applied successfully to solve many other conformable fractional differential problems in various fields. Finally, the classical behaviors of physical problems are recovered when the fractional order α is equal to 1.

Method for Evaluating Fractional Derivatives of Fractional Functions

This paper studies the fractional differential problem of fractional functions, regarding the modified Riemann-Liouvellie (R-L) fractional derivatives. A new multiplication and the fractional power series method are used to obtain any order fractional derivatives of some elementary fractional functions.

A fractional differential quadrature method for fractional differential equations and fractional eigenvalue problems

In this paper, based on the differential quadrature method (DQM), matrix operators are derived for fractional integration and Caputo differentiation. These operators generalize the efficient DQM to fractional calculus. The proposed fractional differential/integral quadrature method (FDIQM) is used to solve various types of fractional ordinary and partial differential equations. FDIQM unifies the solution of multi‐integer fractional‐order differential equations leading to significant simplification in the implementation. Numerous examples are presented to demonstrate the accuracy of the operators. Other examples are presented to solve various fractional differential equations including time‐fractional sub‐diffusion equation, linear/nonlinear, and multiorder fractional differential equations. In addition, numerous boundary conditions are considered including mixed fractional derivatives. Further, a nonlinear fractional eigenvalue problem is solved efficiently, and its bifurcation diagrams are obtained. Comparisons between the proposed method and the existing ones are included, showing the ease of implementation, efficiency, and applicability of FDIQM.

On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems

In this pages, we discuss the problem of equivalence between fractional differential and integral problems. Although the said problem was studied for ordinary derivatives, it makes some troubles in the case of fractional derivatives. This paper dealing with the question revealing the equivalence between the boundary value problems for Caputo-type fractional differential equations and the corresponding integral form. One of our main part is to determine the scope of equivalence of such problems. However, with the help of appropriate examples, we will show that even for the Holderian functions, but being off the space of absolutely continuous functions, the equivalence between the Caputo-type fractional differential problems and the corresponding integral forms can be lost. As a pursuit of this, we will show here that, the main results obtained by many researchers contained a common mathematical error in the proof of the equivalence of the boundary value problems for Caputo-type fractional differential equations and the corresponding integral forms. In this connection, we are going to slightly modify the definition of the Caputo-type fractional differential operator into a more suitable one. We will show that the modified definition is more convenient in studying the boundary value problems for Caputo-type fractional differential equations. As an application, after recalling some properties of the said (new) operator, we summarize our discussion by presenting an equivalence result for differential and integral problems for Caputo-type fractional derivatives and for the weak topology. In order to cover the full scope of this paper, we investigate the equivalence problem in case of multivalued fractional problems.