Sequential Fractional Differential Equations Research Articles

Monotone iterative technique for a sequential delta-Caputo fractional differential equations with nonlinear boundary conditions

Abstract. In this article, we discuss the existence of extremal solutions for a class of nonlinear sequential δ–Caputo fractional differential equations involving nonlinear boundary conditions. Our results are founded on advanced functional analysis methods. To be more specific, we use the monotone iterative approach in conjunction with the upper and lower solution method to create adequate requirements for the existence of extremal solutions. As an application, we give an example to illustrate our results. Mathematics Subject Classification (2010): 34A08, 26A33. Keywords: Sequential δ–Caputo derivative, nonlinear boundary conditions, monotone iterative technique, upper and lower solutions.

Boundary problems of sequential fractional differential equations having a monomial coefficient

The article examines sequential fractional-order differential equations composed of monomial coefficients defined as nonlinear boundary problems. The original problem is converted into an integral equation in an equivalent form. The contraction mapping principle and Krasnoselskii's fixed-point theorem are implemented to obtain two existence conditions. The problem of Ulam-Hyers stability is also investigated. An illustration is presented to showcase practical application of the obtained results.

Analysis of Caputo Sequential Fractional Differential Equations with Generalized Riemann–Liouville Boundary Conditions

This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples.

On the existence of solutions for nonlocal sequential boundary fractional differential equations via ψ-Riemann–Liouville derivative

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder’s and Perov’s fixed-point theorems. Examples are given to validate the theoretical results.

Existence Results for the System of Fractional-Order Sequential Integrodifferential Equations via Liouville–Caputo Sense

We investigate the conditions for the existence and uniqueness of solutions in a nonlinear system of sequential fractional differential equations using the Liouville–Caputo type with varying orders. This system is enriched by nonlocal coupled integral boundary conditions. The desired outcomes are attained by employing traditional fixed-point theorems. It is essential to emphasize that the fixed-point approach proves to be an effective method for establishing the existence of solutions in boundary value problems. Furthermore, we provide constructed examples to illustrate the obtained results.

A novel approach on the sequential type ψ-Hilfer pantograph fractional differential equation with boundary conditions

This article investigates sufficient conditions for the existence and uniqueness of solutions to the ψ-Hilfer sequential type pantograph fractional boundary value problem. Considering the system depends on a lower-order fractional derivative of an unknown function, the study is carried out in a special working space. Standard fixed point theorems such as the Banach contraction principle and Krasnosel’skii’s fixed point theorem are applied to prove the uniqueness and the existence of a solution, respectively. Finally, an example demonstrating our results with numerical simulations is presented.

New Results on the Solvability of Abstract Sequential Caputo Fractional Differential Equations with a Resolvent-Operator Approach and Applications

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative.

Hybrid System of Proportional Hilfer-Type Fractional Differential Equations and Nonlocal Conditions with Respect to Another Function

In this paper, a new class of coupled hybrid systems of proportional sequential ψ-Hilfer fractional differential equations, subjected to nonlocal boundary conditions were investigated. Based on a generalization of the Krasnosel’skii˘’s fixed point theorem due to Burton, sufficient conditions were established for the existence of solutions. A numerical example was constructed illustrating the main theoretical result. For special cases of the parameters involved in the system many new results were covered. The obtained result is new and significantly contributes to existing results in the literature on coupled systems of proportional sequential ψ-Hilfer fractional differential equations.

On boundary values system of sequential hybrid fractional differential equations

On boundary values system of sequential hybrid fractional differential equations

Existence Results for Novel Sequential Phi-Caputo Fractional Differential Equations via Concept of Measures of Noncompactness

Existence Results for Novel Sequential Phi-Caputo Fractional Differential Equations via Concept of Measures of Noncompactness

On Coupled Systems of Hilfer-Hadamard Sequential Fractional Differential Equations with Three-Point Boundary Conditions

On Coupled Systems of Hilfer-Hadamard Sequential Fractional Differential Equations with Three-Point Boundary Conditions

Existence results and stabilization of homogeneous conformable fractional order systems

In this paper, we give results on the stability of homogeneous conformable fractional order systems using assumptions on a family of compact sets and provide the stabilization of an affine control system and given an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists and satisfying a homogeneous condition and we present existence and uniqueness theorems for sequential linear conformable fractional differential equations.

Utilizing Schaefer's fixed point theorem in nonlinear Caputo sequential fractional differential equation systems

<abstract><p>In the present study, established fixed-point theories are utilized to explore the requisite conditions for the existence and uniqueness of solutions within the realm of sequential fractional differential equations, incorporating both Caputo fractional operators and nonlocal boundary conditions. Subsequently, the stability of these solutions is assessed through the Ulam-Hyers stability method. The research findings are validated with a practical example that corroborate and reinforce the theoretical results.</p></abstract>

Coupled systems of nonlinear sequential proportional Hilfer-type fractional differential equations with multi-point boundary conditions

Coupled systems of nonlinear sequential proportional Hilfer-type fractional differential equations with multi-point boundary conditions

Existence results for a system of sequential differential equations with varying fractional orders via Hilfer-Hadamard sense

<abstract><p>We investigate the criteria for both the existence and uniqueness of solutions within a nonlinear coupled system of Hilfer-Hadamard sequential fractional differential equations featuring varying orders. This system is complemented by nonlocal coupled Hadamard fractional integral boundary conditions. The desired outcomes are attained through the application of well-established fixed-point theorems. It is underscored that the fixed-point approach serves as an effective method for establishing both the existence and uniqueness of solutions to boundary value problems. The results obtained are further demonstrated and validated through illustrative examples.</p></abstract>

Existence of Solutions for Coupled System of Sequential Liouville–Caputo-Type Fractional Integrodifferential Equations

The present investigation aims to establish the existence and uniqueness of solutions for a system containing sequential fractional differential equations. Furthermore, boundary conditions that include the Riemann–Liouville fractional integral are taken into consideration. The existence of unknown functions, fractional derivatives, and fractional integrals at lower orders are necessary for the nonlinearity to exist. In order to provide proofs for the results presented in this study, the Leray–Schauder alternative and the Banach fixed-point theorem are utilised. Finally, examples are used to support the main results.

Series Solution Method for Solving Sequential Caputo Fractional Differential Equations

Computing the solution of the Caputo fractional differential equation plays an important role in using the order of the fractional derivative as a parameter to enhance the model. In this work, we developed a power series solution method to solve a linear Caputo fractional differential equation of the order q,0<q<1, and this solution matches with the integer solution for q=1. In addition, we also developed a series solution method for a linear sequential Caputo fractional differential equation with constant coefficients of order 2q, which is sequential for order q with Caputo fractional initial conditions. The advantage of our method is that the fractional order q can be used as a parameter to enhance the mathematical model, compared with the integer model. The methods developed here, namely, the series solution method for solving Caputo fractional differential equations of constant coefficients, can be extended to Caputo sequential differential equation with variable coefficients, such as fractional Bessel’s equation with fractional initial conditions.

Undetermined Coefficients Method for Sequential Fractional Differential Equations

The undetermined coefficients method is presented for nonhomogeneous sequential fractional differential equations involving Caputo fractional derivative of order n\alpha where n-1 n\alpha\le\ n and n\in\mathbb{N}. By employing proposed method, a particular solution of the considered equation is obtained. Some details about estimating the particular solution required to apply this method are explained. This method is shown to be particularly effective for nonhomogeneous fractional differential equations when the fractional differential equations involve some specific right-hand side functions.

Existence and H-U stability of a tripled system of sequential fractional differential equations with multipoint boundary conditions

In this paper, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray–Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers–Ulam type. As an application, we present a few examples to illustrate the main results.

Solution of Inhomogeneous Linear Fractional Differential Equations Involving Jumarie Fractional Derivative

This paper presents a method for solving inhomogeneous linear sequential fractional differential equations with constant coefficients (ILSFDE) involving Jumarie fractional derivatives in terms of Mittag-Leffler functions. For this purpose, the fundamental properties of the Jumarie derivative and Mittag-Leffler functions are given. After this, the successive jumarie fractional derivatives of Mittag-Leffler functions, fractional cosine, and sine functions are obtained. Further, we determined the particular integrals of these functions and then found the complete solutions of ILSFDE. in terms of Mittag-Leffler functions, fractional cosine, and sine functions. We have demonstrated this developed method with a few examples of ILSFDE. This method is similar to the method for finding the complete solutions of classical differential equations with constant coefficients.