Search Results (86)

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators, and $(k_2,{\psi }_2)$- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ${\psi }_1$ and the parameter $\beta$. Also the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and $L^1$-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study. Full article

Since polynomial regression models are generally quite reliable for data that can be handled using a linear system, it is important to note that in some cases, they may suffer from overfitting during the training phase. This can lead to negative values of the coefficient of determination $R^2$ when applied to unseen data. To address this issue, this work proposes the partial implementation of fractional operators in polynomial regression models to construct a fractional regression model. The aim of this approach is to mitigate overfitting, which could potentially improve the $R^2$ value for unseen data compared to the conventional polynomial model, under the assumption that this could lead to predictive models with better performance. The methodology for constructing these fractional regression models is presented along with examples applicable to both Riemann–Liouville and Caputo fractional operators, where some results show that regions with initially negative or near-zero $R^2$ values exhibit remarkable improvements after the application of the fractional operator, with absolute relative increases exceeding $800\%$ on unseen data. Finally, the importance of employing sets in the construction of the fractional regression model within this methodological framework is emphasized, since from a theoretical standpoint, one could construct an uncountable family of fractional operators derived from the Riemann–Liouville and Caputo definitions that, although differing in their formulation, would yield the same regression results as those shown in the examples presented in this work. Full article

Although fractional calculus has evolved significantly since its origin in the 1695 correspondence between Leibniz and L’Hôpital, the numerical treatment of multi-order fractional differential equations remains a challenge. Existing methods are often either computationally expensive or reliant on complex operational frameworks that hinder their broader applicability. In the present study, a novel numerical algorithm is proposed based on orthogonal hybrid functions (HFs), which were constructed as linear combinations of piecewise constant sample-and-hold functions and piecewise linear triangular functions. These functions, belonging to the class of degree-1 orthogonal polynomials, were employed to obtain the numerical solution of multi-order fractional differential equations defined in the Caputo sense. A generalized one-shot operational matrix was derived to explicitly express the Riemann–Liouville fractional integral of HFs in terms of the HFs themselves. This allowed the original multi-order fractional differential equation to be transformed directly into a system of algebraic equations, thereby simplifying the solution process. The developed algorithm was then applied to a range of benchmark problems, including both linear and nonlinear multi-order FDEs with constant and variable coefficients. Numerical comparisons with well-established methods in the literature revealed that the proposed approach not only achieved higher accuracy but also significantly reduced computational effort, demonstrating its potential as a reliable and efficient numerical tool for fractional-order modeling. Full article

The current research study presents a comprehensive analysis of the local discontinuous Galerkin (LDG) method for solving multi-order fractional differential equations (FDEs), with an emphasis on circuit modeling applications. We investigated the existence, uniqueness, and numerical stability of LDG-based discretized formulation, leveraging the Liouville–Caputo fractional derivative and upwind numerical fluxes to discretize governing equations while preserving stability. The method was validated through benchmark test cases, including comparisons with analytical solutions and established numerical techniques (e.g., Gegenbauer wavelets and Dickson collocation). The results demonstrate that the LDG method achieves high-accuracy solutions (e.g., with a relatively large time step size) and reduced computational costs, which are attributed to its element-wise formulation. These findings position LDG as a promising tool for complex scientific and engineering applications, particularly in modeling fractional-order systems such as RL, RLC circuits, and other electrical circuit equations. Full article

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the $\varphi$-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle is applied to determine the existence and uniqueness of the solution. Secondly, continuous dependence findings are presented under the condition that the coefficients satisfy the global Lipschitz criteria, along with regularity results. Thirdly, we establish results for the averaging principle by applying inequalities and interval translation techniques. Finally, we provide numerical examples and graphical results to support our findings. We make two generalizations of these findings. First, in terms of the fractional derivative, our established theorems and lemmas are consistent with the Caputo operator for $\varphi \left(\mathfrak{t}\right)$ = $\mathfrak{t}$, $\mathfrak{a}=1$. Our findings match the Riemann–Liouville fractional operator for $\varphi \left(\mathfrak{t}\right)=\mathfrak{t}$, $\mathfrak{a}=0$. They agree with the Hadamard and Caputo–Hadamard fractional operators when $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$, $\mathfrak{a}=0$ and $\varphi \left(\mathfrak{t}\right)=ln\left(\mathfrak{t}\right)$, $\mathfrak{a}=1$, respectively. Second, regarding the space, we are make generalizations for the case $\mathfrak{p}=2$. Full article

Recently, a new type of derivative has been introduced, known as Caputo proportional derivatives. These are motivated by the applications of such derivatives (which are a generalization of Caputo’s standard fractional derivative) and the need to incorporate such calculus into the research on operators. The investigation therefore focuses on the equivalence of differential and integral problems for proportional calculus problems. The operators are always studied in the appropriate function spaces. Furthermore, the investigation extends these results to encompass the more general notion of Hilfer hybrid derivatives. The primary aim of this study is to preserve the maximal regularity of solutions for this class of problems. To this end, we consider such operators not only in spaces of absolutely continuous functions, but also in particular in little Hölder spaces. It is widely acknowledged that these spaces offer a natural framework for the study of classical Riemann–Liouville integral operators as inverse operators with derivatives of fractional order. This paper presents a comprehensive study of this problem for proportional derivatives and demonstrates the application of the obtained results to Langevin-type boundary problems. Full article

Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann–Liouville and Liouville–Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators. Full article

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter $H\in (0,1)$. The covariance operator Q of the stochastic fractional Wiener process satisfies $\parallel A^{-\rho }{Q}^{1/2}{\parallel }_{HS}<\infty$ for some $\rho \in [0,1)$, where ${\parallel ·\parallel }_{HS}$ denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich’s convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings. Full article

This article aims to explore the existence and stability of solutions to differential equations involving a $\psi$-Hilfer fractional derivative in the Caputo sense, which, compared to classical $\psi$-Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the $\psi$-Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the $\psi$-Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a $\psi$-Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a $\psi$-Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies. Full article

The 1st-level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann–Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the fractional calculus literature. In this paper, we first construct an operational calculus of the Mikusiński type for the 1st-level GFDs. In particular, it includes the operational calculi for the GFDs of the Riemann–Liouville type and for the regularized GFDs as its particular cases. In the second part of the paper, this calculus is applied for the derivation of the closed-form solution formulas to the initial-value problems for the linear fractional differential equations with the 1st-level GFDs. Full article

In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size $p_0$ tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method. Full article

In this paper, we present a general theory for fractional-order sequential differential equations with Riemann–Liouville nabla derivatives and Caputo nabla derivatives on time scales. The explicit solution, in the case of constant coefficients, for both the homogeneous and the non-homogeneous problems, are given using the ∇-Mittag-Leffler function, Laplace transform method, operational method and operational decomposition method. In addition, we also provide some results about a solution to a new class of fractional-order sequential differential equations with convolutional-type variable coefficients using the Laplace transform method. Full article

Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, we constructed Banach spaces required to handle the time-delay terms. To address the challenge of the unbounded nature of the solution operator at the initial moment, we developed an equivalent continuous operator. Subsequently, within the contexts of both compact and non-compact analytic semigroups, we explored the existence and uniqueness of mild solutions, considering various growth conditions of nonlinear terms. Finally, we presented an example to illustrate our main conclusions. Full article

Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method. Full article