Lie symmetries reduction and spectral methods on the fractional two-dimensional heat equation

热传导方程的解析与数值处理：基于时间分数分布阶热传导定律

Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law

本文通过能量平衡方程与分数阶本构热传导定律构建热传导方程，利用傅里叶和拉普拉斯变换及有限差分法分析求解，并通过数值例子验证了两种方法在多项式分布阶热传导定律下的适用性与一致性。

后验误差估计用于热方程的完全离散分数步θ近似

A posteriori error estimates for fully discrete fractional-step $\vartheta$-approximations for the heat equation

本文基于分数步θ方案和有限元方法，推导了热方程初值边值问题的后验误差估计，首次在L^∞(0, T ; L^2(Ω))空间中获得最优阶误差估计。

The sparse representation related with fractional heat equations

This study introduces pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). The method, as the first step, expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as the second step, makes use the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show effectiveness and efficiency of the proposed series solutions.

A new regularisation for time-fractional backward heat conduction problem

It is well-known that the backward heat conduction problem of recovering the temperature $u(\cdot, t)$ at a time $t\geq 0$ from the knowledge of the temperature at a later time, namely $g:= u(\cdot, τ)$ for $τ>t$, is ill-posed, in the sense that small error in $g$ can lead to large deviation in $u(\cdot, t)$. However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for $t>0$ and ill-posed for $t=0$. We use this observation to obtain stable approximate solutions for the TFBHCP for $t=0$, and derive error estimates under suitable source…

一种用于时间分数随机热方程的Galerkin有限元方法

A Galerkin finite element method for time-fractional stochastic heat equation

本文提出了一种用于带有乘性噪声的时间分数随机热方程的Galerkin有限元方法，研究了解的正则性，并通过数值实验验证了理论结果。

Fast method and convergence analysis of fractional magnetohydrodynamic coupled flow and heat transfer model for generalized second-grade fluid

In this paper, we first establish a new fractional magnetohydrodynamic (MHD) coupled flow and heat transfer model for a generalized second-grade fluid. This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law. The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization. The fully discrete scheme is proved to be stable and convergent with an accuracy of $O(τ^2+N^{-r})$, where $τ$ is the time step size and $N$ is …