Search Results (1,899)

In this paper, a finite difference scheme is proposed for the variable-order time-fractional sub-diffusion equation, achieving second-order accuracy in time and sixth-order accuracy in space. For spatial discretization, a newly constructed operator $\mathcal{A}$ is employed to obtain a sixth-order compact approximation of the second derivative. Using an energy analysis method, a priori estimates of the scheme are derived, and the unconditional stability and convergence are rigorously proved. Numerical examples are provided to verify the theoretical accuracy of the scheme. Full article

This paper investigates the fractional calculus of distributions supported on ${\mathbb{R}}^{+}$ in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractional derivative terms with orders in the interval $(0,2)$. The existence and uniqueness of solutions for its nonlinear form are established in a space of continuous functions by applying Banach’s contraction principle, the Leray–Schauder fixed-point theorem, inverse operators, and the multivariate Mittag–Leffler function. Finally, several examples are presented, in which the values of multivariate Mittag–Leffler functions are computed to illustrate the main results. Full article

This paper probes into the propagation characteristics of Rayleigh waves in a partially saturated, porous, thermo-viscoelastic half-space, with full consideration of the fractional viscoelastic effect and thermal coupling effect. A fractional Zener model is introduced to depict the thermo-viscoelastic mechanical behavior of the solid skeleton by constructing a complete set of governing equations that include mass balance, generalized Darcy’s law, momentum balance, and generalized heat conduction. Field equations are derived by means of Helmholtz vector decomposition, and the dispersion equation, and the phase velocity expression of Rayleigh waves are obtained by combining the traction-free and adiabatic boundary conditions of the medium. The impacts of key material properties, such as medium saturation, intrinsic permeability, medium viscoelasticity, and thermal expansion coefficient, on the dispersion feature of Rayleigh waves are discussed in detail. Numerical analysis results show that an increase in the thermal expansion coefficient will lead to a rise in Rayleigh wave phase velocity, in which the increase in P1 compressional wave velocity plays a dominant role among the velocities of various types of waves. Meanwhile, the attenuation coefficient of Rayleigh waves presents a decreasing trend and gradually tends to be stable with the growth of the thermal expansion coefficient. Similarly, the phase velocity of Rayleigh waves also increases with the rise in fractional order index, which is jointly dominated by the velocity enhancement of P1 waves and S waves. In addition, the attenuation coefficient of Rayleigh waves increases first and then decreases with the increase in fractional order index and reaches the peak value when the fractional order index is about 0.4. The research results reveal the influence of laws of thermal expansion characteristics and viscoelasticity on Rayleigh wave propagation and provide theoretical support for the analysis of wave propagation characteristics in porous media in relevant engineering applications. Full article

Nonlinear equations arise extensively in engineering and applied sciences, where efficient and reliable iterative solvers are required. This study introduces two fractional-order iterative schemes based on a common predictor–corrector structure: a Caputo-based method, NCFS${}_1$, and an Atangana–Baleanu–Caputo (ABC)-based variant, NFS${}_1^{abc}$. The proposed schemes incorporate a fractional order and two tunable parameters to improve flexibility in the iterative process. The local convergence behavior of the Caputo-based method is analyzed by means of fractional Taylor expansions, yielding an explicit error equation and convergence order, while analogous asymptotic considerations are discussed for the ABC-based variant. A dynamical-systems analysis is also performed through basins of attraction, the Convergence Area Index, and the Wada measure. Numerical experiments on application-motivated nonlinear models indicate that the proposed methods can provide faster error reduction, smaller residuals, and lower computational cost than selected existing fractional iterative schemes. These results suggest that the proposed framework is a flexible and effective approach for nonlinear root-finding problems, combining local convergence analysis with global dynamical assessment. Full article

In this paper, an efficient numerical framework combined with RK4 method and Richardson extrapolation is proposed to solve nonlinear time-dependent partial differential equations involving the Riesz fractional Laplacian operator ${(-\Delta )}^s$. The RK4 method guarantees fourth-order temporal accuracy and L-stability, whereas the spatial fractional operator is discretized using a second-order central finite difference scheme. Based on the consistency conditions of the underlying spatial discretization, and by constructing a Vandermonde matrix to determine the extrapolation coefficients, novel high-order Richardson extrapolation formulas are derived, achieving a maximum convergence order of $\mathcal{O}\left(h^{2n}\right)$. Numerical experiments, covering 1D variable-coefficient cases, 2D cases with equal/unequal spatial steps, and 3D equidistant differencing cases, demonstrate that the proposed method stably upgrades the convergence order from second-order to fourth-order and further to sixth-order under oscillatory and nonlinear variable-coefficient conditions, with the extrapolated numerical errors reduced to the magnitude of ${10}^{-13}$. Asynchronous convergence observed in 2D unequal-step cases validates Theorem 3, while fourth-order convergence is achieved via extrapolation in 3D complex domains. This method possesses prominent advantages of high accuracy, strong robustness, and high efficiency, breaking through the dimensionality and convergence order limitations of traditional high-precision numerical algorithms. Full article

Solving fractional differential equations using spectral collocation methods based on classical orthogonal polynomials often leads to a reduced convergence rate due to the limited regularity of the solutions. Therefore, spectral collocation methods that employ non-smooth orthogonal functions are frequently preferred for solving various fractional differential equations. This study focuses on solving one- and two-dimensional time-fractional diffusion-wave equations (DWEs). A spectral collocation technique is developed based on fractional-order orthogonal Jacobi functions to approximate the time-fractional derivatives and orthogonal Jacobi polynomials in the spatial directions. For the first time, a fractional-order orthogonal Jacobi functions-based operational matrix is derived and combined with an orthogonal Jacobi polynomials-based operational matrix of second-order derivatives to solve one- and two-dimensional time-fractional DWEs. Three test problems are conducted to evaluate the efficiency of the proposed numerical technique. Full article

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior. Full article

This work develops an analytical framework for nonlinear fractional partial differential equations that combine Kirchhoff-type terms, double-phase operators, and $\psi$-Hilfer fractional derivatives. This paper investigates two classes of problems involving variable-exponent growth conditions. The first problem analyzes general nonlinear sources and formulates the solution as a fixed point of a nonlinear operator. Precisely, by proving that the functional energy is coercive, hemicontinuous, and strictly monotone, we establish the existence and the uniqueness of weak solutions via monotone operator theory. The second problem incorporates a convection-type nonlinearity, which breaks variational structure and requires the more robust theory of pseudomonotone operators. Under suitable growth and mixed-order assumptions on the nonlinearity, we prove the existence of at least one weak solution. The main tools are grounded in variable-exponent Lebesgue and Musielak–Orlicz–Sobolev spaces, with compact embeddings, modular estimates, and fractional integral identities playing a key role in the proofs. We note that the results contribute to the mathematical modeling of phenomena involving nonlocal elasticity, viscoelastic materials, phase-transition media, and fractional dynamical systems where the stiffness of the medium depends on the total deformation (Kirchhoff effect) and the energy density alternates between distinct growth regimes (double-phase). The $\psi$-Hilfer derivative enhances the scope by enabling models with tunable memory and hereditary effects. Full article

In this article, we use truncated M-fractional derivatives to analyze the bifurcation and chaotic behavior of and traveling-wave solutions to the Zhiber–Shabat equation. By introducing truncated M-fractional derivatives, the equation exhibits richer dynamic properties. Based on phase diagram analysis and dynamical system theory, the bifurcation behavior of the equilibrium point of a two-dimensional dynamical system is discussed. At the same time, the dynamical behavior of a two-dimensional dynamical system with periodic disturbances is considered, revealing the complex chaotic phenomena of the system under specific parameters. A planar phase diagram, a three-dimensional phase diagram, a sensitivity analysis, and a maximum Lyapunov exponent diagram of the perturbed two-dimensional dynamical system were employed. Furthermore, various forms of accurate analytical solutions were obtained through traveling-wave transformation and numerical simulation. The three-dimensional, two-dimensional, density, and polar coordinates of the solutions were plotted using mathematical software. The results indicate that the fractional order and system parameters have a significant impact on the morphology and chaotic characteristics of the solution. This study provides new theoretical insights into the nonlinear dynamics of fractional-order Zhiber–Shabat equations. Full article

A 7224C high-speed angular contact ball bearing used in a multi-wire sawing machine is selected as the research object to investigate the cage dynamic characteristics under oil-lubricated operating conditions. First, in order to determine the oil-phase volume fraction on the cage surface, a fluid-domain model of the bearing cavity is established, and numerical simulations are performed using the VOF multiphase-flow method coupled with the RNG k-ε turbulence model. The effects of the guiding clearance, pocket clearance, and rotational speed are analyzed, and a regression equation for the cage-surface oil-phase volume fraction is developed based on a uniform test design. Subsequently, a bearing dynamic model is constructed, in which lubrication-related parameters are determined based on the regression equation, and the force balance and equations of motion for each component are derived. Finally, using the slip ratio and the deviation ratio of the cage-centroid whirl velocity as evaluation indices, the influences of multiple parameters on cage stability are examined. The results indicate that increasing the clearances and rotational speed leads to a higher slip ratio, whereas increasing the axial and radial loads reduces the slip ratio. Moreover, enlarging the guiding clearance and increasing the axial load improve cage stability, while a larger pocket clearance and an excessively high radial load deteriorate cage stability. Full article

This paper studies a nonlinear Schrödinger equation with a variable-order time-fractional derivative. Because classical L1 and L1-2 schemes are not directly applicable to variable-order fractional operators, an improved L1-2 discretization with dynamically updated convolution weights is developed based on the Coimbra-type definition, in which the fractional order is evaluated at the current time. By combining the proposed temporal approximation with the Galerkin finite element method for spatial discretization and a linearized extrapolation technique for the nonlinear terms, a fully discrete numerical scheme is constructed. The unconditional stability of the scheme is rigorously proven, and optimal error estimates are established under a mild time step restriction. Numerical experiments are presented to confirm the theoretical results and to demonstrate the effectiveness of the method in capturing the influence of time-dependent memory effects on wave propagation. A key numerical observation is that stronger memory effects may suppress wave packet evolution, which is qualitatively reminiscent of a Zeno-like inhibition phenomenon. Full article

This study focuses on establishing the existence and uniqueness of solutions for nonlinear fractional differential equations through the application of fixed-point methods in complex-valued suprametric spaces. In order to accomplish this, novel cyclic and interpolative contractive conditions are formulated within the complex-valued suprametric setting, leading to the derivation of several common fixed-point theorems. The obtained results extend and encompass a variety of known fixed-point theorems in complex-valued metric spaces as particular instances. In addition, meaningful and non-trivial examples are presented to highlight the effectiveness and practical relevance of the developed theoretical framework. Full article

Singular behavior near the initial times, arising from the presence of fractional-order derivatives in fractional differential equations, often leads to the deterioration of solutions obtained using spectral methods when applied based on classical orthogonal polynomials. Consequently, instead of relying on smooth polynomials as the basis for spectral approaches, significant attention has been devoted to developing appropriate nonsmooth functions that can form the basis for various spectral methods to address the limitation imposed by the fractional-order derivatives. In this study, we develop a numerical framework for solving one- and two-dimensional time-fractional coupled FitzHugh–Nagumo (FHN) models. We construct a novel, time-nonsmooth but spatially smooth function, called the orthogonal shifted Chebyshev function, in both one- and two-dimensional dimensions, that serves as the basis of the spectral collocation approach. Furthermore, we derive novel operational matrices of second-order and fractional-order derivatives, based on the new basis function in the spatial and time directions, respectively. These matrices are then used in conjunction with the spectral collocation technique in both spatial and time directions to reduce the problem to a system of algebraic equations. The numerical results demonstrate the accuracy of the presented numerical scheme and confirm the superiority of the new basis over the classical shifted Chebyshev polynomials when applied to time-fractional models. Full article

We develop a high-order space-time spectral method for nonlinear convection–diffusion equations with a Riemann–Liouville time-fractional derivative and a spectrally defined space-fractional Laplacian. The spatial discretization uses a Fourier spectral method that diagonalizes the fractional Laplacian under periodic boundary conditions. The temporal discretization employs a Petrov–Galerkin method based on generalized Jacobi functions which capture the initial singularity exactly. The nonlinear convection term is treated pseudo-spectrally, and the resulting algebraic system is solved with a damped Newton iteration. Rigorous error analysis proves exponential convergence in both space and time. Numerical experiments for various fractional orders confirm the spectral accuracy. Simulations of the fractional Burgers equation demonstrate that increasing the viscosity enhances diffusion and stabilizes the solution, while a nonlinear coefficient that significantly exceeds the viscosity leads to error growth over long time intervals. The method provides an efficient and accurate tool for simulating anomalous transport phenomena. Full article

This paper investigates a class of boundary value problems involving Caputo fractional derivatives of order $\nu \in (2,3]$. We begin by establishing two novel comparison principles. Subsequently, by employing the monotone iterative technique coupled with upper and lower solutions, we demonstrate the existence of extremal solutions for the corresponding fractional differential equations. Finally, an illustrative example is provided to validate our main findings. Full article