EXTENDED ANALYSIS ON RIEMANN–LIOUVILLE FRACTIONAL DIFFERENTIAL EQUATIONS: ANTI-PERIODIC BOUNDARY CONDITIONS OF VARIABLE ORDER

PurposeThe purpose of this paper is to investigate the existence, uniqueness and stability of solutions to a class of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order (R-LFDEAPBCVO). The study utilizes standard fixed point theorems (FiPoTh) to establish the existence and uniqueness of solutions. Additionally, the Ulam-Hyers-Rassias (Ul-HyRa) stability of the considered problem is examined. The obtained results are supported by an illustrative example. This research contributes to the understanding of fractional differential equations with variable order and anti-periodic boundary conditions, providing valuable insights for further studies in this field.Design/methodology/approach This paper (1) defines the Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order (R-LFDEAPBCVO); (2) discusses the existence and uniqueness of solutions to these equations using standard FiPoTh; (3) investigates the stability of the considered problem using the Ul-HyRa stability concept (Ul-HyRa); (4) provides a detailed explanation of the design and methodology used to obtain the results and (5) supports the obtained results with a relevant example.FindingsThe authors confirm that no funds, grants or any other form of financial support were received during the preparation of this manuscript.Originality/valueThe originality/value of our paper lies in its contribution to the field of fractional differential equations. Specifically, we address the existence, uniqueness and stability of solutions to a class of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions of variable order. By utilizing standard FiPoTh and investigating Ul-HyRa stability, we provide novel insights into this problem. The results obtained are supported by an example, further enhancing the credibility and applicability of your findings. Overall, our paper adds to the existing knowledge and understanding of Riemann–Liouville fractional differential equations with anti-periodic boundary conditions, making it valuable to the scientific community.

The data presented in this paper are related to the paper entitled “A Numerical Method for Solving Caputo's and Riemann-Liouville's Fractional Differential Equations which includes multi-order fractional derivatives and variable coefficients”, available in the “Communications in Nonlinear Science and Numerical Simulation” journal. Here, data are included for three of the four examples of Fractional Differential Equation (FDE) reported in [1], the other data is already available in [1]. Data for each example contain: the interval of the solution, the solution by using the proposed method, the analytic solution and the absolute error. Data were obtained through Octave 5.1.0 simulations. For a better comprehension of the data, a pseudo-code of three stages and nine steps is included.

In this chapter, we focus on the study of coupled systems of Hadamard and Riemann-Liouville fractional differential equations with coupled and uncoupled Hadamard type integral boundary conditions. Coupled systems of fractional order differential equations are of significant importance as such systems appear in a variety of problems of interdisciplinary fields such as synchronization phenomena [81, 84, 179], nonlocal thermoelasticity [130], bioengineering [119], etc. For details and examples, the reader is referred to the papers [11, 28, 29, 123, 151, 152, 169] and the references cited therein.

In this paper, we study the existence and uniqueness of solutions for a coupled implicit system involving ψ-Riemann–Liouville fractional derivative with nonlocal conditions. We first transformed the coupled implicit problem into an integral system and then analyzed the uniqueness and existence of this integral system by means of Banach fixed-point theorem and Krasnoselskiis fixed-point theorem. Some known results in the literature are extended. Finally, an example is given to illustrate our theoretical result.

We investigate a class of Riemann–Liouville's fractional differential equation with infinite-point boundary conditions. We give some new properties of the Green's function associated with the fractional differential equation boundary value problem. Based upon these new properties and by using Schauder's fixed point theorem, we establish some existence results on positive solutions for the boundary value problem. Further, by using a fixed point theorem of general concave operators, we also present an existence and uniqueness result on positive solutions for the boundary value problem.

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

Variational Lyapunov method for fractional differential equations

Study of nonlinear sequential fractional differential equations of Riemann-Lioville type and Caputo type initial value problem are very useful in applications. In order to develop any iterative methods to solve the nonlinear problems, we need to solve the corresponding linear problem. In this work, we develop Laplace transform method to solve the linear sequential Riemann-Liouville fractional differential equations as well as linear sequential Caputo fractional differential equations of order nq which is sequential of order q. Also, nq is chosen such that (n−1) < nq < n. All our results yield the integer results as a special case when q tends to 1.

This manuscript reanalyses the Bagley–Torvik equation (BTE). The Riemann–Liouville fractional differential equation (FDE), formulated by R. L. Bagley and P. J. Torvik in 1984, models the vertical motion of a thin plate immersed in a Newtonian fluid, which is held by a spring. From this model, we can derive an FDE for the particular case of lacking the spring. Here, we find conditions for the source term ensuring that the solutions to the equation of the motion are bounded, which has a clear physical meaning.

This research is concerned with the existence and uniqueness of solutions for a coupled system of Ψ–Riemann–Liouville fractional differential equations. To achieve this objective, we establish a set of necessary conditions by formulating the problem as an integral equation and utilizing well-known fixed-point theorems. By employing these mathematical tools, we demonstrate the existence and uniqueness of solutions for the proposed system. Additionally, to illustrate the practical implications of our findings, we provide several examples that showcase the main results obtained in this study.

In scattering problems for time-harmonic elastic waves, thin elastic layers are often of interest, e.g., in laminates. Various ways of substituting such layers by some effective boundary conditions have been proposed, and these are briefly reviewed. A rational way of obtaining boundary conditions that are exact to first order in the layer thickness is then described. For a thin spherical layer numerical comparisons are performed between these “exact” first order boundary conditions, the commonly used spring boundary conditions and the exact solution, and it is shown that the “exact” boundary conditions are far superior to the spring boundary conditions in most situations. A drawback with the “exact” boundary conditions is that they are quite complicated.

Using the Laplace transform method and the convolution theorem, we introduce new and more general definitions for fractional operators with non-singular kernels, extending well-known concepts existing in the literature. The new operators are based on a generalization of the Mittag–Leffler function, characterized by the presence of a key parameter p. This power parameter p is important to enable researchers to choose an adequate notion of the derivative that properly represents the reality under study, to provide good mathematical models, and to predict future dynamic behaviors. The fundamental properties of the new operators are investigated and rigorously proved. As an application, we solve a Caputo and a Riemann–Liouville fractional differential equation.

Let the fractional differential equation (FDE) be $$\displaystyle (D^\alpha _{a_+}y)(t) = f[t,y(t)],\hspace {0.2 cm} \alpha > 0,\hspace {0.2 cm} t > a,$$ with the conditions: $$\displaystyle (D^{\alpha - k}_{a+}y)(a+) = b_k,\hspace {0.2 cm} k = 1,\ldots , n,$$ called also Riemann–Liouville FDE.

The novelty of our paper is to establish results on asymptotic stability of mild solutions in $p$th moment to Riemann-Liouville fractional stochastic neutral differential equations (for short Riemann-Liouville FSNDEs) of order $\alpha \in (\frac{1}{2},1)$ using a Banach's contraction mapping principle. The core point of this paper is to derive the mild solution of FSNDEs involving Riemann-Liouville fractional time-derivative by applying the stochastic version of variation of constants formula. The results are obtained with the help of the theory of fractional differential equations, some properties of Mittag-Leffler functions and asymptotic analysis under the assumption that the corresponding fractional stochastic neutral dynamical system is asymptotically stable.

Linear first order Riemann-Liouville fractional differential and perturbed Abel's integral equations