Search Results (509)

Three-dimensional Riemann–Liouville time-fractional integro-differential equations provide useful prototype models for diffusion and transport processes with temporal memory and weakly singular hereditary effects. Their numerical solution is challenging because of the nonlocal fractional derivative, history-dependent fractional integral term, and large-scale discrete systems arising from three-dimensional spatial discretization. In this work, an efficient high-order compact finite difference scheme is developed for solving such problems. The Riemann–Liouville fractional derivative is approximated by the weighted and shifted Grünwald difference formula, the fractional integral term is discretized by the product trapezoidal formula, and the Laplace operator is approximated by compact difference operators. The proposed scheme achieves second-order accuracy in time and fourth-order accuracy in space. Moreover, the solvability, stability, and convergence of the fully discrete three-dimensional scheme are analyzed under suitable regularity assumptions. Numerical experiments, including examples with smooth and non-smooth solutions, verify the theoretical convergence orders and demonstrate the effectiveness of the proposed method for different fractional parameters. Full article

This paper investigates the qualitative and topological behavior of random solutions for a class of partial fractional random differential equations governed by the Darboux problem. Unlike classical configurations that rely on bounded or finite delays, our theoretical framework explicitly addresses systems involving unbounded infinite delay. The dynamics of the state transitions are formulated using left-sided mixed Riemann–Liouville fractional integrals and joint Caputo fractional derivatives of order $\epsilon =({\epsilon }_1,{\epsilon }_2)\in (0,1]\times (0,1]$. Because of the infinite historical horizon, the underlying model is constructed and analyzed within abstract, semi-normed axiomatic phase spaces defined over topological Fréchet spaces. By avoiding restrictive compactness assumptions on the nonlinear operational bounds, we establish novel random mild existence theorems. The structural proofs are achieved through a combination of a regular, sublinear family of axiomatic measures of noncompactness and an advanced generalization of the classical Darbo fixed-point theorem tailored for Fréchet domains. Finally, a concrete mathematical example is systematically analyzed to confirm the validity, consistency, and practical applicability of the established theoretical bounds. Full article

In this paper, we investigate the existence of positive solutions for a class of p-Laplacian tempered Riemann–Liouville fractional equations involving signed measures and sign-changing perturbations with Riemann–Stieltjes boundary conditions. By performing a spectral analysis of the associated linear operator and applying Gelfand’s formula, we establish several fundamental properties of the principal eigenvalue for the corresponding linear fractional-order differential equations. Furthermore, under suitable growth conditions on the nonlinear terms, we employ the fixed point index theory to derive new existence results for positive solutions of the proposed equations. Full article

This investigation is concerned with a time-fractional diffusion equation governed by the Hilfer fractional derivative of order $1<{\rho }_1<2$, subject to homogeneous Neumann boundary data and generalized initial conditions of the Hilfer type. This paper addresses three interconnected problems: a direct problem and two inverse source problems (ISPs). In the first ISP, the objective is to recover an unknown space-dependent source function from measurements taken at a specified final time. In the second ISP, the goal is to determine an unknown time-dependent coefficient through an integral-type over-specification condition. By employing eigenfunction expansions in conjunction with the LT technique, we derive explicit series representations of the solutions in terms of the Mittag-Leffler function. Rigorous existence and uniqueness results for classical solutions are established for all three problems. The second ISP is reformulated as a Volterra integral equation, whose unique solvability is demonstrated via the Banach fixed point theorem. Both ISPs are shown to be ill-posed in the Hadamard sense, indicating instability with respect to data perturbations. Numerical experiments are also presented to validate the theoretical findings and to illustrate the performance of the proposed reconstruction methods. As limiting cases, the formulations corresponding to the Riemann–Liouville and Caputo fractional derivatives are recovered, illustrating the generality of the proposed framework. Full article

This paper investigates inverse spectral problems for a class of discrete Sturm–Liouville operators with multiple transmission interfaces. The presence of several interfaces generates a coupled spectral structure in which the reconstruction of the operator coefficients is determined by a common generalized spectral function. A generalized spectral framework adapted to the multi-interface setting is introduced, and explicit reconstruction formulas for the associated tridiagonal coefficient matrix are derived. Using the corresponding Hankel moment determinants, it is shown that the interface coefficients are spectrally linked through the same moment sequence, producing a globally coupled reconstruction mechanism. In contrast to the single-interface case, the reconstruction cannot be decomposed into independent local procedures. A constructive recovery algorithm for the operator coefficients is obtained, and the role of the transmission parameters in the spectral representation is analyzed. An example illustrating the reconstruction process for multiple interfaces is also presented. Full article

In this paper, we study the k-Kummer hypergeometric function $M_k(a,c;w)$, a generalization of the classical Kummer confluent hypergeometric function, arising as a solution of the k-confluent hypergeometric differential equation. We establish a Kummer-type transformation formula and derive derivative identities, contiguous relations, and addition and multiplication formulas. In addition, we obtain closed-form expressions for the Laplace, Mellin, Stieltjes, Sumudu, and Riemann–Liouville fractional integral transforms involving $M_k(a,c;w)$. The results presented here extend the classical theory of confluent hypergeometric functions to the k-generalized setting and provide analytic tools for further investigations of generalized differential equations and integral transforms. Full article

The Special Issue “Advances in Boundary Value Problems for Fractional Differential Equations—2nd Edition” presents recent advances in the theory and applications of fractional differential equations, fractional inclusions, and systems of fractional differential equations involving Riemann–Liouville, Caputo, Hadamard, Hilfer-Hadamard and other generalized fractional derivatives under a variety of boundary conditions [...] Full article

This paper develops a unified qualitative framework for four classes of Liouville–Caputo fractional difference equations arising from different combinations of fractional sums and integer-order differences. Based on the equivalence between initial/final value problems and Volterra-type summation equations, sufficient conditions for the existence and uniqueness of solutions are established by applying the Banach contraction mapping principle together with refined combinatorial estimates. Furthermore, the continuous dependence of solutions on prescribed initial or final data is investigated. By deriving explicit error estimates through a discrete fractional Gronwall-type inequality, we prove that Lipschitz solutions depend continuously on perturbations of boundary data. Numerical experiments for a representative case are presented to verify the theoretical results, including the influence of the fractional order and the sensitivity with respect to boundary data, while additional examples illustrate the applicability of the framework. The obtained results extend the unified discrete fractional calculus framework by providing a rigorous well-posedness analysis and offering a theoretical foundation for further applications of discrete fractional models with memory effects and diverse boundary conditions. Full article

In this study, we develop a novel variable-order fractional extension of an eight-dimensional (8D) hyperchaotic differential equation system modeled via the Liouville–Caputo operator. Moving beyond constant fractional-order models, our system implements time-variable orders, which enable its historical memory structure to evolve dynamically over time. To numerically approximate the trajectories of this complex 8D system, a second-order Lagrange numerical integration approach is formulated. An extensive dynamic analysis explores the behavior of this variable-order framework under two distinct configurations: a slowly periodic memory function and a smooth, monotonic hyperbolic tangent function. Topological complexity and multidimensional chaos are characterized using parameter-dependent bifurcation diagrams, phase portraits, Kaplan–Yorke fractal dimensions, and Kolmogorov–Sinai metric entropy. Numerical results show that both variable-order configurations display robust hyperchaotic dynamics characterized by four positive Lyapunov exponents. Crucially, the proposed variable-order extension enhances the phase space footprint of the baseline system, achieving a maximum Kaplan–Yorke dimension of 7.100, thereby offering excellent topological density for secure cryptographic applications. Full article

This paper aims to extend the averaging principle for first order fuzzy stochastic differential equations to Riemann–Liouville fuzzy fractional stochastic differential equations, addressing the research gap in this field. This extension enables the averaging principle for fuzzy stochastic systems to cover fractional scenarios with memory, providing a simplified analytical tool for practical systems involving both “randomness” and “fuzziness” uncertainties. Full article

Quantum mechanics is among the most successful physical theories, yet its formulation and empirical testing rely on classical structures. Following Lev Landau and Niels Bohr, this reliance is not merely pragmatic: quantum observables acquire empirical meaning only relative to classical reference frames, and, in practice, quantization starts from classical models. At the same time, the two domains display forms of mutual irreducibility: intrinsically quantum features (that is, spin and exchange statistics) have no counterpart in the phase-space ontology of classical point-particle mechanics, while classical trajectory chaos does not arise straightforwardly from unitary quantum evolution in closed systems. A hierarchy is commonly established between classical and quantum theories, namely, a claim of ontological and explanatory priority according to which quantum mechanics is fundamental and classical mechanics is only a limiting case. This claim is less secure than is often assumed; therefore, the traditional hierarchy deserves to be examined. In this paper, we argue that a quantum–classical framework provides an effective and structurally faithful representation of empirically accessible physical systems in regimes where quantum and classical degrees of freedom coexist within a single, consistent effective dynamical description. To give this point of view a firm theoretical basis, we discuss the quasi-Lie formal structure underlying quantum–classical hybrid dynamics, with applications ranging from gravity and condensed matter to open, driven systems in biology and complex media. Full article

Switched fractional differential equations with Riemann–Liouville fractional derivatives with respect to another function (RLDFs) with orders between zero and one are studied. We provide a detailed algorithm for constructing the solution of the studied switched fractional system, and we illustrate with several examples the influence on both the switching rules and the applied function in the fractional derivative on the behavior of the solution. We define generalized Lipschitz stability in time for the studied system, and this type of stability guarantees that the solutions will be close to the initial values on intervals excluding the points of singularities, i.e., the initial time point and the switching time points. We prove auxiliary results for Lyapunov functions and their RLDFs, including quadratic Lyapunov functions. These results are applied to study generalized Lipschitz stability in time. Full article

The classical Black–Scholes model assumes Markovian dynamics and cannot capture the long-range dependence and gradual memory decay observed in real markets. We formulate the two-dimensional time-fractional Black–Scholes equation for a European put on a weighted basket of two correlated assets under the Atangana–Baleanu–Caputo (ABC) derivative, whose non-singular Mittag-Leffler kernel models distributed, fading memory more faithfully than the singular Riemann–Liouville and Caputo kernels and the localized Caputo–Fabrizio kernel. A closed-form approximate analytical solution is derived via the Laplace homotopy perturbation method. We prove a convergence theorem with an explicit geometric error bound, and show that the series solves the associated Atangana–Baleanu integral equation exactly and the differential equation up to an explicit, decaying initial-layer term that vanishes as $\xi \to 1$. We further prove that, for the basket payoff, the closed-form price is independent of the inter-asset correlation. The solution reduces to the classical two-asset price deep in the money as $\xi \to 1$, agreeing with a Monte Carlo benchmark to within $0.1\%$ in that regime, where the approximation is valid. The contribution combines three elements: the two-asset setting, the non-singular Mittag-Leffler kernel, and a closed-form solution. Full article

In this paper, we employ the Nash–Moser iteration technique and the Saloff–Coste’s Sobolev inequality to study the local and global properties of positive solutions to the Trudinger equation ${\Delta }_{p,f}{u}^{{\displaystyle \frac{1}{p-1}}}+b{u}^q+c{u}^r=0$ on a complete smooth metric measure space with m-Bakry-Émery Ricci curvature bounded from below, where $b,c\in \mathbb{R}$, $p>1$, and $q\le r$ are real constants. We first give universal gradient estimates for the above equation under certain assumptions on b, c, p, q, and r. As their natural corollary, Harnack inequalities and Liouville-type theorems for positive solutions are obtained. Later, we consider the explicit global gradient estimates for such entire solutions through the global gradient estimates obtained. Full article

In this study, a detailed analytical-numerical study of the complex modified Korteweg–De Vries (mKdV) model with truncated M-fractional derivative is carried out to investigate the effects of the fractional order on nonlinear wave propagation. The fractional partial differential equation is solved by an appropriate fractional traveling wave transformation, which transforms it into a nonlinear ordinary differential equation. Two very powerful analytical methods are then used: the modified sub-equation method and the Kumar–Malik method, which give the exact closed-form solutions. The obtained semi-analytical numerical approximations are then obtained from the Differential Transformation Method (DTM). Bright and dark solitons, kink-type waves, periodic and rational solutions, exponential solutions, and Jacobi elliptic functions are found for a variety of parametric regimes. Explicit compatibility conditions and parametric constraints, which control the amplitude, width, and propagation, are derived. The DTM approximations are found to converge to the exact solutions with good accuracy, and the absolute errors are almost negligible, which validates the accuracy of the approximations and reliability of the solution. The three-dimensional visualizations of surface plots, two-dimensional profiles, and contour visualization further illustrate the dispersive dynamics and stability properties. Significance: This study shows that the truncated M-fractional derivative is a good operator to model memory-dependent nonlinear wave propagation. A new precise solution and reliable validation methods have been obtained for high-dimensional fractional nonlinear evolution equations in the hybrid analytical-numerical framework, which can be useful in plasma physics, nonlinear optics, and complex media. The present study contains restrictions for constant coefficients, a specific parametric regime, one fractional derivative definition, and experimental validation is not included. Future directions are limitations on constant coefficients, specific parametric regimes, one fractional derivative definition, and experimental validation is not included. The approach is to be extended in the future to variable coefficients, other fractional operators (Caputo, Riemann–Liouville), and to higher-order nonlinearities, and then to be experimentally tested in optical or plasma systems. Full article