Yujiang Wu

Minghua Chen, Weihua Deng, Yujiang Wu

In this paper, we discuss the time-space Caputo-Riesz fractional diffusion equation with variable coefficients on a finite domain. The finite difference schemes for this equation are provided. We theoretically prove and numerically verify that the implicit finite difference scheme is unconditionally stable (the explicit scheme is conditionally stable with the stability condition $\frac{τ^γ}{(Δx)^α}+\frac{τ^γ}{(Δy)^β} <C$) and 2nd order convergent in space direction, and $(2-γ)$-th order convergent in time direction, where $γ\in(0,1]$.

WenYi Tian, Weihua Deng, Yujiang Wu

This paper discusses the spectral collocation method for numerically solving nonlocal problems: one dimensional space fractional advection-diffusion equation; and two dimensional linear/nonlinear space fractional advection-diffusion equation. The differentiation matrixes of the left and right Riemann-Liouville and Caputo fractional derivatives are derived for any collocation points within any given interval. The stabilities of the one dimensional semi-discrete and full-discrete schemes are theoretically established. Several numerical examples with different boundary conditions are computed to testify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy-Feller advection-diffusion equation are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases.

Shuqin Wang, Jinyun Yuan, Weihua Deng et al.

A hybridized discontinuous Galerkin method is proposed for solving 2D fractional convection-diffusion equations containing derivatives of fractional order in space on a finite domain. The Riemann-Liouville derivative is used for the spatial derivative. Combining the characteristic method and the hybridized discontinuous Galerkin method, the symmetric variational formulation is constructed. The stability of the presented scheme is proved. Theoretically, the order of $\mathcal{O}(h^{k+1/2}+Δt)$ is established for the corresponding models and numerically the better convergence rates are detected by carefully choosing the numerical fluxes. Extensive numerical experiments are performed to illustrate the performance of the proposed schemes. The first numerical example is to display the convergence orders, while the second one justifies the benefits of the schemes. Both are tested with triangular meshes.

Yanyan Yu, Weihua Deng, Yujiang Wu et al.

Power-law probability density function (PDF) plays a key role in both subdiffusion and Lévy flights. However, sometimes because of the finite of the lifespan of the particles or the boundedness of the physical space, tempered power-law PDF seems to be a more physical choice and then the tempered fractional operators appear; in fact, the tempered fractional operators can also characterize the transitions among subdiffusion, normal diffusion, and Lévy flights. This paper focuses on the quasi-compact schemes for space tempered fractional diffusion equations, being much different from the ones for pure fractional derivatives. By using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Some numerical simulations are performed to testify the effectiveness and numerical accuracy of the obtained schemes.

Yanyan Yu, Weihua Deng, Yujiang Wu

In this paper, we design a semi-implicit scheme for the scalar time fractional reaction-diffusion equation. We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time and space stepsizes, and numerically show that the convergent orders are 1 %$2-α$ in time and 2 in space. As a concrete model, the subdiffusive predator-prey system is discussed in detail. First, we prove that the analytical solution of the system is positive and bounded. Then we use the provided numerical scheme to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical scheme preserves the positivity and boundedness.

Yanyan Yu, Weihua Deng, Yujiang Wu

The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has space fractional derivative, which characterizes Lévy flights. Sometimes the infinite variance of Lévy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more `physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high order quasi-compact discretizations for space fractional derivative and tempered space fractional derivative. The fourth order quasi-compact discretization for space fractional derivative is applied to solve space fractional diffusion equation and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart of the fourth order quasi-compact scheme; and the convergence orders are verified numerically.

Jingwei Deng, Lijing Zhao, Yujiang Wu

The tempered evolution equation describes the trapped dynamics, widely appearing in nature, e.g., the motion of living particles in viscous liquid. This paper proposes the fast predictor-corrector approach for the tempered fractional ordinary differential equations by digging out the potential 'very' short memory principle. The algorithms basing on the idea of equidistributing are detailedly described; their effectiveness and low computation cost, being linearly increasing with time $t$, are numerically demonstrated.

Yujiang Wu, Hongjian Song, Jiawen Zhang et al.

The development of effective machine learning methodologies for enhancing the efficiency and accuracy of clinical systems is crucial. Despite significant research efforts, managing a plethora of diversified clinical tasks and adapting to emerging new tasks remain significant challenges. This paper presents a novel paradigm that employs a pre-trained large language model as a universal clinical multi-task decoder. This approach leverages the flexibility and diversity of language expressions to handle task topic variations and associated arguments. The introduction of a new task simply requires the addition of a new instruction template. We validate this framework across hundreds of tasks, demonstrating its robustness in facilitating multi-task predictions, performing on par with traditional multi-task learning and single-task learning approaches. Moreover, it shows exceptional adaptability to new tasks, with impressive zero-shot performance in some instances and superior data efficiency in few-shot scenarios. This novel approach offers a unified solution to manage a wide array of new and emerging tasks in clinical applications.