Search Results (235)

Four-dimensional chaotic systems have garnered significant attention due to their complex nonlinear dynamics, high-dimensional complexity, and wide range of applications in science and engineering. This paper proposes and investigates a novel four-dimensional chaotic system in both constant- and variable-order frameworks to reveal the influence of memory effects on its dynamical behavior. The variable-order formulation is established using the Liouville–Caputo fractional derivative, while an efficient numerical scheme based on Lagrange interpolation is developed to approximate the variable-order derivative accurately. A rigorous local stability analysis is first conducted to characterize the equilibrium points and establish their instability and non-hyperbolic nature under the different derivative formulations. The nonlinear dynamics of the proposed system are then comprehensively examined through phase portraits, time series, bifurcation diagrams, and Lyapunov exponent analysis. The results demonstrate that the variable-order model preserves the fundamental topological characteristics of the chaotic attractors while introducing adaptive transient responses and significantly richer dynamical behaviors than the corresponding constant fractional-order model. Furthermore, the proposed system generates previously unreported chaotic attractors and phase-space patterns, enriching the class of known four-dimensional chaotic systems and demonstrating the variable-order framework’s enhanced capability to produce diverse nonlinear phenomena through adaptive memory effects. 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

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

We consider an impulsive BVP related to the Hilfer fractional derivatives and the nonlinear p-Laplacian operator. The type parameter $\vartheta \in [0,1]$ is kept unchanged in the formulation, and the Riemann–Liouville and Caputo cases are obtained as limiting cases of the formulation, not as separate cases. The variational functional is then built by adding the point-impulse contribution to the distributed potential and the use of an appropriate space of the Hilfer fractional derivative. Using variants of the fountain theorem, we prove the existence of two infinite sequences of weak solutions, one of which is of unbounded energy and another of which is of small energy and tends to zero from below. The weak residual based stability analysis is further developed, and local generalized Hyers–Ulam and Hyers–Ulam–Rassias stability estimates are obtained. Because of multiplicity of solutions, a uniqueness-based argument for stability, Ulam’s approach, is not possible and stability is instead achieved by providing residual-based arguments.The assumptions are verified through illustrative examples. Lastly, we examine the convergence behavior, residual decay, and effect of the Hilfer type parameter in conjunction with a Hilfer-type parameter neural surrogate with boundary constraints based on a discrete Hilfer scheme. The study, in general, proves a link between the solution multiplicity, residual stability, and the numerical realization in one impulsive fractional p-Laplacian framework. 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

We develop a terminal-explicit geometric theory for the algebraic zero sets associated with the Riemann–Liouville and Caputo fractional derivatives of a complex polynomial. Expanding the polynomial about an arbitrary complex terminal a reduces both operators to gamma-weighted polynomial transforms. The coordinate shift $w=z-a$ is algebraically equivalent to a zero-terminal formulation; the terminal dependence re-enters through the shifted coefficients and through the pullback of the resulting geometry to the original z-plane. We derive exact barycenter identities, terminal-centered disk bounds, multiset convergence at the endpoint orders, first-order deformation formulas for simple roots, and solvable examples. The disk bounds provide a star-shaped localization framework rather than a convex-hull theorem. A regular-polygon family yields an exact homothety for the Riemann–Liouville roots, while numerical examples show curved generic trajectories and a nondegenerate Caputo evolution. The terminal therefore acts as a distinguished geometric anchor, with radial attraction occurring under additional algebraic symmetry. 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

The interleaved parallel Buck–Boost converter can reduce output voltage ripple and has been widely used in engineering practice. The application of fractional-order theory has a significant influence on model accuracy and power converter performance. Based on fractional calculus theory and the state space averaging method, this paper establishes a fractional-order mathematical model of the CCM interleaved parallel Buck–Boost converter. The steady-state operating point and ripple characteristics of the converter under the Caputo fractional-order definition are analyzed and compared with those under other fractional-order definitions. Fractional-order energy storage elements are constructed, and a fractional-order circuit simulation model of the converter is established for comparative simulation analysis. Finally, experiments are carried out to verify the effectiveness of the theoretical analysis. Full article

We investigate the existence and uniqueness of solutions for two-point boundary value problems involving left and right Caputo fractional derivatives with possibly complex-valued coefficients on a finite real interval. Under suitable assumptions on the fractional order and the coefficients, we establish the existence and uniqueness of continuous weak solutions. In addition, under further $L^2$-type assumptions on the coefficients, we prove the existence of weak solutions in the Hilbert space $L^2(a,b)$. The analysis is based on an equivalent fractional integral formulation and the Banach–Caccioppoli fixed-point theorem. Full article

In this work, a new 4D chaotic system is presented with Liouville–Caputo fractional derivatives, including constant-order (C) and variable-order (V). In this regard, the paper examines the equilibrium of the system, local stability, and dissipative property. The use of variable order through incorporating time-dependent fractional order in the system, which includes periodic and exponential functions, is an appropriate way of simulating the adaptive memory effect of the fractional-order systems. The analytical and numerical studies reveal that the proposed system is able to show the chaotic behaviour and sensitive dependence on parameters. Specifically, in terms of maximum Lyapunov exponent and Kaplan–Yorke dimension, the performance of the variable-order system is superior to the constant-order one, reaching the values of ${\lambda }_{max}\approx 0.275$ and $D_{KY}\approx 2.223$. The approach presented here offers a more realistic framework to model memory-dependent chaotic systems and finds applications in the design of nonlinear circuits, secure communications, neuromorphic computing, and advanced control systems. 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 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

This study investigates the Mittag–Leffler-type stability properties of fractional perturbed systems with respect to their unperturbed counterparts by incorporating initial time differences into the analysis. In contrast to many existing studies in which initial time effects are neglected, the proposed framework explicitly considers time shifts together with the memory-dependent nature of fractional-order systems. Using Caputo fractional derivatives and Lyapunov-type functionals, new sufficient conditions are established for the stability behavior of perturbed systems relative to the corresponding unperturbed systems under shifted initial times. The obtained results extend existing stability criteria by simultaneously addressing fractional memory effects, perturbation terms, and variations in the initial time. To illustrate the applicability and effectiveness of the theoretical findings, representative examples, numerical simulations, graphical comparisons, and global error analyses are presented. The numerical part is based on the Caputo framework and is further supported by benchmark comparisons involving Riemann–Liouville and shifted Grünwald–Letnikov approaches. The proposed results provide a useful framework for the stability analysis of memory-dependent dynamical systems arising in engineering and applied sciences. Full article

A non-homogeneous fractional differential equation with a variable coefficient involving a Caputo fractional derivative is considered. The equation is first transformed into an integral equation and then solved using the method of successive approximations. The obtained general solution involves a generalized Mittag–Leffler-type function and Meijer G-functions. Example solutions corresponding to particular choices of the non-homogeneous term are presented. As an application of the considered non-homogeneous equation, direct and inverse source problems are studied. The solutions are expressed in the form of series expansions using an orthogonal basis obtained through separation of variables. Illustrative examples for the direct and inverse problems are also presented for specific choices of the initial and final time data and the source function. Full article

This paper investigates the h-stability of nonlinear Hilfer nabla fractional difference equations, which interpolate between the Riemann–Liouville and Caputo nabla fractional differences through an additional type parameter. To the best of our knowledge, this is the first work devoted to the h-stability analysis of nonlinear Hilfer nabla fractional difference equations. By employing the properties of generalized nabla fractional sums and Taylor monomials together with a comparison-based approach, we establish new sufficient conditions guaranteeing the h-stability of the zero solution under suitable growth assumptions on the nonlinear term. Furthermore, we show that the associated solution map is differentiable with respect to the initial condition, providing a sensitivity framework for Hilfer-type discrete fractional systems. As another contribution, we derive a discrete variation in parameters formula for perturbed Hilfer nabla fractional difference equations, yielding an explicit representation of perturbed solutions. The obtained results extend the existing stability theory for discrete fractional systems and provide a unified framework encompassing both the Riemann–Liouville and Caputo nabla settings as particular cases. Full article