Search Results (93)

This paper studies nonlinear time-fractional reaction–diffusion equations with Caputo memory and non-local integral boundary conditions on a bounded interval. The aim is to formulate a boundary-compatible well-posedness framework and to construct a high-order temporal approximation that can be coupled with a constraint-preserving spectral spatial discretization. The analytical part proves boundedness of the non-local boundary functionals, states compatibility assumptions, and introduces a finite-dimensional nondegeneracy condition for an explicit polynomial lifting. Under a sectorial non-local elliptic realization and a global Lipschitz reaction term, existence, uniqueness, stability, and continuous dependence of mild solutions are obtained by fractional resolvent estimates and fractional Gronwall inequalities. The main novelty is the combined construction of an explicit polynomial lifting for integral boundary constraints, a constraint-preserving Legendre–Galerkin basis, and a high-order Beta-window temporal quadrature together with a discrete stability condition that accounts for sign-changing weights. The numerical evidence shows high-order behavior for smooth Caputo benchmarks, accurate enforcement of the non-local boundary constraints, and improved accuracy over the classical $L1$ approximation in the reported tests. The stability discussion identifies the discrete coercivity condition required for the sign-changing Beta-window weights. 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

Heat transport in heterogeneous materials can deviate markedly from classical Fourier behavior when microstructural disorder, trapping effects, nonlinear mobility, and long-range temporal correlations interact across multiple spatial and temporal scales. These mechanisms may produce delayed relaxation, persistent thermal footprints, front deformation, and non-classical spreading patterns that are not adequately represented by conventional integer-order diffusion models. In this study, a modeling and simulation framework is developed for anomalous heat transport in heterogeneous media using a two-dimensional time-fractional porous medium equation. The model combines a Caputo fractional time derivative, which represents thermal memory, with nonlinear degenerate porous-medium diffusion, spatially heterogeneous conductivity, localized volumetric heating, and Robin-type convective boundary exchange. A conservative fully discrete numerical scheme is constructed using flux-based finite differences for the heterogeneous nonlinear diffusion operator and an $L1$ approximation for the Caputo derivative. The nonlinear algebraic system at each time level is solved using an under-relaxed Picard frozen-coefficient iteration with non-negativity enforcement and sparse direct solution of the resulting linear systems. The numerical implementation is verified through a manufactured-solution convergence study, and additional analyses are performed to examine computational cost, Picard iteration behavior, coefficient-regularization sensitivity, strong-source effects, heterogeneous conductivity structures, and long-time thermal-footprint persistence. The results show that heterogeneous conductivity mainly redirects heat through preferential pathways and enlarges the spatial footprint while producing negligible changes in global heat content. Stronger fractional memory, represented by smaller fractional order, increases the persistence and spatial reach of moderate heating, whereas larger porous-medium exponents confine heat near the source and preserve higher local peaks. Source amplitude increases the thermal burden and footprint monotonically over the tested range, including strong forcing, without producing an abrupt localization-spreading transition. Boundary exchange remains secondary in the short-time interior-heating regime considered. These findings demonstrate that the proposed two-dimensional time-fractional porous medium framework provides a verified and physically interpretable model for non-Fourier heat transport in heterogeneous materials, where local intensity, global heat retention, and spatial thermal exposure must be assessed jointly. Full article

Time-fractional diffusion equations (TFDEs) are essential for modeling anomalous transport in heterogeneous media, but high-fidelity long-time simulations face two bottlenecks: the $O\left(N^2\right)$ complexity of non-local fractional derivatives, and the spatial truncation error of polynomial-based finite element methods (FEMs) when resolving oscillatory plumes or singular sources. We propose a framework combining a C-Bézier FEM for spatial approximation with a fast L1 temporal discretization. By coupling the shape parameter of the C-Bézier basis to the mesh size ($\mu =\pi h$), the scheme reproduces trigonometric profiles of the corresponding frequency exactly; for solutions whose spatial part lies in the C-Bézier space this eliminates the spatial truncation error and drives the associated error constant to near zero. A sum-of-exponentials (SOE) approximation reduces the temporal complexity from $O\left(N^2\right)$ to $O\left(N\right)$ and storage to $O\left(1\right)$, enabling scalable 3D simulation. We prove the optimal $O({\tau }^{2-\alpha }+h^{k+1})$ convergence, and numerical experiments confirm these rates. For profiles matched by the basis, the method yields substantially smaller errors than Lagrange FEM; for a general solution outside the C-Bézier space, the two methods share the same order and comparable error magnitudes, so the gains are specific to fields reproduced by the basis. We further examine low-regularity scenarios, including discontinuous interfaces and Dirac-delta injections. 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

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

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 survey presents an overview of recent developments in the analysis and numerical treatment of spectral fractional diffusion equations. Particular attention is devoted to efficient strategies for solving spectral fractional diffusion problems, including approaches based on rational approximation that enable efficient numerical realization of fractional powers of elliptic operators. Building on these approximations, we discuss adaptive finite element discretization techniques for polygonal domains, where singularities and geometric irregularities require carefully designed mesh refinement strategies. The survey also highlights the role of fractional diffusion operators in the preconditioning of coupled and multiphysics problems, where they can significantly improve the robustness and convergence of iterative solvers. Furthermore, we review recent results on maximum principles and monotonicity preservation for spectral fractional diffusion–reaction equations, which are essential for ensuring physically meaningful numerical solutions. Finally, we discuss current efforts aimed at improving robustness and computational efficiency through reduced and multilevel iteration methods. These approaches provide scalable algorithms for large-scale problems while maintaining accuracy and stability. The survey concludes by outlining several open problems and promising directions for future research in the numerical analysis of fractional diffusion models. Full article

Thisarticle develops and analyzes a fractional-order Leslie–Gower prey–predator–parasite system incorporating two discrete delays and nonlocal spatial diffusion. The model’s central novelty lies in the simultaneous integration of three biologically realistic features that have not previously been combined: (i) fractional-order memory effects via a Caputo derivative of order $\alpha \in (0,1]$, (ii) two distinct biological delays—an infection transmission delay ${\tau }_1$ and a predator handling delay ${\tau }_2$—and (iii) nonlocal spatial dispersal modeled through fractional Laplacian operators ${(-\Delta )}^{\gamma /2}$. This triple integration enables the model to capture long-range temporal memory, delayed biological responses, and nonlocal spatial interactions simultaneously, offering insights into dynamics that are challenging to capture with classical integer-order or single-delay formulations. The fractional Laplacian generalizes classical diffusion by allowing long-range dispersal events (Lévy flights), where individuals can occasionally move over large distances with heavy-tailed step-size distributions—a phenomenon observed in many animal movement patterns but absent from standard diffusion models. We provide rigorous proofs of solution existence, uniqueness, non-negativity, and boundedness in both temporal and spatiotemporal settings. Local asymptotic stability conditions are derived for all feasible equilibrium states via characteristic equation analysis. The coexistence equilibrium undergoes a Hopf bifurcation when either delay crosses a critical threshold, with fractional order $\alpha$ modulating the bifurcation point and post-bifurcation oscillation frequency. A Lyapunov functional demonstrates global asymptotic stability of the infection-free equilibrium under biologically interpretable conditions. Turing instability analysis reveals conditions for spontaneous pattern formation, with the fractional exponent $\gamma$ controlling pattern wavelength and correlation length. Numerical simulations validate theoretical predictions, including spatial patterns, traveling waves, and chaos. To bridge theory with potential applications, we outline a statistical framework for parameter estimation and uncertainty quantification, suggesting that $\beta$, $\alpha$, and ${\tau }_1$ may be priority targets for parameter estimation. Full article

In this paper, an efficient and accurate framework for nonlinear spacetime fractional diffusion equations is proposed. The methods are based on the spectral deferred correction technique, which employs a compact difference scheme as the preconditioner via the Picard integral collocation formulation. The nonlinear term is incorporated into the preconditioner in a way similar to linear systems without using Newtonian methods. The preconditioner is proven to be a stable operator, and the resulting spectral deferred correction method maintains an arbitrary order of accuracy and excellent stability. Due to the dense property of the central finite difference approximation of the fractional Laplacian ${(-\Delta )}^s$, a dual accelerated algorithm for the exact computation of the matrix–vector product is presented by introducing the discrete sine transform. The numerical results demonstrate that the proposed new methods are highly efficient and precise. Full article

Time–space fractional diffusion equations are widely used to model anomalous transport in heterogeneous biological tissues, where memory effects, spatial nonlocality, and coefficient variability are intrinsically coupled. However, existing numerical approaches typically treat these aspects in isolation, and a fully discrete framework that simultaneously accounts for heterogeneity, long-memory effects, and computational efficiency remains lacking. In this work, a fully discrete numerical method is developed and analyzed. The method integrates heterogeneous diffusion coefficients and memory-efficient temporal discretization within a unified variational framework. It combines a finite element approximation of a spectral fractional elliptic operator with an implicit $L^1$ discretization of the Caputo derivative enhanced by a sum-of-exponentials approximation of the memory kernel. Unconditional stability, preservation of a discrete energy structure, and a fully discrete error estimate are established, explicitly separating temporal, spatial, and kernel approximation errors. The proposed approach reduces memory complexity from $\mathcal{O}\left(N\right)$ to $\mathcal{O}(logN)$ without compromising accuracy. Numerical experiments confirm the theoretical convergence rates, demonstrate stable behavior across all tested configurations, and illustrate the impact of heterogeneous coefficients on anomalous transport dynamics. Full article

In this paper, the nonlinear time-fractional convection-diffusion equations are solved by the Legendre spectral method. The Caputo time-fractional derivative is discretized by the $L2$–$1_{\sigma }$ scheme. A priori estimates of the fully discrete scheme are derived, and the existence and uniqueness of the numerical solution are analyzed. It is rigorously proved that the fully discrete scheme is unconditionally stable, and the convergence order of the numerical scheme is $O(N^{1-m}+{\tau }^2)$. Finally, numerical results are presented to verify the theoretical analysis. Full article

In this work, we propose a nonlinear fractional partial differential equation model incorporating a Caputo fractional derivative in time, a second-order spatial derivative, and a nonlinear Fredholm integral term. This model accounts for memory effects, anomalous diffusion, and nonlocal interactions, offering a more realistic description of complex transport phenomena compared to classical integer-order models. To solve the model numerically, we develop a fully discrete scheme that combines Lagrange interpolation-based approximation for the Caputo fractional derivative in time with central difference discretization for the spatial derivative. This approach ensures accuracy and flexibility in handling both the fractional derivative and the nonlinear integral term. A comprehensive convergence and stability analysis is conducted, establishing second-order accuracy in space and nearly second-order accuracy in time. Rigorous error estimates confirm the reliability and robustness of the proposed scheme for practical computations. Finally, a numerical example with a known exact solution is solved to validate the method. Errors are computed in both the $L_2$ and maximum norms, and the temporal and spatial convergence orders are verified. The results, summarized in tables, demonstrate the effectiveness of the fully discrete scheme and underscore the practical utility of the proposed fractional model in complex physical and engineering systems. Full article

This paper presents a finite difference method for solving the Caputo generalized time fractional diffusion equation. The method extends the $L1$ scheme to discretize the time fractional derivative and employs the central difference for the spatial diffusion term. Theoretical analysis demonstrates that the proposed numerical scheme achieves a convergence rate of order $2-\alpha$ in time and second order in space. These theoretical findings are further validated through numerical experiments. Compared to existing methods that only achieve a temporal convergence of order $1-\alpha$, the proposed approach offers improved accuracy and efficiency, particularly when the fractional order $\alpha$ is close to zero. This makes the method highly suitable for simulating transport processes with memory effects, such as oil pollution dispersion and biological population dynamics. Full article

Time-fractional interface problems arise in systems where interacting materials exhibit memory effects or anomalous diffusion. These models provide a more realistic description of physical processes than classical formulations and appear in heat conduction, fluid flow, porous media diffusion, and electromagnetic wave propagation. However, the presence of complex interfaces and the nonlocal nature of fractional derivatives makes their numerical treatment challenging. This article presents a numerical scheme that combines radial basis functions (RBFs) with the finite difference method (FDM) to solve time-fractional partial differential equations involving interfaces. The proposed approach applies to both linear and nonlinear models with constant or variable coefficients. Spatial derivatives are approximated using RBFs, while the Caputo definition is employed for the time-fractional term. First-order time derivatives are discretized using the FDM. Linear systems are solved via Gaussian elimination, and for nonlinear problems, two linearization strategies, a quasi-Newton method and a splitting technique, are implemented to improve efficiency and accuracy. The method’s performance is assessed using maximum absolute and root mean square errors across various grid resolutions. Numerical experiments demonstrate that the scheme effectively resolves sharp gradients and discontinuities while maintaining stability. Overall, the results confirm the robustness, accuracy, and broad applicability of the proposed technique. Full article