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1. Regularity of solutions to space–time fractional wave equations: A PDE approach

   https://doi.org/10.1515/fca-2018-0067  

   Otárola, Enrique; Salgado, Abner J. (October 2018, Fractional Calculus and Applied Analysis)

   Abstract We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic. 

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2. Residual-based data-driven variational multiscale reduced order models for parameter-dependent problems

   https://doi.org/10.1007/s40314-025-03273-0  

   Koc, Birgul; Rubino, Samuele; Chacón_Rebollo, Tomás; Iliescu, Traian (September 2025, Computational and Applied Mathematics)

   Abstract In this paper, we propose a novel residual-based data-driven closure strategy for reduced order models (ROMs) of under-resolved, convection-dominated problems. The new ROM closure model is constructed in a variational multiscale (VMS) framework by using the available full order model data and a model form ansatz that depends on the ROM residual. We emphasize that this closure modeling strategy is fundamentally different from the current data-driven ROM closures, which generally depend on the ROM coefficients. We investigate the new residual-based data-driven VMS ROM closure strategy in the numerical simulation of three test problems: (i) a one-dimensional parameter-dependent advection-diffusion problem; (ii) a two-dimensional time-dependent advection-diffusion-reaction problem with a small diffusion coefficient ($$\varepsilon = 1e-4$$ $\epsilon =1e-4$); and (iii) a two-dimensional flow past a cylinder at Reynolds number$$Re=1000$$ $Re=1000$. Our numerical investigation shows that the new residual-based data-driven VMS-ROM is more accurate than the standard coefficient-based data-driven VMS-ROM. 

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3. A semi-adaptive preservative scheme for a fractional quenching convective-diffusion problem

   https://doi.org/10.1016/j.camwa.2023.09.043  

   Liu, Nabing; Zhu, Lin; Sheng, Qin (December 2023, Computers & Mathematics with Applications)

   A preservative scheme is presented and analyzed for the solution of a quenching type convective-diffusion problem modeled through one-sided Riemann-Liouville space-fractional derivatives. Properly weighted Grünwald formulas are employed for the discretization of the fractional derivative. A forward difference approximation is considered in the approximation of the convective term of the nonlinear equation. Temporal steps are optimized via an asymptotic arc-length monitoring mechanism till the quenching point. Under suitable constraints on spatial-temporal discretization steps, the monotonicity, positivity preservations of the numerical solution and numerical stability of the scheme are proved. Three numerical experiments are designed to demonstrate and simulate key characteristics of the semi-adaptive scheme constructed, including critical length, quenching time and quenching location of the fractional quenching phenomena formulated through the one-sided space-fractional convective-diffusion initial-boundary value problem. Effects of the convective function to quenching are discussed. Numerical estimates of the order of convergence are obtained. Computational results obtained are carefully compared with those acquired from conventional integer order quenching convection-diffusion problems for validating anticipated accuracy. The experiments have demonstrated expected accuracy and feasibility of the new method. 

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4. Efficient discretization and preconditioning of the singularly perturbed reaction-diffusion problem

   https://doi.org/https://doi.org/10.1016/j.camwa.2022.01.031  

   Constantin Bacuta; Daniel Hayes; Jacob Jacavage (January 2022, Computers mathematics with applications)

   Elsevier (Ed.)

   We consider the reaction diffusion problem and present efficient ways to discretize and precondition in the singular perturbed case when the reaction term dominates the equation. Using the concepts of optimal test norm and saddle point reformulation, we provide efficient discretization processes for uniform and non-uniform meshes. We present a preconditioning strategy that works for a large range of the perturbation parameter. Numerical examples to illustrate the efficiency of the method are included for a problem on the unit square. 

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5. Neural Network Method for Integral Fractional Laplace Equations

   https://doi.org/10.4208/eajam.010122.210722  

   Hao, Zhaopeng; Park, Moongyu; Cai, Zhiqiang (January 2023, East Asian Journal on Applied Mathematics)

   A neural network method for fractional order diffusion equations with integral fractional Laplacian is studied. We employ the Ritz formulation for the corresponding fractional equation and then derive an approximate solution of an optimization problem in the function class of neural network sets. Connecting the neural network sets with weighted Sobolev spaces, we prove the convergence and establish error estimates of the neural network method in the energy norm. To verify the theoretical results, we carry out numerical experiments and report their outcome. 

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