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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. The regular free boundary in the thin obstacle problem for degenerate parabolic equations

   Banerjee, A.; Danielli, D.; Garofalo, N.; Petrosyan, A. (June 2020, Algebra i analiz)

   In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial_t - \Delta_x)^s$$ for $$s \in (0,1)$$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as $$H^{1+\gamma,\frac{1+\gamma}{2}}$$ regularity of the free boundary near such regular points. 

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3. A C ^α finite difference method for the Caputo time‐fractional diffusion equation

   https://doi.org/10.1002/num.22686  

   Davis, Wesley; Noren, Richard; Shi, Ke (November 2020, Numerical Methods for Partial Differential Equations)

   Abstract We begin with a treatment of the Caputo time‐fractional diffusion equation, by using the Laplace transform, to obtain a Volterra integro‐differential equation. We derive and utilize a numerical scheme that is derived in parallel to the L1‐method for the time variable and a standard fourth‐order approximation in the spatial variable. The main method derived in this article has a rate of convergence ofO(k^α + h⁴)foru(x,t) ∈ C^α([0,T];C⁶(Ω)),0 < α < 1, which improves previous regularity assumptions that requireC²[0,T]regularity in the time variable. We also present a novel alternative method for a first‐order approximation in time, under a regularity assumption ofu(x,t) ∈ C¹([0,T];C⁶(Ω)), while exhibiting order of convergence slightly more thanO(k)in time. This allows for a much wider class of functions to be analyzed which was previously not possible under the L1‐method. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques. 

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4. Low Regularity Integrators for the Conservative Allen–Cahn Equation with a Nonlocal Constraint

   https://doi.org/10.1007/s10915-024-02703-1  

   Doan, Cao-Kha; Hoang, Thi-Thao-Phuong; Ju, Lili (December 2024, Journal of Scientific Computing)

   In contrast to the classical Allen-Cahn equation, the conservative Allen-Cahn equation with a nonlocal Lagrange multiplier not only satisfies the maximum bound principle (MBP) and energy dissipation law but also ensures mass conservation. Many existing schemes often fail to preserve all these properties at the discrete level or require high regularity in time on the exact solution for convergence analysis. In this paper, we construct a new class of low regularity integrators (LRIs) for time discretization of the conservative Allen-Cahn equation by repeatedly using Duhamel's formula. The proposed first- and second-order LRI schemes are shown to conserve mass unconditionally and satisfy the MBP under some time step size constraints. Temporal error estimates for these schemes are derived under a low regularity assumption that the exact solution is only Lipschitz continuous in time, followed by a rigorous proof for energy stability of the corresponding time-discrete solutions. Various numerical experiments and comparisons in two and three dimensions are presented to verify the theoretical results and illustrate the performance of the LRI schemes, especially when the interfacial parameter approaches zero. 

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5. Optimal Control, Numerics, and Applications of Fractional PDEs

   https://doi.org/10.1016/bs.hna.2021.12.003  

   Antil, H.; Brown T.; Khatri, R.; Onwunta, A.; Verma, D.; Warma, M. (February 2022, Handbook of numerical analysis)

   Trélat, E.; Zuazua, E. (Ed.)

   This chapter provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps, and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems. 

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