Types Of Fractional Differential Equations Research Articles

Stability of positive solutions for a class of nonlinear Hadamard type fractional differential equations with $p$-Laplacian

Stability of positive solutions for a class of nonlinear Hadamard type fractional differential equations with $p$-Laplacian

Generalized Hukuhara Weak Solutions for a Class of Coupled Systems of Fuzzy Fractional Order Partial Differential Equations without Lipschitz Conditions

As is known to all, Lipschitz condition, which is very important to guarantee existence and uniqueness of solution for differential equations, is not frequently satisfied in real-world problems. In this paper, without the Lipschitz condition, we intend to explore a kind of novel coupled systems of fuzzy Caputo Generalized Hukuhara type (in short, gH-type) fractional partial differential equations. First and foremost, based on a series of notions of relative compactness in fuzzy number spaces, and using Schauder fixed point theorem in Banach semilinear spaces, it is naturally to prove existence of two classes of gH-weak solutions for the coupled systems of fuzzy fractional partial differential equations. We then give an example to illustrate our main conclusions vividly and intuitively. As applications, combining with the relevant definitions of fuzzy projection operators, and under some suitable conditions, existence results of two categories of gH-weak solutions for a class of fire-new fuzzy fractional partial differential coupled projection neural network systems are also proposed, which are different from those already published work. Finally, we present some work for future research.

On the Generalized Liouville–Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions

In this study, we examine the existence and Hyers–Ulam stability of a coupled system of generalized Liouville–Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray–Schauder alternative and Krasnoselskii’s fixed point theorem are used to demonstrate the existence of a solution. The Banach fixed point theorem’s concept of contraction mapping is used in the second theorem to emphasise the analysis of uniqueness, and the results for Hyers–Ulam stability are established in the next theorem. We establish the stability of Ulam–Hyers using conventional functional analysis. Finally, examples are used to support the results. When a generalized Liouville–Caputo (ρ) parameter is modified, asymmetric results are obtained. This study presents novel results that significantly contribute to the literature on this topic.

Solution of Fuzzy Volterra Integral and Fractional Differential Equations via Fixed Point Theorems

In this paper we present fuzzy coupled fixed point results in the turf of complete b-metric spaces via nonlinear F-contraction; in follow we derive some interesting results as byproducts. Eventually, we apply our results in solving fuzzy Volterra integral equations and Caputo-Hadamard type of fractional differential equations.

Some Results on Backward Stochastic Differential Equations of Fractional Order

In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order \(\alpha \in (\frac{1}{2},1)\) whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution.

Analytical Investigation of Nonlinear Fractional Harry Dym and Rosenau-Hyman Equation via a Novel Transform

We use a new integral transform approach to solve the fractional Harry Dym equation and fractional Rosenau-Hyman equation in this work. The Elzaki transform and the integral transformation are combined in the suggested method (ET). To handle two nonlinear problems, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD). The ultimate purpose of this study is to find an error analysis that demonstrates that our final result converges to the exact and approximate result. The convergent series form solution demonstrates the method’s efficiency in resolving several types of fractional differential equations. Furthermore, the solutions obtained in this study agree well with the exact solutions; thus, this strategy is powerful and efficient as an alternate way for obtaining approximate solutions to both linear and nonlinear fractional differential equations.

Fractional-View Analysis of Space-Time Fractional Fokker-Planck Equations within Caputo Operator

In this article, we investigate the fractional-order Fokker-Planck equations with the help of the Yang transform decomposition method (YTDM). The YTDM combines Yang transform, Adomian decomposition method, and Adomian polynomials into one method. In the Caputo sense, fractional derivatives of space and time are studied. The convergent series form solution demonstrates the method’s efficiency in resolving several types of fractional differential equations. Compared to other methods of finding approximate and exact solutions for nonlinear partial differential equations, this technique is more efficient and time-consuming.

Existence results for nonlocal Cauchy problem of nonlinear $\psi-$Caputo type fractional differential equations via topological degree methods

This manuscript is devoted to the investigation of the existence results of fractional Cauchy problem forsome nonlinear ψ−Caputo fractional differential equations with non local conditions. By applying fixedpoint theorems, some results of topological degree theory for condensing maps and some fractional analysistechniques, we establish some new existence theorems. As application, a nontrivial example is given toillustrate our theoretical results.

Stability of a Non-Lipschitz Stochastic Riemann-Liouville Type Fractional Differential Equation Driven by Lévy Noise

Stability of a Non-Lipschitz Stochastic Riemann-Liouville Type Fractional Differential Equation Driven by Lévy Noise

Study on the existence and nonexistence of solutions for a class of nonlinear Erdélyi-Kober type fractional differential equation on unbounded domain

The aim of the present paper is to investigate the existence and nonexistence of solutions for a nonlinear Erdélyi-Kober (EK) fractional-order differential equation associated with EK fractional integral boundary conditions on the half-line. The considered equation contains a large amount of linear and nonlinear differential equations with EK fractional operator. In order to consider this equation, we first formulated the linear problem in integral equation. Our approach is based on recent fixed point theorem which uses the strongly positive-like operators and eigenvalue criteria for existence and nonexistence of positive solutions.

Convergence and Stability of a Novel M ‐Iterative Algorithm with an Application

In this manuscript, we propose a novel three - step iteration scheme called M - iteration to approximate the invariant points for the class of weak contractions in the sense of Berinde and obtain that M - iteration strongly converges to one and only one fixed point for Berinde mappings. Proving ‘almost - stability’ of M - iteration, we compare the M - iterative scheme with other chief iterative algorithms, namely. Picard, Mann, Ishikawa, Noor, S, normal-S, Abbas, Thakur, Varat, Ullah and F ∗ and claim that the framed iterative procedure converges to the invariant points of weak contractions at faster rate than other vital algorithms. Some numerical illustrations are adduced to strengthen our claim. We further ascertain data dependency through M - iteration. Finally, we establish the solution of Caputo type fractional differential equation as an application and exhibit that M - fixed point procedure converges to the solution of a fractional differential equation. The obtained results are not only new but, also, extend the scope of previous findings.

Atomic Solution of Poisson Type Equation

In this paper we find a certain solution of Poisson type fractional differential equation using theory of tensor product of Banach spaces.

On the periodic boundary value problems for fractional nonautonomous differential equations with non-instantaneous impulses

In this paper, we investigate periodic boundary value problems for Caputo type fractional semilinear nonautonomous differential equations with non-instantaneous impulses. By using semigroup theory combined with the measure of noncompactness and some fixed point theorems, the existence of PC-mild solutions for the equations is established. At the end, an example is presented to illustrate the application of our main results.

Gronwall inequality and existence of solutions for differential equations with generalized Hattaf fractional derivative

The classical Gronwall inequality is one of the basic tools in the theory of differential and integral equations. In this paper, a new version of this inequality is presented and extended to differential equations with the generalized Hattaf fractional derivative involving non-singular kernel. The existence and uniqueness of solutions for such last type of fractional differential equations are rigorously investigated. Furthermore, an application is presented to study the Ulam-Hyers stability of certain equations.

On the Existence and Stability of Variable Order Caputo Type Fractional Differential Equations

In the theory of differential equations, the study of existence and the uniqueness of the solutions are important. In the last few decades, many researchers have had a keen interest in finding the existence–uniqueness solution of constant fractional differential equations, but literature focusing on variable order is limited. In this article, we consider a Caputo type variable order fractional differential equation. First, we present the existence–uniqueness of a solution of the considered problem. Secondly, By borrowing the idea from the theory of ordinary differential equations, we extend the continuation theorem for the variable order fractional differential equation. Further, we prove the global existence results. Finally, we present different types of Ulam–Hyers stability results, which have never been studied before for the Caputo type variable order fractional differential equation.

The construction of solutions to $ {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $ type FDEs via reduction to $ \left({}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)}\right)^n $ type FDEs

<abstract><p>A scheme for the integration of $ \, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $-type fractional differential equations (FDEs) is presented in this paper. The approach is based on the expansion of solutions to FDEs via fractional power series. It is proven that $ \, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $-type FDEs can be transformed into equivalent $ \left(\, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)}\right)^n $-type FDEs via operator calculus techniques. The efficacy of the scheme is demonstrated by integrating the fractional Riccati differential equation.</p></abstract>

Existence of solutions of a class of p-Laplacian type fractional impulsive differential equation with boundary value problem

Existence of solutions of a class of p-Laplacian type fractional impulsive differential equation with boundary value problem

On degree theory for non-monotone type fractional order delay differential equations

<abstract><p>In this paper, we establish a qualitative theory for implicit fractional order differential equations (IFODEs) with nonlocal initial condition (NIC) with delay term. Because area related to investigate existence and uniqueness of solution is important field in recent times. Also researchers are using existence theory to derive some prior results about a dynamical problem weather it exists or not in reality. In literature, we have different tools to study qualitative nature of a problem. On the same line the exact solution of every problem is difficult to determined. Therefore, we use technique of numerical analysis to approximate the solutions, where stability analysis is an important aspect. Therefore, we use a tool from non-linear analysis known as topological degree theory to develop sufficient conditions for existence and uniqueness of solution to the considered problem. Further, we also develop sufficient conditions for Hyers- Ulam type stability for the considered problem. To justify our results, we also give an illustrative example.</p></abstract>

Results on controllability for Sobolev type fractional differential equations of order $ 1 < r < 2 $ with finite delay

<abstract><p>In this article, exact controllability results for Sobolev fractional delay differential system of $ 1 < r < 2 $ are investigated. Fractional analysis, cosine and sine function operators, and Schauder's fixed point theorem are applied to verify the main results of this study. To begin, we use sufficient conditions to explore the controllability for fractional evolution differential system with finite delay. Lastly, an example is provided to illustrate the obtained theoretical results.</p></abstract>

Positive solutions for a p-Laplacian type fractional differential equation with mixed boundary conditions

Positive solutions for a p-Laplacian type fractional differential equation with mixed boundary conditions