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1. 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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2. 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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3. An efficient positive‐definite block‐preconditioned finite volume solver for two‐sided fractional diffusion equations on composite mesh

   https://doi.org/10.1002/nla.2372  

   Dai, Pingfei; Jia, Jinhong; Wang, Hong; Wu, Qingbiao; Zheng, Xiangcheng (March 2021, Numerical Linear Algebra with Applications)

   Abstract It is known that the solutions to space‐fractional diffusion equations exhibit singularities near the boundary. Therefore, numerical methods discretized on the composite mesh, in which the mesh size is refined near the boundary, provide more precise approximations to the solutions. However, the coefficient matrices of the corresponding linear systems usually lose the diagonal dominance and are ill‐conditioned, which in turn affect the convergence behavior of the iteration methods.In this work we study a finite volume method for two‐sided fractional diffusion equations, in which a locally refined composite mesh is applied to capture the boundary singularities of the solutions. The diagonal blocks of the resulting three‐by‐three block linear system are proved to be positive‐definite, based on which we propose an efficient block Gauss–Seidel method by decomposing the whole system into three subsystems with those diagonal blocks as the coefficient matrices. To further accelerate the convergence speed of the iteration, we use T. Chan's circulant preconditioner³¹as the corresponding preconditioners and analyze the preconditioned matrices' spectra. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method and its strong potential in dealing with ill‐conditioned problems. While we have not proved the convergence of the method in theory, the numerical experiments show that the proposed method is convergent. 

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4. Modeling Contacts and Hysteretic Behavior in Discrete Systems Via Variable-Order Fractional Operators

   https://doi.org/10.1115/1.4046831  

   Patnaik, Sansit; Semperlotti, Fabio (September 2020, Journal of Computational and Nonlinear Dynamics)

   Abstract The modeling of nonlinear dynamical systems subject to strong and evolving nonsmooth nonlinearities is typically approached via integer-order differential equations. In this study, we present the possible application of variable-order (VO) fractional operators to a class of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. Fractional operators are intrinsically multiscale operators that can act on both space- and time-dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or VO. In the latter case, the order can be function of either independent or state variables. We show that when using VO equations to describe the response of dynamical systems, the order can evolve as a function of the response itself; therefore, allowing a natural and seamless transition between widely dissimilar dynamics. Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. Within this context, we present a physics-driven strategy to define VO operators capable of capturing complex and evolutionary phenomena. Specific examples include hysteresis in discrete oscillators and contact problems. Despite using simplified models to illustrate the applications of VO operators, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex dynamical systems. 

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5. An Analysis and Global Identification of Smoothless Variable Order of a Fractional Stochastic Differential Equation

   https://doi.org/10.3390/fractalfract7120850  

   Li, Qiao; Zheng, Xiangcheng; Wang, Hong; Yang, Zhiwei; Guo, Xu (December 2023, Fractal and Fractional)

   We establish both the uniqueness and the existence of the solutions to a hidden-memory variable-order fractional stochastic partial differential equation, which models, e.g., the stochastic motion of a Brownian particle within a viscous liquid medium varied with fractal dimensions. We also investigate the inverse problem concerning the observations of the solutions, which eliminates the analytic assumptions on the variable orders in the literature of this topic and theoretically guarantees the reliability of the determination and experimental inference. 

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