Sequential Fractional Differential Equations Research Articles

Coupled system of three sequential Caputo fractional differential equations: Existence and stability analysis

Recently, many studies on fractional coupled systems involving different sequential fractional derivatives have appeared during the past several years. The paper is dealing with a coupled system of three sequential Caputo fractional differential equations, and the designed system absorbs none of the commutativity and the semigroup properties. The Banach contraction principle is used for proving the existence and uniqueness results. We prove the existence of at least one is obtained by using the Leray–Schauder alternative. The Ulam–Hyers–Rassias stability of the considered system is defined and discussed. An illustrative example is also presented.

Systems of Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations with Fractional Integro-Differential Nonlocal Boundary Conditions

In this paper, we introduce and study a new class of coupled and uncoupled systems, consisting of mixed-type ψ1-Hilfer and ψ2-Caputo fractional differential equations supplemented with asymmetric and symmetric integro-differential nonlocal boundary conditions (systems (2) and (13), respectively). As far as we know, this combination of ψ1-Hilfer and ψ2-Caputo fractional derivatives in coupled systems is new in the literature. The uniqueness result is achieved via the Banach contraction mapping principle, while the existence result is established by applying the Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.

Solvability of Sequential Fractional Differential Equation at Resonance

The sequential fractional differential equations at resonance are introduced subject to three-point boundary conditions. The emerged fractional derivative operators in these equations are based on the Caputo derivative of order that lies between 1 and 2. The vital target of the current contribution is to investigate the existence of a solution for the boundary value problem by using the coincidence degree theory due to Mawhin which is basically depending on the Fredholm operator with index zero and two continuous projectors. An example is given to illustrate the deduced theoretical results.

Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations

In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ1-Hilfer and ψ2-Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results.

A Novel Implementation of Dhage’s Fixed Point Theorem to Nonlinear Sequential Hybrid Fractional Differential Equation

In this work, the existence and uniqueness of solutions to a sequential fractional (Hybrid) differential equation with hybrid boundary conditions were investigated by the generalization of Dhage’s fixed point theorem and Banach contraction mapping, respectively. In addition, the U-H technique is employed to verify the stability of this solution. This study ends with two examples illustrating the theoretical findings.

Boundary value problems for sequential Hilfer fractional differential equations and inclusions with integro-multistrip-multipoint boundary conditions

Boundary value problems for sequential Hilfer fractional differential equations and inclusions with integro-multistrip-multipoint boundary conditions

Solution of Homogeneous Linear Fractional Differential Equations Involving Conformable Fractional Derivative

In this paper, we have found the solution of linear sequential fractional differential equations involving conformable fractional derivatives of order with constant coefficients. For this purpose, we first discussed fundamental properties of the conformable derivative and then obtained successive conformable derivatives of the fractional exponential function. After this, we determined the analytic solution of linear sequential fractional differential equations (L.S.F.D.E.) in terms of a fractional exponential function. We have demonstrated this developed method with a few examples of homogeneous linear fractional differential equations. This method gives a conjugation with the method to solve classical linear differential equations with constant coefficients.

On a general class of $ n $th order sequential hybrid fractional differential equations with boundary conditions

<abstract><p>This manuscript is related to consider a general class of $ n $th order sequential hybrid fractional differential equations (S-HFDEs) with boundary conditions. With the help of the coincidence degree theory of topology, some appropriate results for the existence theory of the aforementioned class are developed. The mentioned degree theory is a powerful tool to investigate nonlinear problems for qualitative theory. A result related to Ulam-Hyers (U-H) stability is also developed for the considered problem. It should be kept in mind that the considered degree theory relaxes the strong compact condition by some weaker one. Hence, it is used as a sophisticated tool in the investigation of the existence theory of solutions to nonlinear problems. Also, an example is given.</p></abstract>

Existence results by Mönch's fixed point theorem for a tripled system of sequential fractional differential equations

<abstract><p>In this paper, we study the existence of the solutions for a tripled system of Caputo sequential fractional differential equations. The main results are established with the aid of Mönch's fixed point theorem. The stability of the tripled system is also investigated via the Ulam-Hyer technique. In addition, an applied example with graphs of the behaviour of the system solutions with different fractional orders are provided to support the theoretical results obtained in this study.</p></abstract>

Mönch's fixed point theorem in investigating the existence of a solution to a system of sequential fractional differential equations

<abstract><p>In this article, the existence of a solution to a system of fractional equations of sequential type was investigated via Mönch's fixed point theorem. In addition, the stability of this solutions was verified by the Ulam-Hyers method. Finally, an applied example is presented to illustrate the theoretical results obtained from the existence results.</p></abstract>

Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications

It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding integer model, we need to compute the solutions of the fractional differential equations and compare them with an integer model relative to the data available. In this work, we will illustrate that the linear nq-order sequential Caputo fractional differential equations, which are sequential of order q where q<1 with fractional initial conditions and/or boundary conditions, can be solved. The reason for choosing sequential fractional dynamic equations is that linear non-sequential Caputo fractional dynamic equations with constant coefficients cannot be solved in general. We used the Laplace transform method to solve sequential Caputo fractional initial value problems. We used fractional boundary conditions to compute Green’s function for sequential boundary value problems. In addition, the solution of the sequential dynamic equations yields the solution of the corresponding integer-order differential equations as a special case as q→1.

Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions

Abstract In this paper, we analyses the existence and Hyers–Ulam stability of a coupled system of three sequential fractional differential equations with coupled integral boundary conditions. This manuscript can be categorized into three parts: The Leray–Schauder alternative is used to prove the existence of a solution in the first section. The second section emphasizes the analysis of uniqueness, which is based on the Banach fixed point theorem’s concept of contraction mapping, and the third section establishes the Hyers–Ulam stability results. In addition, we provide examples to demonstrate our findings.

Uniqueness and stability of coupled sequential fractional differential equations with boundary conditions

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Caputo-type sequential fractional differential equations supplemented with boundary conditions. The existence of solutions is derived by applying the Leray–Schauder alternative, while the uniqueness of solutions is established via Banach’s contraction principle. Moreover, some necessary conditions for the Hyers–Ulam-type stability to the solutions of the boundary value problem are developed. Finally the results are supported by example.

The Existence and Uniqueness of Solution to Sequential Fractional Differential Equation with Affine Periodic Boundary Value Conditions

The solution to a sequential fractional differential equation with affine periodic boundary value conditions is investigated in this paper. The existence theorem of solution is established by means of the Leray–Schauder fixed point theorem and Krasnoselskii fixed point theorem. What is more, the uniqueness theorem of solution is demonstrated via Banach contraction mapping principle. In order to illustrate the main results, two examples are listed.

Existence and Stability Results for Caputo-Type Sequential Fractional Differential Equations with New Kind of Boundary Conditions

In this paper, we present the existence and the stability results for a nonlinear coupled system of sequential fractional differential equations supplemented with a new kind of coupled boundary conditions. Existence and uniqueness results are established by using Schaefer’s fixed point theorem and Banach’s contraction mapping principle. We examine the stability of the solutions involved in the Hyers–Ulam type. A few examples are presented to illustrate the main results.

Applicability of Mönch’s Fixed Point Theorem on Existence of a Solution to a System of Mixed Sequential Fractional Differential Equation

In this paper, we study the existence and uniqueness of the solution for a coupled system of mixed fractional differential equations. The main results are established with the aid of “Mönch’s fixed point theorem.” In addition, an applied example that supports the theoretical results reached through this study is included.

Existence and Stability Results for a Tripled System of the Caputo Type with Multi-Point and Integral Boundary Conditions

In this paper, we introduce and investigate the existence and stability of a tripled system of sequential fractional differential equations (SFDEs) with multi-point and integral boundary conditions. The existence and uniqueness of the solutions are established by the principle of Banach’s contraction and the alternative of Leray–Schauder. The stability of the Hyer–Ulam solutions are investigated. A few examples are provided to identify the major results.

Existence of general mean square solutions for a sequential random fractional differential system with nonlocal conditions

In this study, we discuss the existence of solutions for a coupled system of two- sequential random fractional differential equations involving nonlocal periodic boundary conditions. Using a version of Leray-Schauder alternative theorem, we present recent results on the existence of coupled solutions for the problem. The obtained solutions as well as their Caputo derivatives satisfy some stochastic processes properties. An illustrative example is also discussed at the end of this paper.

On System of Nonlinear Sequential Hybrid Fractional Differential Equations

In this study, the existence and uniqueness of the solution for a system consisting of sequential fractional differential equations that contain Caputo–Hadamard (CH) derivative are verified. To study the existence and uniqueness of these solutions, some of the most important results from the fixed point theorems in Banach space were used. A practical example is also given to support the theoretical side that was obtained.

The Unique Solution for Sequential Fractional Differential Equations with Integral Multi-Point and Anti-Periodic Type Boundary Conditions

In this paper, we obtain the existence of the unique solution of anti-periodic type (anti-symmetry) integral multi-point boundary conditions for sequential fractional differential equations. We apply the Banach contraction mapping principle to get the desired results. Our results specialize and extend some existing results.