插值法解时间分数阶扩散方程

Interpolation method for the time fractional diffusion equation

针对一类时间分数阶扩散方程提出了一种新的隐式差分格式,空间导数直接采用中心差分格式离散,为了近似Caputo型时间分数阶导数,在小区间[t_(n-1),t_n](2≤n≤N)上使用三点u(x,t_(n-2))、u(x,t_(n-1))、u(x,t_n)二次插值近似u(x,t)的值,在小区间[t_0,t_1]上使用线性插值近似u(x,t)的值,并利用能量范数证明该格式的无条件稳定性和收敛性,最后通过数值实验验证该格式的有效性。

A new implicit difference approximation to solve a time fractional derivative equation is proposed. The spatial derivative is directly discretized by central difference scheme. To approxi- mate the Caputo fractional derivative, it is established by means of the quadratic interpolation approximation. Using three points U（X,tn-2）,U（X,tn-1）,U（X,tn） for the integrand u（x,t） on each small interval [tn-1,tn]（2≤n≤N）, while the linear interpolation approximation is applied on the first small interval [t0 ,tl]. Using the energy norm, the unconditional stability and convergence of the scheme are proved. Finally, a numerical experiment shows that the scheme is efficient.