【摘要】 为了研究时间分数阶扩散方程的高精度的数值方法,得到高阶的数值格式,采用Caputo分数阶导数的差分公式——L2-1_σ公式离散时间分数阶导数,得到了时间分数阶扩散方程的半离散格式,并证明了半离散格式是无条件稳定的,且收敛阶为O(τ~2).空间导数采用拟小波方法离散,构造出了时间分数阶扩散方程的一种新的全离散数值格式.最后,通过数值算例验证了理论分析的正确性和数值解的有效性,而且结果表明这种算法收敛快、误差小,是一种高效的数值算法.更多还原

【Abstract】 In order to study the high-precision numerical method of the time fractional diffusion equation,the high-order numerical scheme is obtained.By using the difference formula of Caputo fractional derivative——L2-1_σ formula to discrete the time fractional derivative,the time semi-discrete scheme of the time fractional diffusion equation is obtained,it is proved that the semi-discrete scheme is unconditionally stable and convergence with O(τ~2).The quasi-wavelet method is used to discretize the spatial derivative,then a new fully discrete numerical scheme for the time fractional diffusion equation is constructed.Finally,numerical examples are given to verify the correctness of the theoretical analysis and the validity of the numerical solution.It is also shown that the proposed algorithm has fast convergence and small error,and it is an efficient numerical algorithm.更多还原