Search Results (473)

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard, $\psi$-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives. Full article

In this paper, we aim to extend the bicomplex two-parameter Mittag-Leffler (M-L) function by introducing a new k-parameter. This results in the definition of the bicomplex k-M-L function with two parameters. This generalization offers more flexibility and broader applicability in modeling complex fractional systems. We explore its key properties, develop new theorems, and establish the corresponding k-Riemann–Liouville fractional calculus within the bicomplex setting for the extended function. Furthermore, we solve several fractional differential equations using the bicomplex k-M-L function with two parameters. The results prove the enhanced flexibility and generality of the proposed function, particularly in deriving fractional kinetic equations, offering novel insights beyond existing bicomplex fractional models. Full article

This paper aims to develop a rheological model with fewer parameters that accurately describes the primary and secondary creep behavior of wood materials. The models studied are grounded in Riemann–Liouville fractional calculus theory. A comparison was conducted between the constant-order fractional Zener model and the variable-order fractional Maxwell model, with four parameters each. Using experimental creep data from four-point bending tests on two tropical wood species, along with an optimization algorithm, the variable-order fractional model demonstrated greater effectiveness. The selected fractional derivative order, modeled as a linearly increasing function of time, helped to elucidate the internal mechanisms in the wood structure during creep tests. Analyzing the parameters of this order function enabled an interpretation of their physical meanings, showing a direct link to the material’s mechanical properties. The Sobol indices have demonstrated that the slope of this function is the most influential factor in determining the model’s behavior. Furthermore, to enhance descriptive performance, this model was adjusted by incorporating stress non-linearity to account for the effects of the variation in constant loading level in wood. Consequently, this new formulation of rheological models, based on variable-order fractional derivatives, not only allows for a satisfactory simulation of the primary and secondary creep of wood but also provides deeper insights into the mechanisms driving the viscoelastic behavior of this material. 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

This paper investigates the analytical properties of multiplicative generalized proportional $\sigma$-Riemann–Liouville fractional integrals and the corresponding Hermite–Hadamard-type inequalities. Central to our study are two key notions: multiplicative $\sigma$-convex functions and multiplicative generalized proportional $\sigma$-Riemann–Liouville fractional integrals, both of which serve as the foundational framework for our analysis. We first introduce and examine several fundamental properties of the newly defined fractional integral operator, including continuity, commutativity, semigroup behavior, and boundedness. Building on these results, we derive a novel identity involving this operator, which forms the basis for establishing new Hermite–Hadamard-type inequalities within the multiplicative setting. To validate the theoretical results, we provide multiple illustrative examples and perform graphical visualizations. These examples not only demonstrate the correctness of the derived inequalities but also highlight the practical relevance and potential applications of the proposed framework. Full article

This study aims to explore the effect of spatial-fractional derivatives and the multiplicative standard Wiener process on the solutions of the stochastic fractional regularized long-wave equation (SFRLWE) and contribute to its analysis. We introduce a new systematic method that combines the auxiliary function method with the complete discriminant polynomial system. This method proves to be effective in discovering precise solutions for stochastic fractional partial differential equations (SFPDEs), including special cases. Applying this method to the SFRLWE yields new exact solutions, offering fresh insights. We investigated how noise affects stochastic solutions and discovered that more intense noise can result in flatter surfaces. We note that multiplicative noise can stabilize the solution, and we show how fractional derivatives influence the dynamics of noise. We found that the noise strength and fractional derivative affect the width, amplitude, and smoothness of the obtained solutions. Additionally, we conclude that multiplicative noise impacts and stabilizes the behavior of SFRLWE solutions. 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

Starting from ${\psi }_k$-Raina’s fractional integrals (${\psi }_k$-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve ${\psi }_k$-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), k-Riemann–Liouville (k-R-L), $\psi$-Riemann–Liouville ($\psi$-R-L), ${\psi }_k$-Riemann–Liouville (${\psi }_k$-R-L), Raina’s, k-Raina’s, and $\psi$-Raina’s fractional integrals ($\psi$-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function. Full article

Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial fractional derivative is also defined, and its properties are discussed. The Laplace transforms (LTs) of the introduced fractional operators are evaluated. The Hyers–Ulam stability and the existence of a novel tempered fractional differential equation are examined. Moreover, a fractional integro-differential kinetic equation is formulated, and the LT is used to find its solution. A growth model and its graphical representation are established, highlighting the role of novel fractional operators in modeling complex dynamical systems. The developed mathematical framework offers valuable insights into solving a range of scenarios in mathematical physics. Full article

This paper investigates the oscillatory behavior of a class of mixed fractional-order nonlinear differential equations incorporating both the Liouville right-sided and conformable fractional derivatives. Symmetry plays a key role in understanding the oscillatory behavior of these systems. The motivation behind this study arises from the need for a more generalized framework to analyze oscillatory behavior in fractional differential equations, bridging the gap in the existing literature. By employing the generalized Riccati technique and the integral averaging method, we establish new oscillation criteria that extend and refine previous results. Illustrative examples are provided to validate the theoretical findings and highlight the effectiveness of the proposed methods. Full article

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter $H\in (1/2,1)$. The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order $\alpha \in (0,1)$ and the Riemann–Liouville time-fractional integral of order $\gamma \in (0,1)$, respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order $O({\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,\alpha \}}),\epsilon >0$. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method. Full article

The structure, bonding, and properties of diradicals, triradicals, and polyradicals have been investigated using broken symmetry (BS) molecular orbital (MO) and BS density functional theory (DFT) methods, which are regarded as the first steps in the mean-field approach toward strongly correlated electron systems (SCES). The natural orbital (NO) analyses of the BS MO and BS DFT solutions were performed to elucidate the natural orbitals of their occupation numbers, which are used for derivations of the diradical character (y) and several chemical indices for the open-shell molecules under investigation. These chemical indices are also obtained using SCES, the next theoretical step, which uses symmetry-recovered resonating BS (RBS) and multi-determinant methods such as multi-reference (MR) configuration interaction (CI) and MR-coupled cluster (CC) methods that employ the NOs generated in the first step. The nonlinear optical response properties of organic open-shell species were theoretically investigated with several procedures, such as MR CI (CC), the numerical Liouville, and Monte Carlo wavefunction methods, as the third step to SCES. The second-order hyperpolarizability (γ) of diradicals such as a phenalenyl radical dimer were mainly investigated in relation to the generation of quantum squeezed lights, which are used for the construction of the quantum entangled states for quantum optical devices such as quantum sensing and quantum computation. Basic quantum mechanical concepts, such as the Pegg–Barnett quantum phase operator, were also revisited in relation to the design and chemical synthesis of stable diradicals and polyradicals such as optical quantum molecular materials and future molecular qubits materials. Full article

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler–Lagrange equations for the FOCPs. Full article

In this work, we present the semi-analytical solution for the early-time recession phase of the horizontal unconfined aquifer of finite length for steady-state initial conditions. This is a case where self-similarity arguments are not applicable. The solution is built as linear perturbations from the initial steady state. The solution is determined via a Sturm–Liouville eigenvalue problem, which should be solved numerically. On the other hand, the immediate response of the aquifer to the sudden switching off the recharge, i.e., in the earliest times, is obtained by deducing analytically the large eigenvalue asymptotic solutions of the problem. We find analytically that in this time regime, the outflow Q is given by Q = Q₀ − 1.4Q₀^5/3L^−4/3n^−2/3t^2/3, where Q₀ is the initial outflow rate, L is the length of the aquifer, n is the porosity of the formation and t is the time from the start of the recession. The stated result is very accurate for times t up to ~0.01 nL^3/2k^−1/2Q₀^−1/2, where k is the hydraulic conductivity of the formation. The analytical and quantitative relation of the presented solution with the classical recession phase asymptotic solutions derived in the past by Polubarinova (early-time solution) and Boussinesq (separable, late-time solution) is discussed in detail. The presented results can be used as a benchmark solution for modeling or numerical validation purposes. Full article

We investigate how novelly generated solutions of the stochastic space-fractional Davey–Stewartson equations are affected by spatial-fractional derivatives and multiplicative Brownian motion (in the Stratonovich sense). These equations model the behavior of weakly nonlinear water waves on a fluid surface. By applying the qualitative theory of planar systems, some new fractional and stochastic solutions are obtained. These solutions gain significance from the application of Davey–Stewartson equations to the theory of turbulence for plasma waves, as they can explain several fascinating physical phenomena. Some solutions are graphically displayed to illustrate the influence of noise strength and fractional derivatives on the obtained solutions. These effects influence the solution’s amplitude and width, as well as its smoothness. Full article