Search Results (124)

This paper develops and rigorously analyzes a fully discrete upwind interior penalty discontinuous Galerkin (IPDG) scheme for simulating solute transport in two-dimensional variable-order fractional mobile–immobile media. The temporal variable-order Caputo derivative is discretized via a Grünwald–Letnikov approximation in conjunction with a first-order backward difference, while the spatial discretization employs an IPDG method featuring an upwind numerical flux for the convection term and a penalty formulation for the diffusion operator. Under the physically relevant assumption of a divergence-free velocity field, we establish the unconditional stability of the proposed scheme. A comprehensive error analysis in the $L^2$ norm yields a convergence rate of $O(\Delta t+h^{min(k+1,s)-\chi -1/2})$, explicitly linking the polynomial degree k, solution regularity s, and the penalty variant $\chi$. Numerical experiments in two dimensions are conducted to verify the accuracy and robustness of the proposed scheme in simulating anomalous transport phenomena in subsurface environments. Full article

Permeability models are essential for quantitatively describing coal permeability evolution and predicting gas migration during deep mining. Deep coal subjected to mining disturbance commonly exhibits pronounced nonlinear changes in permeability, limiting the applicability of conventional models. In this study, coal is idealized as a dual-component medium comprising the matrix and fractures, and deformation of both components induced by mining-related stress changes and gas adsorption is incorporated into the model. The conventional Weibull statistical damage variable is generalized to a fractional-order form using the Caputo derivative, yielding a Mittag–Leffler-type damage evolution law. By coupling this formulation with matrix–fracture deformation and an exponential damage–permeability term, a fractional-order damage-based permeability model is established to describe the complete evolution from elastic deformation through pre-peak damage to post-peak failure. The model parameters are calibrated separately using published datasets for protective-seam mining, top-coal caving, no-pillar mining, and a full-process loading case. The calibrated model yields coefficient of determination (R²) values of 0.9374, 0.9625, 0.9875, and 0.9980, respectively. The identified fractional order is λ = 1 for the three mining-disturbance datasets, whereas the full-process dataset yields λ = 0.7734. For the full-process dataset, the fractional-order model reduces root mean square error (RMSE) and mean absolute error (MAE) by approximately 31.4% and 34.7%, respectively, compared with its integer-order counterpart. Sensitivity analysis shows that λ, p, ε_d, and γ play distinct roles in permeability evolution. At an axial strain of 0.8%, increasing ε_d from 0.721% to 1.121% decreases k/k₀ from 2.6919 to 1.3433, whereas increasing γ from 0 to 2.543 increases k/k₀ from 1.0003 to 3.0334, indicating that ε_d and γ strongly affect the strain level and magnitude of permeability enhancement, respectively. The proposed model provides an effective tool for characterizing the nonlinear permeability evolution of deep coal under mining disturbance. Full article

Time-fractional diffusion equations describe subdiffusive transport in heterogeneous media, but their numerical treatment is complicated by the nonlocal Caputo derivative and by the weak singularity that the solution develops at the initial time. We study a global-in-time discretization that combines spectral collocation in time—on the fractional power basis ${\left\{t^{\mathsf{\ell }\alpha }\right\}}_{\mathsf{\ell }=0}^N$, evaluated at Chebyshev–Gauss–Lobatto nodes, which reproduces the leading terms of the singular expansion of the solution—with a second-order conservative finite-difference stencil in space that uses harmonic averaging of the diffusivity at the cell faces and therefore remains accurate across discontinuous media. The resulting fully discrete problem is a large, nonsymmetric, dense-in-time linear system whose two-norm condition number grows like the inverse square of the spatial mesh size, so that Krylov subspace iteration without preconditioning stalls under refinement. Exploiting the Kronecker sum structure of the discrete operator, we build a preconditioner by fast diagonalization of the spatial factor through the discrete sine transform. For constant diffusivity the preconditioner reproduces the operator exactly and yields a direct solver; for variable diffusivity it is spectrally equivalent to the operator, and we prove that the eigenvalues of the preconditioned system cluster in a disk centered at one whose radius depends only on the coefficient contrast, and not on the mesh, the number of temporal degrees of freedom, or the fractional order. Numerical experiments in one and two space dimensions confirm second-order spatial accuracy and a preconditioned iteration count that stays flat—twelve iterations from $M=32$ up to $M=1024$ in one dimension and eleven up to $M=256$ per direction in two—while the unpreconditioned count grows by more than two orders of magnitude. In time, the accuracy is spectral until round-off in the ill-conditioned Vandermonde matrix of the power basis takes over: the barrier is reached at $N=9,10,13$ for $\alpha =0.3,0.5,0.7$, where the attainable error is about ${10}^{-6}$. A benchmark against the L1 scheme on uniform and graded meshes, the Alikhanov L2-$1_{\sigma }$ scheme and Grünwald–Letnikov convolution quadrature quantifies when the global approach pays: on forced problems and on modes with $\kappa \lambda T^{\alpha }\lesssim 2$ it reaches a prescribed accuracy one to two orders of magnitude faster and with several times less memory, while for strongly damped modes the fractional power basis converges only algebraically and graded time marching is preferable below a relative error of ${10}^{-2}$. Full article

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

In this paper, we consider a nonlinear fractional superdiffusion equation involving a Caputo derivative of order $\alpha \in (1,2)$. Solutions to this class of problems typically exhibit weak singularities near the initial time. To handle this singularity, we introduce an auxiliary variable $p:=D_t^{\alpha /2}(u-t{u}_1)$ and reformulate the original problem as an equivalent coupled system. This system is then discretized using the nonuniform Alikhanov scheme in time, the conforming Galerkin finite element method in space, and Newton linearization for the nonlinear term. We prove that the numerical solutions remain uniformly bounded, independent of the temporal and spatial mesh parameters. Combining this result with a discrete fractional Grönwall inequality and a temporal-spatial splitting argument, we establish unconditional optimal error estimates of order $O(N^{-min\{2,r\alpha /2\}}+h^k)$. Finally, numerical experiments are presented to verify the sharpness of the theoretical convergence rates. Full article

This work constructs high-order numerical methods for a class of nonlinear fractional differential equations featuring the Caputo fractional derivative with respect to another function. Through variable transformation of the fractional differential operator, the original governing problem is converted into a weakly singular Volterra integral equation. After conducting the variable substitution $x=\varphi \left(t\right)$, the fractional integral term relative to $\varphi$ is restated into a standard fractional integral in the transformed variable. We then construct quadratic and cubic Lagrange interpolation approximations for the fractional integral operator on graded meshes. Special treatments are introduced near the initial point and on the last subinterval so that the resulting schemes can be implemented explicitly. Under suitable regularity assumptions allowing weak singularities, detailed error estimates are derived. The theoretical results show that, by choosing an appropriate grading parameter, the proposed methods can recover the expected convergence orders $\mathcal{O}\left(N^{-(3+\alpha )}\right)$ and $\mathcal{O}\left(N^{-4}\right)$, where $\alpha \in (0,1)$ is the order of the fractional derivative. Several numerical test cases are presented to validate the theoretical error estimates and demonstrate the effectiveness of the constructed schemes. Full article

In this paper, the convergence-to-zero and finite-horizon guaranteed-cost synchronization criteria are developed for linear Caputo–Hadamard fractional-order systems with an admissible time-varying delay. The delay assumption is expressed in such a way that is consistent with the Caputo–Hadamard Halanay inequality and the memory structure of the model, which is logarithmic in time. This analysis includes a quadratic Caputo–Hadamard Lyapunov method, Schur-complement bounds for the delayed channel and a supremum argument in logarithmic time. An important novelty in the proposed approach is that the current state, the delayed state and the Caputo–Hadamard derivative are not considered as independent augmented variables; this prevents the structural feasibility obstacle from occurring when using full-space residual LMI formulations. The convergence-to-zero condition is first established for the drive system. Next, a fixed-gain guaranteed-cost synchronization theorem is established and, by using a standard change of variables, a convex controller-synthesis condition is arrived at. An explicit logarithmic-time form of the finite-horizon cost estimate is derived. The criteria are further extended to systems with several admissible delays and to systems with norm-bounded parametric uncertainty. Four numerical examples are reported, in which feasible matrices, the controller gain, strict eigenvalue margins and a comparison of the simulated cost and the theoretical upper bound are given, together with a quantitative comparison against augmented-state linear matrix inequality formulations, a scalability study up to a dimension of 30 and a sensitivity study. The simulations show the dynamics that the theory predicts; the convergence to the asymptotics is valid for the LMI certificates checked in the simulations and for the Caputo–Hadamard Halanay inequality. In conclusion, the paper delivers a complete and numerically verifiable design chain for Caputo–Hadamard synchronization: admissibility of a possibly unbounded time-varying delay is checked directly, a stabilizing gain is obtained from a convex program whose largest block has size $2n$ instead of $3n$, and an a priori cost certificate $J_T^{*}$ is produced from the same feasible variables; on the reported benchmark, the method retains $95.7\%$ of the admissible delay-channel gain of an augmented-state formulation while solving up to 21 times faster at dimension 30. 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

In this paper, we investigate the stability analysis for a class of novel variable-order Caputo fractional-order dual switching systems. First, the short memory principle is adopted to construct the studied system model, where the Caputo fractional order is randomly time-varying, and the outer deterministic switching signal governs the overall dwell-time scheduling of subsystems. Under the designed event-triggered deterministic switching strategy, each fractional-order subsystem is characterized by an internal Markov random jumping processing. Secondly, combining the multiple Lyapunov functions method, fractional-order comparison lemma and average dwell time (ADT) technique, the corresponding sufficient stability criteria are established to guarantee the globally asymptotic stability almost surely (GAS a.s.) and the global Mittag–Leffler stability almost surely (GMLS a.s.). Finally, a numerical simulation example is presented to verify the feasibility and effectiveness of the derived theoretical results. Full article

In this paper, the issue of robust sensor fault estimation for Takagi–Sugeno (T–S) fuzzy systems with Caputo–Hadamard fractional-order dynamics subject to monotone nonlinearities is addressed. An adaptive observer is designed for the joint estimation of the system state and a globally constant sensor bias fault. The Caputo–Hadamard operator is used to handle logarithmic memory effects, and the T–S fuzzy representation is used for multi-regime nonlinear dynamics through a convex interpolation structure. Sufficient linear matrix inequality (LMI) conditions are obtained to ensure generalized Mittag–Leffler stability of the augmented estimation error system under a constant-fault assumption, by combining a sector inequality for strongly monotone nonlinearities with a fractional Lyapunov approach. The stability conditions are directly posed in the decision variables and the observer gains are recovered through a standard change of variables. To broaden the engineering applicability of the result, a finite-horizon practical Mittag–Leffler stability theorem is also derived for absolutely-continuous time-varying sensor faults whose Caputo–Hadamard derivative is bounded on the operating horizon $[t_0,T]$, in which the augmented estimation error remains in a residual ball whose radius is proportional to that bound. An alternative design, called a polynomial gain-scheduled observer, is also developed to reduce the conservatism of the constant-gain design, with observer gains given as polynomials of a measurable, fault-free scheduling vector. Quantitative root-mean-square performance metrics, LMI feasibility margins and an adaptation-gain sensitivity study are reported, and the polynomial matrix inequality is certified both by a dense grid check and by a sum-of-squares (SOS) feasibility argument so that the polynomial design is supported by a constructive certificate over the admissible scheduling set. Three numerical scenarios with fractional order $0.8$ are provided: a strict constant-bias scenario that exactly validates the LMI theorem, a bounded-derivative ramp scenario that validates the practical Mittag–Leffler theorem, and a polynomial gain-scheduled scenario that validates the polynomial observer. Full article

In this study, a comprehensive stochastic and fractional-order modeling framework is developed to investigate the dynamic behavior of a shunt DC motor under random disturbances and memory effects. The motor dynamics are formulated as a system of stochastic differential equations incorporating Gaussian noise to represent uncertainties in the electrical and mechanical subsystems. The existence, stochastic ultimate boundedness, stationary distribution, and ergodic properties of the proposed model are established. To further enhance modeling capabilities, a modified Atangana–Baleanu–Caputo (mABC) fractional operator is introduced, enabling the incorporation of nonlocal memory effects inherent in electromechanical systems. The series solution is derived using the Laplace transform and the Adomian decomposition method to handle nonlinearities. Qualitative analysis of the solution is performed through fixed-point theory, while stability assessments utilize the T-Picard method. The results of the numerical simulation indicate that the stochastic model exhibits limited variability around the operating regimes, whereas the fractional-order representation is more effective at smoothing transient responses and limiting oscillatory behavior. The study proposes a realistic and adaptable method to analyze the dynamics of shunt DC motors with uncertainty and also presents useful information for the design and control of electromechanical systems. Full article

Fractional optimal control problems provide an effective mathematical framework for modeling dynamical systems with memory, hereditary behavior, and anomalous diffusion effects. However, the nonlocal nature of Caputo fractional operators and the reduced regularity of fractional solutions pose significant challenges for the development of accurate and efficient computational methods. In this paper, we develop a spectral-fractional Physics-Informed Neural Network (Spectral-fPINN) framework for solving fractional optimal control problems governed by Caputo fractional differential equations. The proposed methodology combines normalized shifted Legendre spectral approximations, fractional operational matrix formulations, and physics-informed optimization within a unified computational framework. Unlike conventional PINN and fPINN approaches, which directly approximate the unknown solution variables, the proposed framework predicts the spectral coefficient vectors associated with the shifted Legendre basis functions, yielding a low-dimensional global representation with improved approximation efficiency. Caputo fractional derivatives are evaluated through spectral operational matrices, while the resulting optimization problem is discretized using Gauss–Legendre quadrature and solved through gradient-based optimization. In addition, a theoretical analysis of the proposed Spectral-fPINN framework is presented, including approximation, consistency, stability, and convergence results, together with error estimates and residual control properties. Several benchmark linear and nonlinear fractional optimal control problems are investigated to validate the proposed methodology. The numerical results demonstrate excellent agreement with exact solutions, very small residual errors, and rapid spectral coefficient decay, confirming the high-order accuracy and robustness of the proposed approach. Overall, the proposed Spectral-fPINN framework provides an accurate, stable, and computationally efficient methodology for solving a broad class of fractional optimal control problems. 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

Arnold tongues are wedge-shaped regions in parameter space associated with mode locking and synchronization phenomena in nonlinear dynamical systems. The Caputo fractional standard map extends the classical standard map by incorporating long-memory effects through fractional derivatives and is known to generate Arnold tongue structures as the fractionality parameter approaches unity. In this paper, we investigate the fractional standard map applied to matrix-valued state variables, with particular emphasis on systems governed by high-dimensional nilpotent matrices. We show that the interplay between fractional memory and nilpotent algebra produces hierarchical families of Arnold tongues associated with divergent dynamics. This phenomenon is not observed in either the classical standard map or the non-fractional standard map of nilpotent matrices alone. For idempotent matrices, the fractional standard map retains the same level of dynamical complexity as its scalar counterpart. For nilpotent matrices, higher-order terms induce coupling between the map coefficients, giving rise to substantially richer dynamical behavior. This combination of fractional memory and nilpotent algebra provides a systematic framework for studying higher-dimensional nonlinear dynamics beyond the scalar setting. To support numerical investigations, an efficient computational scheme for the auxiliary parameters is derived and calibrated using the H-rank algorithm, which provides a concise measure of algebraic complexity in sequences generated by dynamical systems. Numerical simulations reveal hierarchical structures of Arnold tongues of divergence together with characteristic divergence rates of the auxiliary parameters. The hierarchical level of a given auxiliary parameter is identified as a key quantity determining the algebraic complexity of the transient dynamics, with potential implications for information encoding in applications exploiting transient dynamical processes. Full article

This paper is concerned with the numerical solution of the variable-order time fractional advection–diffusion–reaction equation (VO-TFADRE) in two space dimensions. We first propose a Crank–Nicolson (C-N) discretization scheme based on central difference operators and L1 formula for space and time variables, respectively. Then, we apply the C-N scheme to construct a new algorithm, namely the explicit group (EG) method, for the model problem under consideration. The EG method utilizes the idea of small fixed-size groups of mesh points and comes with computational merits as compared with the C-N scheme. Stability and convergence analyses are given in this work. The resulting discretization leads to large sparse linear systems, which are solved using the Bi-CGSTAB iterative method. Numerical experiments demonstrate that both the C–N and EG schemes achieve accurate approximations, while the EG method significantly reduces computational time. To economize further on the computational cost, we propose a parallelized version of the EG method for solving the VO-TFADRE. Carried out numerical simulations reveal that the parallel algorithm is more efficient than the serial algorithm for solving the problem under consideration. Full article