Search Results (191)

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

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

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 study, we investigate the existence and uniqueness of weak solutions for a stochastic Ginzburg–Landau equation involving the fractional Laplacian. The primary focus is on establishing a proper mathematical framework to handle the coexistence of the nonlocal fractional Laplacian and stochastic perturbations. By employing the Galerkin method, we prove that the initial-boundary value problem admits a unique global weak solution for any ${\mathcal{F}}_0$-measurable $L^2\left(I\right)$-valued random initial value with a finite second moment. We also utilize the properties of the fractional Laplacian and fractional Sobolev spaces to provide a proof of the existence of the uniqueness theorem. These results extend the analysis of the Ginzburg–Landau equations to models incorporating stochastic terms and the fractional Laplacian. 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

While superdeterministic and fractal spacetime models offer compelling alternative perspectives on quantum foundations, the simulation and validation of effective wave dynamics in such non-differentiable, deterministic settings remain computationally and theoretically challenging. To address this, a framework built around the Fractional Nonlinear Klein–Gordon Equation (FNKGE), defined through the spectral fractional Laplacian, was developed. This equation was solved and benchmarked through a comparative study between Physics-Informed Neural Networks (PINNs) with Fourier features and Physics-Informed Graph Neural Networks (PI-GNNs). Additionally, detection patterns were simulated via deterministic agents, and theoretical links between fractal geometry, computational irreducibility, and deviations from statistical independence were formalized. Regarding the computational evaluation, superior accuracy was achieved by the PI-GNNs, yielding a mean relative error of $0.5\%$ ($\overline{ϵ}=0.005$), alongside faster convergence and a more well-conditioned Hessian spectrum compared to PINNs. Crucially, a continuous power-law decay ($S\left(k_y\right)\sim k_y^{-1.8}$) was revealed by the spectral analysis of the simulated detection patterns, confirming the emergence of classical optical caustics rather than discrete quantum-interference peaks. Furthermore, a modified dispersion relation that accurately predicts linear instability regimes was derived, and specific boundary artifacts in non-periodic domains were identified. Taken together, the FNKGE is validated by these results as a viable effective model for fractal wave phenomenology and as a robust benchmark for physics-informed learning architectures. Full article

Binding affinity prediction in computational drug discovery is confounded by trivial correlations between molecular size and measured potency. We introduce cellular sheaf Laplacians as descriptors of ligand molecular geometry that quantify geometric frustration independent of system size. Sheaves are constructed over molecular graphs by assigning three-dimensional coordinate spaces to atoms and projection operators encoding ideal bonding geometry to edges; eigendecomposition of the resulting Laplacian yields spectral features measuring inconsistencies between local geometric constraints and global topology. Applied to 14,050 protein-ligand complexes from the PDBbind v2020 refined set, MW-residualized Sheaf features capture a statistically significant geometric signal ($r_{\mathrm{partial}} = 0.171$, $p<{10}^{-70}$) that is orthogonal to the Wiener index ($r=0.013$) and persists after controlling for both molecular weight and classical graph-theoretic descriptors ($r_{\mathrm{partial}} = 0.390$, $p<{10}^{-9}$). Sheaf spectral features alone achieve predictive performance ($R^2=0.403$) approaching that of fourteen classical cheminformatics descriptors ($R^2=0.446$), and their combination yields consistent improvements across the binding affinity spectrum (RMSE $=1.43$$p{K}_d$). Permutation importance analysis confirms the Sheaf Frobenius norm as the second most influential descriptor after molecular weight. We introduce Topological Binding Efficiency as a size-normalized quality metric identifying ligands that achieve potent binding through geometric complementarity rather than molecular bulk. Gaussian mixture analysis of the maximum eigenvalue distribution among strong binders reveals two distinct spectral modes corresponding to planar aromatic and three-dimensional sp3-rich scaffolds, confirmed by significant differences in fraction of sp3 carbons and aromatic ring counts ($p<{10}^{-8}$). As an intentionally ligand-centric framework, our approach complements rather than replaces protein-aware co-modelling architectures. This work establishes cellular sheaf theory as a principled framework for encoding molecular topology with statistically significant associations with binding affinity, providing interpretable geometric insights that are inaccessible to conventional molecular descriptors. Full article

Subdiffusion on graphs is often modeled by time-fractional diffusion equations; yet, its structural and dynamical consequences remain unclear. We show that subdiffusive transport on graphs is a memory-driven process generated by a random time change that compresses operational time, produces long-tailed waiting times, and breaks Markovianity while preserving linearity and mass conservation. While the subordination representation and complete monotonicity properties of the Mittag-Leffler function are classical, we develop a graph-based synthesis in which Mittag-Leffler dynamics admit an exact convex, mass-preserving representation as a superposition of Laplacian semigroups evaluated at rescaled times. This perspective reveals fractional diffusion as ordinary diffusion acting across multiple intrinsic time scales and enables new structural and dynamical interpretations of graphs. This framework uncovers heterogeneous, vertex-dependent memory effects and induces transport biases absent in classical diffusion, including algebraic relaxation, degree-dependent waiting times, and early-time asymmetries between sources and neighbors. These features define a subdiffusive geometry on graphs, enabling the recovery of global shortest paths, in contrast to the graph exploration of diffusive geometry, while simultaneously favoring high-degree regions. Finally, we show that time-fractional diffusion can be interpreted as a singular limit of multi-rate diffusion, in an appropriate asymptotic sense. Full article

We prove the existence of a positive ground state solution for a fractional $(p,q)$-Laplacian Choquard equation that features both a singularity and an upper critical exponent. The proof relies on a combination of the Nehari manifold technique and Ekeland’s variational principle. Full article

Recently, a great amount of attention has been focused on the study of fractional and nonlocal operators of the elliptic type both for pure mathematical research and in view of concrete real-world applications. We are interested in proving the existence and nonexistence of solutions of a minimizing problem involving a fractional Laplacian with weight. We consider the nonlocal minimizing problem on ${\mathbb{H}}_0^s(\Omega )\subset L^{q_s}(\Omega )$, with $q_s:={\displaystyle \frac{2n}{n-2s}}$, $s\in (0,1)$, and $n\ge 3$${\displaystyle \underset{\begin{array}{c}u\in {\mathbb{H}}_0^s(\Omega )\\ {\left|\right|u\left|\right|}_{L^{q_s}(\Omega )}=1\end{array}}{inf}{\int }_{{\mathbb{R}}^n}{p\left(x\right)|(-\Delta )}^{\frac{s}{2}}{u\left(x\right)|}^{2}dx-\lambda {\int }_{\Omega }{\left|u\left(x\right)\right|}^{2}dx,}$ where $\Omega$ is a bounded domain in ${\mathbb{R}}^n, p:{\mathbb{R}}^n\to \mathbb{R}$ is a given positive weight presenting a global positive minimum $p_0>0$ at $a\in \Omega$, and $\lambda$ is a real constant. The objective of this paper is to show that minimizers do not exist for some $k,s,\lambda$, and n. After that, we show some nonexistence results thanks to a fractional Pohozaev identity and fractional Hardy inequality. Full article

This paper investigates non-fractional operators, a type of nonlocal operator, within the framework of Orlicz spaces. Using inclusions between certain function spaces, we prove the continuity and/or compactness of generalized operators in Orlicz spaces and show that solutions exist for integral equations of fractional order. We also introduce a generalized Hilfer-type derivative and examine the equivalence of differential and integral problems. Finally, we relate these results to the study of compositional p-Laplacian fractional problems involving generalized Hilfer fractional derivatives. Among other things, we prove the existence of solutions to such problems in Orlicz and Orlicz–Sobolev spaces. Full article

In this paper, we investigate a class of coupled fractional differential systems involving Caputo derivatives and nonlinear $\varphi$-Laplacian operators subject to nonlocal boundary conditions. By transforming the problem into an equivalent integral system via appropriate Green’s functions, the existence of solutions is studied within a generalized Banach space framework. Using a Leray–Schauder type fixed point theorem and suitable growth conditions on the nonlinear terms, we establish the existence of at least one bounded solution. Furthermore, we prove that the solution set is compact. An illustrative example involving the p-Laplacian operator is provided to demonstrate the applicability of the obtained theoretical results. Full article

In this paper, we investigate the properties of the boundedness of fractional integral operators $K_{\alpha }$ defined on general measure metric spaces. We study their action in Lebesgue spaces $L^p\left(Y\right)$, Morrey spaces $L_{\phi }^p\left(Y\right)$, and extend our analysis to fractional Sobolev spaces $W^{\alpha ,p}\left(Y\right)$. Using classical dyadic decomposition and the Hardy–Littlewood maximal operator, we establish sharp bounds for $K_{\alpha }$ in terms of kernel parameters and the geometric structure of the space. A significant contribution of this work is the proof that $K_{\alpha }$ is bounded from $W^{\alpha ,p}\left(Y\right)$ to $L^q\left(Y\right)$, where thus linking our operator-theoretic framework with the theory of nonlocal and fractional partial differential equations. These results provide valuable tools for studying regularity, a priori estimates, and solution mappings in nonlocal problems involving the fractional Laplacian and related operators on irregular or non- Euclidean domains. Full article

This paper focuses on establishing the existence of infinitely many solutions for non-local fractional equations characterized by unbalanced growth and Hardy potentials. We prove that these solutions converge to zero in the $L^{\infty }$-norm, requiring conditions on the nonlinearity only near the origin and dispensing with assumptions at infinity. As far as we are aware, results for non-local fractional $(p,q)$-Laplacian problems with singular coefficients such as Hardy potentials have not been extensively studied. To address this gap, we employ the dual fountain theorem together with the modified functional method. Full article

In this paper, we study a fractional Kirchhoff problem with a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient: ${\displaystyle M\left({\int \int }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}\frac{{|u\left(x\right)-u\left(y\right)|}^q}{{|x-y|}^{N+qs}}dxdy\right){(-\Delta )}^{s}u=\lambda \frac{u}{{\left|x\right|}^{2s}}+f(x,u,\nabla u)\mathrm{in}\Omega ,}$ where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain containing the origin, $s\in (0,1)$, $q\in (1,2]$ with $N>2s$, $\lambda >0$, and f is a measurable non-negative function satisfying suitable hypotheses. The main objective is to establish the existence of positive solutions for the largest possible class of nonlinearities f without imposing restrictions on $\lambda$. Two main cases areconsidered: $\left(I\right)-f(x,u,\nabla u)=u^p+\mu ,\mathrm{and}\left(II\right)-f(x,u,\nabla u)={|\nabla u|}^p+\mu g.$ Existence is proved under suitable hypotheses on $q,p$ and the data $g,\mu$. The results are new, including for the local case $s=1$. Full article