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1. 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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2. Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order

   https://doi.org/10.1051/m2an/2020072  

   Li, Buyang; Wang, Hong; Wang, Jilu (January 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   We prove well-posedness and regularity of solutions to a fractional diffusion porous media equation with a variable fractional order that may depend on the unknown solution. We present a linearly implicit time-stepping method to linearize and discretize the equation in time, and present rigorous analysis for the convergence of numerical solutions based on proved regularity results. 

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3. An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion

   https://doi.org/10.1088/1361-6420/ab6503  

   Feng, X.; Li, P.; Wang, X. (January 2020, Inverse problems)

   This paper is concerned with the mathematical analysis of an inverse random source problem for the time fractional diffusion equation, where the source is driven by a fractional Brownian motion. Given the random source, the direct problem is to study the stochastic time fractional diffusion equation. The inverse problem is to determine the statistical properties of the source from the expectation and variance of the final time data. For the direct problem, we show that it is well-posed and has a unique mild solution under a certain condition. For the inverse problem, the uniqueness is proved and the instability is characterized. The major ingredients of the analysis are based on the properties of the Mittag–Leffler function and the stochastic integrals associated with the fractional Brownian motion. 

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4. Inverting the variable fractional order in a variable-order space-fractional diffusion equation with variable diffusivity: analysis and simulation

   https://doi.org/10.1515/jiip-2019-0040  

   Zheng, Xiangcheng; Li, Yiqun; Cheng, Jin; Wang, Hong (April 2021, Journal of Inverse and Ill-posed Problems)

   null (Ed.)

   Abstract Variable-order space-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena, including anomalously superdiffusive transport of solutes in heterogeneous porous media, long-range spatial interactions and other applications, as well as eliminating the nonphysical boundary layers of the solutions to their constant-order analogues.In this paper, we prove the uniqueness of determining the variable fractional order of the homogeneous Dirichlet boundary-value problem of the one-sided linear variable-order space-fractional diffusion equation with some observed values of the unknown solutions near the boundary of the spatial domain.We base on the analysis to develop a spectral-Galerkin Levenberg–Marquardt method and a finite difference Levenberg–Marquardt method to numerically invert the variable order.We carry out numerical experiments to investigate the numerical performance of these methods. 

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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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