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This article deals with a solvability for a problem of random fractional differential equations with n sequential derivatives and nonlocal conditions. The existence and uniqueness of solutions for the problem is obtained by using Banach contraction principle. New random data concepts for the considered problem are introduced and some related definitions are given. Also, some results related to the dependance on the introduced data are established for both random and deterministic cases.

Gronwall's inequalities are important in the study of differential equations and integral inequalities. Gronwall inequalities are a valuable mathematical technique with several applications. They are especially useful in differential equation analysis, stability research, and dynamic systems modeling in domains spanning from science and math to biology and economics. In this paper, we present new generalizations of Gronwall inequalities of integral versions. The proposed results involve ((ρ, φ)-Riemann-Liouville fractional integral with respect to another function. Some applications on differential equations involving ((ρ, φ)-Riemann-Liouville fractional integrals and derivatives are established. [ABSTRACT FROM AUTHOR]

In this work, we investigate a novel nonlinear differential problem involving Caputo derivatives. The problem features sequential derivatives that do not adhere to the semi-group and commutativity properties. Under certain specific conditions, the problem simplifies to a fourthorder ordinary problem, representing the static equilibrium of an elastic beam. Using the Banach contraction principle followed by Schaefer's fixed point theorem, we establish two key results regarding the uniqueness of solutions and the existence of at least one solution. We provide an example to validate one of these results. Additionally, we discuss Ulam-Hyers stability, including noteworthy limiting-case examples. [ABSTRACT FROM AUTHOR]

This paper extends classical results on integral inequalities involving monotone functions to the domain of Riemann-Liouville fractional integrals with positive arbitrary order α. By employing a unified framework, our approach provides a more generalized understanding of the interplay between monotonicity and integrability in the case of fractional integration. We review classical results, introduce Riemann-Liouville integrals, and establish the fractional integral extensions. Our main results are presented, with discussions on their applications, contributing to a broader comprehension of this type of inequalities in mathematical analysis and its applications. [ABSTRACT FROM AUTHOR]

In this paper, we combine the Riemann-Liouville integral operator and Caputo derivative to investigate a nonlinear time-singular differential equation of Lane Emden type. The considered problem involves n fractional Caputo derivatives under the conditions that neither commutativity nor semi group property is satisfied for these derivatives. We prove an existence and uniqueness analytic result by application of Banach contraction principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions related to this result are established by means of the fixed point theorem of Schaefer. We end the paper by presenting to the reader some illustrative examples.

In this paper, we apply the Laplace decomposition and the conformable variational iteration methods to derive new numerical solutions of time and space-conformable fractional Wu-Zhang system. Some graphs for the obtained solutions are plotted. [ABSTRACT FROM AUTHOR]

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems. [ABSTRACT FROM AUTHOR]

In the current paper, we look at the existence, uniqueness, and stability of solutions for a new pantograph problem with three sequential derivatives of Caputo-Hadamard type. The proposed problem admits the third-order pantograph problem as a limiting case. So, based on Banach contraction principle and Leray-Schauder fixed point theorems, two main theorems are proved. Another main result for the Ulam-Hyers stability of solutions for the problem is established. Furthermore, an illustrative example is presented to show the applicability of the existence and uniqueness result as well as the Ulam stability one. [ABSTRACT FROM AUTHOR]

The present work deals with a coupled system of fractional differential equations involving four sequential Caputo derivatives in each of its components. The fractional differential system gives rise to a standard coupled system of two ordinary differential equations of order four, which has practical applications in some real-world phenomena such as robotics, aerospace, and electrical engineering. The existence of a unique vector solution for our sequential system is studied. The existence of at least one vector solution for the considered system is also investigated. Some illustrative examples are discussed in detail to show the main results’ applicability. The stabilities in the sense of Ulam Hyers for the system is discussed. A conclusion follows at the end. [ABSTRACT FROM AUTHOR]

In this work, we suggest a new numerical scheme called the fractional higher order Taylor method (FHOTM) to solve fractional differential equations (FDEs). Using the generalized Taylor's theorem is the fundamental concept of this approach. Then, the local truncation error generated by the suggested FHOTM is estimated by proving suitable theoretical results. At last, several numerical applications are given to demonstrate the applicability of the suggested approach in relation to their exact solutions. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the application of the Laplace decomposition method (LDM) to find approximate solutions for some classes of time and space-conformable fractional evolution equations. Some illustrative examples are presented showing that the LDM is an efficient method for finding solutions of nonlinear conformable fractional problems. [ABSTRACT FROM AUTHOR]

In this work, we are interested to investigate the existence and uniqueness of random solutions for a class of fractional differential equation (FDE) in the sense of sample path Caputo-Hadamard operator with boundary conditions. Our main results will be carried out with the help of some approporiate stochastic versions of fixed point theorems. Two other results on the Ulam-Hyers (UH) and the Ulam-Hyers-Rassias (UHR) stabilities of stochastic fractional differential equations are established (SFDE). At the end, two examples are discussed. [ABSTRACT FROM AUTHOR]

We propose an alpha-beta-point non-linear boundary value problem that involves both Caputo derivatives and RL integrals, such that its limiting-case on the parameters alpha and beta product a fourth order problem that arises in bridge design. We begin first by presenting a uniqueness of solution result for the problem. Then, the Ulam-Hyers stability is discussed. An example is then discussed. Also, in a second part, we apply the tanh method for finding travelling waves for a nonlinear problem that involves Khalil derivatives in time and space. Some graphs are plotted for the obtained travelling waves. [ABSTRACT FROM AUTHOR]

The type of symmetry exhibited by a travelling wave can have important implications for its behaviour and properties, such as its polarization, dispersion, and interactions with other waves or boundaries. The fractional differential Duffing problem refers to the mathematical modelling of nonlinear, damped oscillations of a system with fractional derivatives. It is a generalization of the classical Duffing equation, which describes the behaviour of a nonlinear, damped oscillator (the equation becomes symmetric under time-reversal). The fractional derivatives allow for a more accurate description of the system's memory and hereditary properties. The solution of the fractional Duffing equation can provide insight into the complex dynamic behaviour of various physical, biological, and engineering systems. We are concerned with studying a new differential Duffing fractional problem. It involves some sequential Caputo derivatives with an infinite series of Riemann-Liouville integrals and some other functions. We begin by proving a first existence and uniqueness result, then we discuss two types of stability for the obtained uniqueness result. An illustrative examples is given to show the applicability of the result. We are also concerned with applying the Tanh method to obtain new classes of travelling wave solutions for three important classes of (Khalil) fractional conformable problems; the generalized equation of Duffing, the Landau-Ginzburg-Higgs equation and the Sine-Gordon one. Some numerical simulations are plotted and a conclusion is given at the end. [ABSTRACT FROM AUTHOR]

We apply the sine-Gordon expansion method, SGEM for short, to derive solutions for some classes of time and space-conformable fractional evolution equations. Some examples are discussed and their graphs are plotted to illustrate the efficiency of the SGEM method. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag–Leffler kernel. Firstly, we establish the existence of solutions to the studied problem by using the Banach contraction theorem. Then, by means of the Dhage fixed‐point principle, we discuss the existence of mild solutions. Finally, we study the Ulam–Hyers stability of the introduced fractional hybrid problem. [ABSTRACT FROM AUTHOR]

We study a new Jerk equation involving three fractional derivatives and anti periodic conditions. By Banach contraction principle, we present an existence and uniqueness result for the considered problem. Utilizing Krasnoselskii fixed point theorem we prove another existence result governing at least one solution. We provide an illustrative example to claim our established results. At the end, an approximation for Caputo derivative is proposed and some chaotic behaviours are discussed by means of the Runge Kutta 4th order method. [ABSTRACT FROM AUTHOR]

In this paper, we study new classes of differential equations of fractional order. The first considered problem involves the derivative of Caputo, while the second one involves conformable Khalil derivative. For the first class, we prove an existence and uniqueness result, then, we discuss an example to show the applicability of the result. For the second one, we apply the exp-function method to find new traveling wave solutions for a generalised conformable fractional partial differential equation, then some examples on Ostrovsky and KdV equations are given to illustrate the efficiency and accuracy of the method. Some graphs are plotted and discussed to show more the importance of the obtained results. [ABSTRACT FROM AUTHOR]