Sequential Fractional Derivatives Research Articles

Beam deflection coupled systems of fractional differential equations: existence of solutions, Ulam–Hyers stability and travelling waves

In this paper, we study a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo fractional derivatives. Under flexible/fixed end-conditions, two main theorems on the existence and uniqueness of solutions are proved by using two fixed point theorems. Some examples are discussed to illustrate the applications of the existence and uniqueness of solution results. Another main result on the Ulam–Hyers stability of solutions for the introduced system is also discussed. Some examples of stability are discussed. New travelling wave solutions are obtained for another conformable coupled system of beam type that has a connection with the first considered system. A conclusion follows at the end.

Solvability and Mittag–Leffler–Ulam stability for fractional Duffing problem with three sequential fractional derivatives

In this paper, we discuss the existence, uniqueness, and the Mittag–Leffler–Ulam stability of solutions for fractional Duffing equations involving three fractional derivatives. Uniqueness result for solution of the underlying Duffing problem is given with the help of Banach's fixed point theorem, where the existence result is computed using Leray–Schauder's alternative. Also, the Mittag–Leffler–Ulam stability results are computed by employing generalized singular Gronwall's inequality. An illustrative example is also given.

General fractional integrals and derivatives and their applications

In this survey paper, some recent results obtained for the general fractional integrals and derivatives and for the regularized general fractional derivatives are discussed. We start with a short historical survey of the general Fractional Calculus operators with the Sonin kernels and continue with a presentation of the recent developments regarding the general fractional derivatives and their applications. In particular, we introduce the first and the second fundamental theorems of Fractional Calculus for the general fractional derivatives of arbitrary order, for the regularized general fractional derivatives of arbitrary order, and for the sequential general fractional derivatives. As an application of these results, the generalized convolution Taylor formulas and the generalized convolution Taylor series with the coefficients and remainders in terms of the general fractional derivatives and the regularized general fractional derivatives are presented. Finally, we consider some Cauchy problems for the fractional differential equations with the general fractional derivatives.

Analytical and Numerical Methods for Solving Second-Order Two-Dimensional Symmetric Sequential Fractional Integro-Differential Equations

In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and L2 errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of 10−14, while the L2 error is on the order of 10−13. Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one.

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.

A new sequential proportional fractional derivative of hybrid di?erential equations with nonlocal hybrid condition

In this paper, we study the existence of solutions for a new problem of hybrid differential equations with nonlocal integro multi point boundary conditions by using the proportional fractional derivative. The presented results are obtained by using hybrid fixed point theorems for three Dhage operators. The application of theoretical conclusions is demonstrated through an example.

Existence and Mittag-Leffler-Ulam-Stability Results for Duffing Type Problem Involving Sequential Fractional Derivatives

Existence and Mittag-Leffler-Ulam-Stability Results for Duffing Type Problem Involving Sequential Fractional Derivatives

On a coupled system of pantograph problem with three sequential fractional derivatives by using positive contraction-type inequalities

This paper aims to establish conditions for the existence, uniqueness and Ulam–Hyers stability of solutions for a coupled system of pantograph problem with three sequential fractional derivatives. Two results on the uniqueness and existence of solutions are proved to utilize the Leray–Schauder and Banach fixed point theorems and positive contraction-type inequalities. Also, the stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias are studied. An illustrative example with graphical and numerical simulations is also proposed.

Operational Calculus for the General Fractional Derivatives of Arbitrary Order

In this paper, we deal with the general fractional integrals and the general fractional derivatives of arbitrary order with the kernels from a class of functions that have an integrable singularity of power function type at the origin. In particular, we introduce the sequential fractional derivatives of this type and derive an explicit formula for their projector operator. The main contribution of this paper is a construction of an operational calculus of Mikusiński type for the general fractional derivatives of arbitrary order. In particular, we present a representation of the m-fold sequential general fractional derivatives of arbitrary order as algebraic operations in the field of convolution quotients and derive some important operational relations.

Sequential generalized Riemann\u2013Liouville derivatives based on distributional convolution

Sequential generalized fractional Riemann–Liouville derivatives are introduced as composites of distributional derivatives on the right half axis and partially defined operators, called Dirac-function removers, that remove the component of singleton support at the origin of distributions that are of order zero on a neighborhood of the origin. The concept of Dirac-function removers allows to formulate generalized initial value problems with less restrictions on the orders and types than previous approaches to sequential fractional derivatives. The well-posedness of these initial value problems and the structure of their solutions are studied.

Convolution series and the generalized convolution Taylor formula

In this paper, we deal with the convolution series that are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler type functions. Special attention is given to the most interesting case of the convolution series generated by the Sonine kernels. In this paper, we first formulate and prove the second fundamental theorem for the general fractional integrals and the $n$-fold general sequential fractional derivatives of both the Riemann-Liouville and the Caputo types. These results are then employed for derivation of two different forms of a generalized convolution Taylor formula for representation of a function as a convolution polynomial with a remainder in form of a composition of the $n$-fold general fractional integral and the $n$-fold general sequential fractional derivative of the Riemann-Liouville and the Caputo types, respectively. We also discuss the generalized Taylor series in form of convolution series and deduce the formulas for its coefficients in terms of the $n$-fold general sequential fractional derivatives.

Existence and stability results of pantograph equation with three sequential fractional derivatives

<abstract><p>The subject of this work is the existence and Mittag-Leffler-Ulam (MLU) stability of solutions for fractional pantograph equations with three sequential fractional derivatives. Sufficient conditions for the existence and uniqueness of solutions are constructed by utilizing well-known classical fixed point theorems such as the Banach contraction principle, and Leray-Schauder nonlinear alternative. The generalized singular Gronwall's inequality is used to show the MLU stability results. An illustrated example is provided to support the main findings.</p></abstract>

Nonlinear Random Differential Equations with n Sequential Fractional Derivatives

Abstract 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.

Two hybrid and non-hybrid k-dimensional inclusion systems via sequential fractional derivatives

Some complicated events can be modeled by systems of differential equations. On the other hand, inclusion systems can describe complex phenomena having some shocks better than the system of differential equations. Also, one of the interests of researchers in this field is an investigation of hybrid systems. In this paper, we study the existence of solutions for hybrid and non-hybrid k-dimensional sequential inclusion systems by considering some integral boundary conditions. In this way, we use different methods such as α-ψ contractions and the endpoint technique. Finally, we present two examples to illustrate our main results.

Sequential Abstract Generalized Right Side Fractional Landau Inequalities

We give uniform and Lp Caputo-Bochner abstract sequential generalized right fractional Landau inequalities over $\mathbb{R}_{-}$. These estimates the size of second and third sequential abstract generalized right fractional derivatives of a Banach space valued function over $\mathbb{R}_{-}$. We give an application when the basic fractional order is 1

Existence of positive solutions for a semipositone boundary value problem with sequential fractional derivatives

We investigate the existence and multiplicity of positive solutions for a system of Riemann‐Liouville fractional differential equations with sequential derivatives, positive parameters and sign‐changing singular nonlinearities, subject to nonlocal uncoupled boundary conditions which contain various fractional derivatives and Riemann‐Stieltjes integrals. We apply the nonlinear alternative of Leray‐Schauder type and the Guo‐Krasnosel'skii fixed point theorem in the proof of our main existence results.

A new nonlinear duffing system with sequential fractional derivatives

By considering the Caputo fractional derivative and Riemann-Liouville integral, in the present work, we are concerned with a nonlinear sequential fractional differential system of Duffing oscillator type. The considered system has neither the commutativity nor the semi group properties, since the sum of the two orders of derivatives, of the left hand side of the problem, are outside the interval [0,1]. With the absence of these two properties, we have to find other arguments to obtain the integral representation of the problem, to be able thereafter to present the other main results. Then, using the contraction mapping principle and Scheafer theorem, two main theorems on the uniqueness and existence of solutions are proved. Finally, some examples are given to illustrate the proposed main results.

Existence Results for a Nonlocal Coupled System of Sequential Fractional Differential Equations Involving ψ -Hilfer Fractional Derivatives

In this article, we discuss the existence and uniqueness of solutions for a new class of coupled system of sequential fractional differential equations involving ψ -Hilfer fractional derivatives, supplemented with multipoint boundary conditions. We make use of Banach’s fixed point theorem to obtain the uniqueness result and the Leray-Schauder alternative to obtain the existence result. Examples illustrating the main results are also constructed.

Maximum Principle for Space and Time-Space Fractional Partial Differential Equations

In this paper we obtain new estimates of the sequential Caputo fractional derivatives of a function at its extremum points. We derive comparison principles for the linear fractional differential equations, and apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the initial-boundary-value problem for nonlinear anomalous diffusion possesses at most one classical solution and this solution depends continuously on the initial and boundary data. This answers positively to the open problem about maximum principle for the space and time-space fractional PDEs posed by Luchko in 2011. The extremum principle for an elliptic equation with a fractional derivative and for the fractional Laplace equation are also proved.

Positive Solutions for a System of Coupled Semipositone Fractional Boundary Value Problems with Sequential Fractional Derivatives

We study the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with sequential derivatives, positive parameters and sign-changing singular nonlinearities, subject to nonlocal coupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main existence results we use the nonlinear alternative of Leray–Schauder type and the Guo–Krasnosel’skii fixed point theorem.