Recently, I come to a question about the 2nd order centered finite difference approximation of a fractional function, more precisely,we set $\delta^{2}_x u(x,t) = u(x_{i-1},t)-2u(x_i,t) + u(x_{i+1},t)$. Then how to derive the following difference approximation, we have a given function $D(x)$, $u(x,t)$ is the unknown solution of 1D time-dependent PDE:$\delta^{2}_{x}\Big(\frac{u(x,t)}{D(x)}\Big)=\ ?$In fact, it belongs to variable coefficients. Who can help me? Moreover, can we write this resulted scheme into a matrix form?such as $\delta^{2}_{x}\Big(\frac{u(x,t)}{D(x)}\Big)\rightarrow D*T*u$, $D$ is a diagonal matrix, $T$ is a tridiagonal matrix ?? $u = [u(x_i,t),u(x_2,t),...,u(x_n,t)]^T$.
The centered difference operator for fractional function
pde;finite difference